THE SCALES OF COMMERCE AND TRADE: Balancing betwixt the Buyer and Seller, Artificer and Manufacture, Debtor and Creditor, the most general Questions, artificial Rules, and useful Conclusions incident to Traffic: Comprehended in two Books. The first states and Ponderates to Equity and Custom, all usual Rule●, legal Bargains and Contracts, in Wholesale or Retail, with Factorage, Returns, and Exchanges of Foreign Coin, of Interest-Money, both Simple and Compounded, with Solutions from Natural and Artificial Arithmetic. The second Book treats of Geometrical Problems and Arithmetical Solutions, in dimensions of Lines, Superficies and Bodies, both solid and concave, viz. Land, Wainscot, Hang, Board, Timber, Stone, gauging of Casks, Military Propositions, Merchants Accounts by Debtor and Creditor; Architectonice, or the Art of Building. By THOMAS WILLSFORD Gent. LONDON, Printed by J. G. for Nath: Brook, at the Angel in Cornhill. 1660. A DEDICATORY EPISTLE TO THE Illustrious and most Ingenuous MERCHANTS, The Patrons of Commerce and Trade, wishing Success may crown their good Endeavours. RIGHT HONOURABLE, ADVENTURERS (the Primum mobile of this Subject) whose Negotiations are dilated beyond the Sun's annual progress, as both the Indies and Polar Stars can clearly witness; to your candid censures I address myself, as competent judges of Commerce and Trade; and to avoid obstructions from fond Informers (with licence of the Court) I will render here a breviate of some occurrences, and the motive wherefore I have harboured this so long, not becalmed, as some suppose. When first I framed this Abstract of Commerce and Trade, (and having shipped the chiefest Rules depending upon Arithmetic) I thought good to balance them with Geometrical Problems and Propositions of Magnitude, with sundry Questions mixed and applied to both those liberal Sciences. Thus fraught with variety (according to the Vessels capacity) rigged and made ready for a voyage to the public prospect of the world's inhabitants, indigent of Protectors for a Convoy, until reflecting upon the Right Honourable Societies, under whose colours I weighed Anchor, and stood to Sea, willing to strike sail to men of Art, in Peace or War, yet scorning to submit unto a Fleet of vapouring Romancers, or empty Liters, whose Topsails are filled with vain glorious words: these verbal Rovers (perhaps) will question and charge me of Piracy, or surreptitious Goods from manual Trades, and ingenious Artificers. To all rash and Malignant censurers I shall plead Not guilty; animated, that this noble Consistory of Senators will vindicate my endeavours, and write upon the Plaintiffs Bill Ignoramus; since all Humane Knowledge depends upon Time (the World's grave Tutor) ratified by the sage experience of yourselves and others: and as for these, they have been lawfully gained, selected and recollected by my Industry, under the conduct of Art, and registered upon the account of many years' revolutions, applied to Practice, seconded by Reason, and inserted in this form, modelized to my Sense, without prejudice, or entrenching myself upon any others Ground, Claim, Title, or Prerogative. Neither have I set forth one Adventure (as some have done) and never appeared upon this Theatre again, nor heard of, as if exploded, or cast away at Sea by some evil steered course, not Regulated by Demonstration, shaped by Experience, nor rectified by Compass, or lost by spreading too great a Sail, unable to stem a tumultuous current of the Times without a Pilot, Masters and Mates now termini covertibiles; yet notwithstanding them, the acceptance of my former Labours hath given me fair hopes of an Insurance for these purchased by Barter, Exchange, or the expense both of Time and Money, transported from several Regions, under your protection now arrived and delivered out, for the benefit or pleasure of others. My request is (worthy Patrons in this Metropolis of Trade) that ye will sign these with your Magisteriall Impressions, and condescend to own my Adventure, although intended as a guide to young beginners only, and to attend your vacant hours, as an Index to your memories, and a Directory to your dilated courses, since Nature hath provided the biggest Whale but a little Pilot, and Art a small Rudder, to steer the greatest Ship. From hence if this may find a reception in your Society, the Charges are all defrayed, but Custom to the Stationer's Office, my intentions (to a serene Auditor) will balance the other Accounts, and the next Voyage or Return, the Tare and Errors shall be deducted, and ye (Noble Merchants) represented with an ampler Cargazone, if I be admitted, and this to your Patronage (which Hope bids me to believe) so I will subscribe myself, DEBTOR to ye All, THOMAS WILLSFORD. A GENERAL PREFACE TO ALL Adventurers & Negotiators, conversant in Commerce & Trade, with a compendious Discourse, which is to be preferred in an happy Republic, viz. the Lawyer, Merchant, or Soldier; wishing well to all honest Endeavours. MAn is a sociable Creature (according to his natural inclination) and in respect of temporal employments, the noble Merchant transcends all others, as the Superior to the Corporation of Tradesmen; they being the supporters of Traffic, conservers of Amity, friends to Peace, patrons of Plenty, and grand coadjutors to All, by supplying the indigency or defect of one Country, with the excess or superfluity of some others. For what part of the habitable world is so sterile, but can export Commodities and Necessaries useful and advantageous, for the Inhabitants resident in pregnant and fertile Soil●? And thus Virgil says: — Non omnis fert omnia tellus. If Adam had not transgressed, his Race had never been expulsed Paradise, nor humane Industry urged by Necessity, terrified by Poverty, solicited by Ambition, or prompted by Riot: 'tis probable several countries had diversity of endowments out of Nature's treasury conferred on them, whereby to attract exiled men (as brothers) either with a coercent or an obligent Fraternity, by a continual league of Amity, and intercourse of Trade, whether situated unde● the temperate, frigid, or torrid Zone; all which the honourable Society of Merchants do perform, keeping a correspondency with the habitable world, surveying Neptune's watery Regions, discovering the Bounds, embarking the Indie and treasury of the Seas, transporting their Magazines over the proud and angry Surges, slighting dangers and the fury of impetuous Storms, thereby to support themselves, please and supply the defects of others. Poverty renders Man despicable unto those who honoured him in Prosperity, it makes him a stranger to his intimate acquaintance; perhaps pitied by many, but relieved by few; it divorces the society of friends (like some mortiferous infection) especially the Parasites of the Rich, or the Idolaters of Fortune, who are the world's mercenary Slaves in golden chains; she appears (to the enamoured of temporal blessings) a more formidable disguised Spectrum then Death; the first is Fortune's Praeludium to Contempt, the other Catastrophe to a Tragedy of Cares, or a Comedy of Errors, where true Hope attends their Transmigrations from this Scene of Miseries, ushering them to his fatal attiring-room, with a Plaudite for those whose parts were acted well; yet Penury in the Interludes often provokes noble minds to act ignoble things, and usually expulseth the fear of any enterprise, and exasperates some to the contempt of penal Laws; therefore humane Industry is urgently necessary, whereby to shun those mischiefs entailed on want; for all desperate and unjust courses are to be abhorred, as the endeavours of Societies, and the lawful adventures of Merchants are to be commended. And thus writeth the ingenious Poet Horace: Impiger extremos Mercator currit ad Indos, Per mare pauperiem fugiens per saxa, per ignes. In English thus: The nimble Merchant runs to th' Indian shore Through fire and water, fearing to be poor. The ingenious Syrians, Phoenicians and Chaldaeans restored Arithmetic, conducing much to the furtherance of Commerce and Trade, and the Merchants since, have illustrated the practic part of Astronomy, applied to the perfecting of Navigation, by which Art new Territories have been happily discovered, and the Colonies since the General Deluge traced into America, whereby sacred Religion hath made procession thither, embraced and received in climates unknown before to Europe, Asia, or Africa, not inferior to them all three together. Thus God hath graciously pleased to make them instruments of his Mercy, but to keep within their sphere, or in the circumference of mundane affairs; the mighty Monarches, with the Peers of this British Island, by custom condescended to have been made free of some Trade, Company, or society of their Subjects, in the Metropolitan of England: the Lucitanian Kings, with their Nobility, have not only accepted the title of Merchant, but have really employed adventures at Sea, to encourage their subjects in advancing their public good, and strengthening themselves both by Sea and Land. States by Traffic have risen out of a Fisherboat, dilating their jurisdictions equal to the potent Princes of the Earth; and to descend unto particulars, how many great and eminent Families in all Countries have been raised out of the dirt, or from despicable degrees, and yet elevated to the most illustrious Titles that subjects could be capable of, or Princes could confer on them, sitting at the helm of their Republics? others again with shipwrecked conditions, emerged in their estates, through their own extravagant courses (not steered within compass) the injustice of others, or the injuries of Fortune; yet it hath often pleased God to allot this means whereby they have been buoyed up again, and prosperous gales have filled their sails, until they have anchored in the wealth harbours of their eminent Predecessors? The Pen, Sword and Merchandizing, have been generally in all ages the instrumental means of accumulating worldly blessings, and preferring men to eminency in Honours, and temporal possessions; as for those who rise by the exposition of political or municipal Laws, I do not deny but it is of itself an honourable employment, especially to all candid breasts, in whom Justice keeps her Courts of Judicatory, her Scales not used to weigh their gold, but the cause; where bribed Rhetoric is not allowed to gild over an unjust Process, to make the richest cause (though the foulest) seem fair, to procrastinate with Demurs, or Fines and Recoveries without end or recovery; when as the controversy is but about Meum & Tuum; and the general reason wherefore, the sins of the people, and civil dissensions; yet cases may be doubtful, though often visibly unjust. Those Ovid seems to check. Turpe reos empta miseros defendere lingua, Quod faciat magnas turpe Tribunal opes. Base is that bribed tongue which guilt defends, Base that Tribunal which seeks private ends. Secondly, for the Sword, grave Cicero prefers the unjustest Peace before the justest War; besides, it raises but a few by the suppression of many, and the common Soldier, or general part, are but the back stairs for others to climb up by to eminency, and all the others have nothing to glory in, but how Princes and States are indebted to them; or in their badges of grinning Honour, which they bore before them as Cognizanses of their valour, engraved in their bodies with capital Characters by the fatal Steel, where one may read (as in an History at large) the storming of Forts, razing of Towns, or affronting the Canon, disgorging Death's Commissions, wrapped up in Fire and Smoke; some with half faces, or dismembered bodies, repaired with wooden limbs, like weatherbeaten Statues, that have stood century many years in the open air, until defaced by the hand of Time. O wretched state or Principality, where crooked justice must be rectified in iron bodies, her scales thrown away, and appeals made to the Sword as Umpire, licenced by incisions to cure Wounds, contentions to decide Controversies, injuries to support Right, Injustice Innocency, and tumults Peace! These are dire punishments from above, phlebotomising distempered Commonwealths, or politic Bodies falling into a Frenzy: as in humane actions we often see, desperate Diseases have desperate Cures, and by Medicines or Corrosives, many times more violent for the present then the malignancy of the Malady, fearing a relapse, or a total subversion; whereas the adventurous Merchant is beneficial to all, in Barter or currant Coin, and by his happy arrival inricheth himself and those he trade's withal, exchanging his Cargazones, and distributing his Treasure for the accommodation of others. Thus raiseth many, ruins few, a free Trade, by compulsion nothing, only damnifying others. Volenti non fit injuria. Here I have touched at the three chief Employments, tending to Riches and Honour, as the sole scope aimed at by worldly men, one of them no friend to Peace, nor necessary at all times, of Necessity rarely; litigious men are the Lawyer's Stewards; and which of these three Vocations ought to be preferred in a glorious Commonwealth, most candid Reader, judge, I will not be one of the Inquest to give my verdict, because most will censure according to their educations, or as their natural dispositions incline them, which renders man partial, by being prone to what he loves or follows: yet that I may not be censured for what is past, I will thus far expose an account of myself. Before the uncivil Scots commenced a civil War, I had served both by Sea and Land in a just defensive quarrel, which the Law of God and Nature does allow; the Maritine Employment pleased me best, by reason my former studies had initiated me in the grounds of Mathematical Sciences, whereby Navigation seemed facile and delightful: as for the Common Law, I never loved contention, nor put on the Gown, but only have observed the practice of men, and found all these courses good and laudable, if not abused by sinister ends, and the lawful endeavours of every man (in honest vocations) highly to be commended: so I will make no farther a progress herein, but retreat to my subject and intended course. Many will start Objections (as Huntsmen Hares) and pursue them over others unwarrantable grounds, which incited me to make my former books of Arithmetic better for use, and more perspicuous to the understanding by this Treatise, once before attending the Press, and now you, with the regulation of Commerce and Trade, accommodated to all ingenious capacities; prescribed rules being equally necessary to all; for those who know not how to buy, will be ignorant how to sell, or how to borrow, that know not how to lend. Besides these, here are divers Geometrical Propositions appertaining to Manufacture Trades, and some for the Surveyer, Soldier, Engineer, and Accountants, all of good use, and convenient for the illustration of my former books of Arithmetic, proportioned with Lines and Numbers, composed more for speculation than practice, and this designed more for practice then the Theory; whereby none shall be deluded with words, nor deviated with doubtful directions in diversity of ambiguous Tracts, or bewildered in Mazes, out of which these Rules shall be your conduct, if you please to accept them for a guide: In witness whereof I give you here my hand, by the subscription of Your benevolent Friend, Thomas Willsford. To the Tyron of Merchant's Accounts, short Advertisements, as to the Debtor and Creditor, with some precautions to prevent mistakes, for the right use of it. THou hast here presented thee for thy practice what is really promised in the Title, viz. Merchants Accounts epitomised; yet is it so furnished with variety of useful, practical and necessary Resolutions, as may render it to be nothing deficient for thy initiation into the famous art of Accountantship by way of Debtor and Creditor, here being both the Introductory part and Practical, so fitted to the meanest capacity, that the more common Trades may hereby be informed to keep their Books Merchant like, and an ordinary capacity may in a small time hereby learn a method whereby they may be rendered capable of keeping any accounts after the Italian manner. This Book is so complete, that I thought it unnecessary to annex a Wast-book to the Journal, it being complete enough without; by the reading of the Introductory part, thou wilt be able of thyself to frame a wast-book, whose office is nothing else but to set down at large and explain the time when we buy or sell, the person of whom we bought or to whom we sold, and what, and in what nature, whether for time, or for ready money, or exchange, thereby to refer every particular parcel of Wares and Contracts to their proper places in the Journal, there to be inserted their true Debtor and Creditor; the use of which Book and all others necessary, thou hast in the Introduction page 206, 207, and what volume they ought to be of. Now before you proceed to put any thing in practice, you are desired to amend the Erratas committed in the printing of this Deb. & Cr. (some of them being occasioned by absence from the Press, and the unusual printing things of this nature, the greatest being the misfoljoing, which are insufferable in books of this kind, by reason of the several referring the Journal & Leaguer hath to each others true place or folio. You shall find the second folio in order of the Leaguer to be by the printer numbered folio (1) the reason was because that stock which is there placed was in the original copy in folio (1) but by reason in the printing it could not be brought into one folio, it was put into one by itself, bearing its original folio, by reason of its several references to the Journal, which in all places has it noted in fol. (1) the rest of the errors you have in the Errata following. ERRATA. THe pages 203, 204, 205, 206, 207, 208, 209. should be entitled, An Introduction to Merchant's Accounts. p. 217. line 7. deal Creditor, in fol. 1. of the Journal, l. 7. deal to, l. 3. in the 2. column it should be ●/1. fol. 4. in the first l. of the 4 last, read or for our, but that which is fol. 2. in the 6. fol. should be 6. in line 5. in the col. of pence, for 8. r. 4 d. l. 21. for 1659. r. 1658. l. 6. in the col. of pence r. 8 d. that which is fol. 7. in order is f. 3. and for that 3 r. f. 7. & in dito f. l. 19 just against A.B. in the col. of l. s. d. insert 1 l. 2 s. 8 d. l. 22. just against Middlesex, for 1 l. 2 s. 8 d. r. 12 l. l. 26. just against A.M. in the col. of l. s. d. for 12 l. r. 200 l. f. 9 l. 1. for foe 1. for. l. 6. for Vigmys r. Virginia. f. 10. in the col. of ls. for 430 l. r. 304 l. l. 11. in the col. of l. for 3 l. r. 4 l. In the Leaguer. Fol. 4. Debtor side, l. 3. in column 2. on the Debtor side to the left hand, 7. deal 3. fol. 4. Cr. side col. 2. to the left hand, for 8. r. 9 in l. 4. f. 6. the total and last sum of profit and loss on the Debtor side should be 588 l. 19 s. 4 d. f. 9 col. 2. the Debtor side, l. 1. deal 5. Reader, ALthough the benefit of my Country and my own Recreation hath put me upon the study, and publishing of these Curiosities, for the knowledge of these Arts, wherein all things cannot be so plain, but that there may be some need of the further assistance of an Artist, I here make bold to acquaint thee with the perfections of Mr. Nathanael Sharp, who writeth all the usual hands writ in this Nation, the Art of Arithmetic, Integers and Fractions, and decimal Merchants Accounts, also youth boarded, and made fit either for Foreign or Domestic Employments. He lives in Chain Alley in Crutchet-Friars. To his Honoured UNCLE, M. Thomas Willsford, etc. WHat sacred Apathy confirms your breast? And in loud storms rocks you to peaceful rest? Calm Studies, and the gentler Arts you ply; No outward airs untunes your Harmony; Resolved how bad or mad soe'er we be, Not to revolt from your loved Industry; The great Archimedes 'mongst blood and rage, Smoke, and the cries of every sex and age, Smiled on the face of Horror, and was found Tracing his mystic figures on the ground. Thus He and you seem to look down on Fate, For 'tis not Life, but Time, we ought to rate: Which you improve to Miracle, each sand Attests the labour of your head or hand: While Arts Arcana, and new Worlds you find; The blessed discoveries of a trav'ling mind. Nor are you to one Science only known, For every Muse, all Phoebus is your own. Edward Boteler. AN INDEX TO THE FIRST BOOK, Divided into three Parts. PART I. OF Wholesale and Retail, without gain or loss, or the contrary, whether relating to the whole parcel, or part, or to any Interest per cent. per ann. as in page 1. to the 15. Propositions. Equation of payment, p. 19 prop. 16. Barter, with the dirivation, p. 20. pr. 17. & 18. Tear, Neat, Tret, and Cloff, p. 22. pr. 19, 20, & 21. Exchanges of foreign Coin, Assurances and Returns of Money, p. 27. pr. 22. to 28. Reduction of Weights and Measures, p. 35. pr. 29. to 32. Factorage, p. 39 pr. 33. to 36. Cambio Maritimo, Sea-hazard, or Bottomree, p. 44. pr. 37. PART II. Definitions and Etymologies of Usury and Interest-money, p 47. Simple Interest in Forbearance, where the Principal, Interest and Time, are of whole or mixed denominations, p. 50. pr. 38. to 45. Discount or rebate of money for any Interest or Time, p. 59 pr. 46, & 47. Tables of forbearance and discount in compound Interest, calculated by decimal Arithmetic, from 1 day to 25 years inclusive, at 6 l. per cent. per ann. p. 64, 65, 66. The construction of decimal Tables, p. 67. in forbearance and discount of Money, Rents, Pensions, Annuities and Reversions, with the purchase of them, to p. 78. The application of these Tables, p. 79. quest. 1. to qu. 21. PART III. Rules of Practice by memory, and the assistance of one Table, p. 103. & 105. The description and use of this Table in 9 Examples, p. 107. A Julian Calendar for the receipt and payment of money, or other business, as to find what day of the week any day of the month shall fall upon for 11 succeeding years, p. 112. A Gregorian Calendar, for the receipt or payment of money beyond sea, where that account is received, p. 114. The Contents of the second Book, divided into five parts. PART I. The dimension of all plain or right-lined Triangles, pag. 117. Problem 1, 2, 3. The dimension of Board, Glass, Hang, Wainscot, Pavements, Land, etc. p. 122. pr. 4, 5, 6. Reduction of any squared Superficies, from a greater to a less, and the contrary, p. 129. pr. 7. How to make the Carpenter's Ruler, for the dimension of any Superficies, measured by the foot or yard square, p. 136. pr. 8. To find the content of a Square, including any Circle propounded, p. 134. pr. 9 The Diameter of any Circle known, to find the greatest circumscribed Quadrangle, p. 135. pr. 10. To find the nearest Quadrature of a Circle, p. 137. pr. 11. The dimension of solid bodies, p. 139. prob. 12. to pr. 14. A Demonstration in the commensuration of tapering Timber, p. 147. To divide the Carpenter's Ruler, for the measuring of solid bodies. p. 151. PART II. The dimension of round and concave Measures, p. 155. prob. 1. The gauging or measuring of Casks, from the Runlet to the Tun, either of Wine or Beer, p. 158. pr. 2. By the diameter of any Circle, to find the Circumference, p. 160. pr. 3. With the diameters of two Circles, and the circumference of the one how to find the other, or the contrary, p. 161. pr. 4. With the diameter and superficies of one Circle, to find the content of any other, the diameter being known, p. 161. pro. 5. With the superficial content of two Circles given, and one Diameter to find the other, p. 162. pr. 6. To find the convex superficies of any Sphere or Globe, whose diameter or circumference is known, and that four several ways, p. 163. pr. 7. By the diameter or circumference of any Sphere or Globe known, to find the solid content 3 several ways, p. 194. pr. 8. With the diameter and weight of any sphere or Globe, to find the weight of any other, whole diameter is known, p. 166. pr. 9 PART III. With the diameters of 2 Bullets known, with the weight of one to discover the other, p. 169. proposit. 1. By the weight of 2 Bullets known, and diameter of the one to find the other, p. 70. prop. 2. Compendious Rules for martialling of Soldiers, in all right angled forms of battles, with their definitions, p. 172. prop. 3, 4, 5, 6. The encamping of Soldiers in their several quarters, p. 176. pr. 7. The height of any Wall or Tower being known, to find the length of a scaling Ladder, p. 177. pr. 8. To find the height of any Wall or Tower that is accessible, p. 178. To find the distance to any Fort or place, though not accessible, p. 179. pr. 10, 11, 12. In a City, Castle, or Fort, to be beleaguered, how to proportion the Men and Victuals, Guns and the Powder, whereby to make the Works tenentable for any time limited, p. 182. pr. 13. to pr. 16. General Rules and Observations of experienced Engineers and Gunners, p. 189. PART IU. The form of keeping Merchants Books of Account after the Italian manner, in form of Debtor and Creditor, p. 233 Architectonice, or the art of Building, as an Introduction to young Surveyors, of the Estimates, Valuations, and Contracts, from p. 1. to p. 5. The manner in taking of a survey of Mason's work by the great, p. 6, 7, 8, 9 A bill of Measure, with the charges in money according to the articles of agreement. p. 10. Estimates, Contracts, Rates, Rules and Proportions, observed by Carpenters, p. 11. Proportions and dimensions of a Roof, p. 12. The Materials, Valuations and Proportions in covering of Structures, and finishing of them, to make them tenentable and commodious, by sundry Artificers, p. 15, to 30. The five orders of Columns or Pillars described, with the Artificers and Inventors of them, p. 31. to 34. FINIS. THE FIRST BOOK. PROPOSITION I. A Grocer bought 5 ¾ C gross weight of Wares, which lay him in (with all charges defrayed) 163 lb: 13 ss: 8 d. sterling: and it is demanded what one lb cost: or how to sell it by the pound without gain or loss. The Rule. As the quantity of any one Commodity or wares is unto the total price with the cost and charges, so will a l, or an unite of the first denomination be in proportion unto the rate it may be sold for. An Explanation of Wholesale and Retail: Lib. 2. Parag. 8. Observe in all Commodities where a hundred gross is mentioned, it is 112 lb usually noted with a C. for Centum, as in this Example, where five hundred and three quarters is given, which 5 C multiplied by 112 lb and ¾, or 84 lb added unto it, the product with the addition will make the sum of 644 lb for the subtle weight, and the first number; the second number in proportion, is the price, viz. 3273 ⅔ 8: the third number is 1 lb, on which the demand is made: these compound f●actions you may reduce into the least denomination, or the least but one, as in the Table; where by the Rule of Proportion in either way you will find th●t one lb of those Wares stood the buyer in 5 ss: 1 d: or as in the Table 5 2/●2 ss: with all charges defrayed, according to the demand and state of the Question propounded. PROPOSITION II. How to sell a●y Wares or Commodities by retail, the pr●ce or value of the whole par●el being known, and to gain a certain sum of money in the whole quantity required. The RULE. As the quantity of the whole Commodity bought is unto the sum of the price and gain required, so will 1 C. 1 lb; or 1 yard of the first denomination be proportional unto the price it must be sold at. An explanation where gain is imposed upon the parcel: Lib. 2. Parag. 8. Admit a Draper bought 58 yards of Cloth, which stood him in (with all charges) 13 lb: 1 ss: at what rate by the yard must he sell it for, whereby to gain 1 lb 9 ss: add the required profit unto the price, the sum will be 14 lb: 10 ss: that is 58 Crowns, or in shillings 290: to avoid fractions, the least denomination is usually best: here the fourth proportional found will be 5 ss, the price of one yard according to the gains required, as by the operation in the Table is made evident. PROPOSITION III. A Tradesman bought a whole piece of Cloth containing 28 ¾ yards, which did stand him in (with all charges defrayed) 19 lb: 3 ss: 4 d. sterling, how should he sell it, whereby to gain 1/10 part in every yard, or forced unto so much loss in retail. The RULES. RULE 1. As the Denominator is to the price known, so the fractions sum. As the quantity given, RULE 2. so an Unite of the same. An explanation, where gain or loss is imposed upon a part, Lib. 2. Parag. 10. This Question is stated according to the Double Rule of Proportion, either for gain or loss, by changing the extremes in the first Rule, viz. in this 10 for 11, the fraction in all such cases making two terms; the Denominator in the first place being Divider, the price of the Wares or Merchandizes the second term, and the sum both of Numerator and Denominator must possess the third place, if for gain; but must be made Divider, if the Proposition be for loss: the first number in the second Rule aught to be the quantity propounded, either in Number, Weight, or Measure; and the last Number an Unite on which the query is made, of gain or loss: or, which is all one, if an improper fraction as in the Table 4/4, the Denominators being made equal, viz. 135/4 and 4/4 and consequently may be omitted, one being a Multiplier, the other a Divider; their Products are these 1150/4 · 115/6 · 44/4 · the first and third Numerators may be reduced by 2, and their Denominators canceled; they will stand thus, as in the third Table, viz. As 575 to 115/6 so 22 unto 253/345 which is in money 14 ss. 8 d, the price of one yard with the profit required. The reason is evident; for if 20 ss were the Integer, the Numerator would have been 2 ss and consequently the proportion as 20 to the middle term, so 228 the sum of Numerator and Denominator to the gains required. This question may be easily solved without a Double Rule, as thus: by the first Proposition you may find that one yard cost 13 ss. 4 d. 1/10 part of it is 1 ss. 4 d, the sum 14 ss. 8 d. as before: but this may be of good use in other questions, and therefore conveniently inserted. PROPOSITION IU. How a Commodity must be sold by retail, upon any certain loss of money in the whole parcel or quantity. The RULE. As the quantity of any Commodity or Parcel is to the difference betwixt the Price and Loss, so shall 1 C, 1 lb, or one yard of the Commodity itself be proportionable unto the rate it must be sold at. An Explanation upon Loss sustained in any Commodity. Lib. 2. Parag. 8. A Grocer bought 340 lb subtle of a Commodity which cost him in ready money with his charges 13 lb. 16 ss. 3 d. and by this parcel he lost 1 lb. 1 ss. 3 d. what was it sold for a pound? the loss in the whole subtracted from the price it cost, the remainder or difference is 12 lb. 15 ss. which in pence is 3060 d. so the proportion will be as 340 lb, is to 3060 d, so shall 1 lb be to the price of it; which is 9 d, as in the Table appears: and as for the trial of it, the proportion will be as 1 lb is unto 9 d, so will 340 lb be unto 3060 d, or 12 lb. 15 ss. as by the first Proposition. PROPOSITION V. Upon the price of any Commodity known, how to sell it by wholesale or retail, with gain or loss, at any rate in the 100 lb that shall be requi●ed. The RULE. As the sum of 100 lb sterling is in proportion unto the price of the Wares, so shall the rate in money for gain or loss be in proportion to a fourth number, which added to the price of the Commodity, gives the gain, and subtracted from it, shows the loss sustained at the rate required. An Explanation of gain or loss as any ●ate per Cent. Lib. 2. Parag. 8. Suppose 1 Gro●e bought 4 C weight of Prunes, at 16 ss. 8 d. the hundred, how must he sell them by wholesale again, and to make of his money 20 lb in the 100 lb. or after that rate? The answer will be 3 ss. 4 d. which if taken from the price at which they were bought, the remainder or difference is 13 ss. 4 d. and if sold at that price, there will be after the rate of 20 lb in the 100 lb lost: and if 3 ⅓ ss. be added to the price, at which 'twas bought, the sum is 20 ss, and if vended at that rate, it will bring the desired profit. If this had been Cloth, and the whole Piece had contained yards 28 ½, which co●t in money 23 lb. 15 ss: by the first Proposition find the price of one yard (if it must be sold by retail) the answer will be 16 ⅔ ss. now the question at 20 lb per centum will be the same by retail, as was the former in wholesale. Many of these Questions may be performed without calculation, as in this Example, where 20 lb per cent. is required. The profit in money here is ⅕, and so the gain or loss in the Commodity must also be ⅕: the price in this was 16 ss. 8 d. that is 50 groats, and ⅕ part of it is 10 groats to be added or subtracted accordingly as it is gain or loss. PROPOSITION VI. The price of any Wares or Merchandise by which the said Commodity was bought and sold: what gain was made, or loss sustained in the 100 lb: or after what rate or proportion. The RULE. As the price (by which any Merchandise was bought) shall be in value or proportion unto 100 lb sterling, so will the price of the Wares by which they were sold, be in proportion to the true gain or loss sustained. An Explanation of Gain or Loss sustained, at any rate per Cent. Lib. 2. Parag. 8. A Draper bought Kerseys at 6 lb. 13 ss. 4 d. the Piece, and sold them all again for 7 lb. 10 ss. how much he gained, and after what rate in the 100 lb is the thing required: the price by which 'twas bought, and likewise the rate at which 'twas sold must be reduced into one denomination, or by the Rule of Fractions, viz. As 20/3 lb the price is to 100 lb, so 15/2 unto 112 ½: by which it is apparent that he gained 12 lb, 10 ss. in the 100 l. or after that rate; for 100 l. thus employed will return 112 ½ l. If any question of this kind should depend upon Loss, the Price at which 'twas sold must be less than that by which the Commodity was bought at, so the fourth proportional number will be discovered by the same Rule; the state of the Question not differing in any thing, either by Wholesale, or Retail, so it requires no Precedent or Rule but this, which will bring your stock short home, as unfortunately true, as prosperously with increase. PROPOSITION VII. By the Price which any Wares or Merchandizes were sold at, with the rate of Gain or Loss in one Piece, how to discover what the whole Commodity cost. The RULE. As 1 Piece, 1 Hund. 1 Yard, or 1 Pound weight, etc. shall be in proportion unto the price thereof, so will the number of Pieces, or quantity sold, be proportionable to the price of them all together. An Explanation of Gain or Loss in one Parcel, to find the rest, Lib. 2. Parag. 8. Admit 15 Clothes or Pieces were sold for 340 l; then was the price of one Piece 22 l: 13 ss: 4 d, as by the first Proposition; in this there was present gain 19 ss. 4 d, upon every Piece, which subtracted from the Price 'twas sold at, viz. 22 l. 13 ss. 4 d. the difference is 21 l. 14 ss. for the price it cost: then will the proportion be as 1 whole Cloth is to 21 7/10 l, so shall 15 Clothes be unto 325 l, 10 ss, as in the Table appears. If this Commodity had been sold to loss, the differences betwixt the prices makes it evident, and then what one Piece, or any pa●t had co●●, will be discovered as before, with all the whole loss sustained; and if it should be required after what rate in the 100 l. the last proposition will unfold it according to the Rule of Trade. PROPOSITION VIII. To find the Gain or Loss upon Merchandizes bought and sold, with time agreed upon betwixt the Debtor and Creditor for payment of the money at any rate per cent. per an. The RULES. Rule 1. As 100 l sterling is to any interest so a sum given If for 12 Month's Rule 2. What for the time. An explanation of Gain or Loss with time at any rate per Cent. Lib. 2. Parag. 10. Admit a Tradesman had bought a Commodity at 5 d the pound, and after 6 Month's time sold it again for 6 d the l. or suppose the Merchandise was bought at 5 ss the yard, and sold it presently again for 6 s the yard, but with 6 Months given for day of payment, or to abate so much as the interest should come unto at 8 l per cent. per annum, by the sixth Proposition, the gain of those Wares will be discovered after the rate of 20 l per cent. if present pay; but here is to be rebate of money, or forbearance of the stock and profit for six Months: suppose 100 l disbursed for these Wares at first, which would make 120 l if paid down on the nail; but here use is to be considered for that sum, and six month's time with the increase to be deducted: the interest of which sum is thus found: in this Proposition 'tis six Months, and 8 l per centum, as in the first row or rule in the Table; in the second row under 100 l stands the term for a year, in the same denomination with the time given, viz. 12 months; and under the third term, the time limited for payment, viz. 6 Months, the products of them (according to the double rule of Proportion) in the third line is, as 1200 to 8 l. so 720. these are again reduced in the operation of the fourth Table, as 120 to 8. so 72 unto 4 l. 16 s. and might have been reduced again to 5.1.24. which will also produce 4 l. 16 s. that subtracted from 120 l. the remainder will be 115 l. 4 s. which shows 15 l. 4 s. clear gains in relation to the rate by which 'twas bought and sold at, with the interest for the forbearance, agreed upon according to custom and contract, but not exactly true. PROPOSITION IX. By the price of any Wares bought and sold, with the time limited for payment, to find the gain made, or loss sustained, and at what rate per cent. per Annum. THE RULES. Rule 1. As the first price shall be unto 100 l. so the gain or loss. If for 12 months. Rule 2. So the time limit. An Explanation in Gain or loss with Time. Lib. 2. Parag. 10. A Merchant bought Mace at 6 s. 4 d. the l. ready money, and he sold the same again unto a Grocer for 7 s. the l. at this rate, the Mace was delivered, and upon condition to be paid at the end of 4 month's next ensuing the receipt thereof; and it is required what gain the Merchant made of his money, and at what rate per cent. per Annum. In all questions of this kind make the price at which 'twas bought, and as 'twas sold, of one denomination, the difference shall be the third term in the first rule, 100 l. the second number, and the price for which 'twas bought, the first term: in the second rule under the first number I place the magnitude of a year, in that denomination, in which the time limited is given; as in Months, Weeks, or days: in this 'tis Months, as the Letter M denotes; the space of time given for payment is 4 Months, subscribe that under the third number; then draw a line from thence towards 19 G, and that cross with another, as from 12 M t 2 G in this Example; these multiplied crosse-wise (the second rule being reversed) for the less time is given for payment, the profit will be the greater: in the third row stand the products in the Rule of 3 direct; and in the fourth Row or Table is placed the form of operation, wherein the desired product is discovered to be 31 1●/19 l that is, 31 l: 11 ss: 6 d 1●/19: the profit required at the rate per cent. per annum. PROPOSITION X. A Grocer bought Cloves at 4 ss 3 d the l. and after 6 Month's time sold them again for 4 ss the l, what loss did the Grocer sustain, and how much per cent. per ann. by the last proposition you will find his loss to be 11 l. 15 ss. 3 9/17 d. PROPOSITION XI. By the difference of prices in any one Commodity bought and sold, by wholesale, or retail, to find what time must be allowed for to gain after any rate per Centum per annum that shall be assigned. The RULES. Rule 1. As 100 l sterling is unto 12 Months, so the rate propounded Unto the 1 price. Rule 2. so the gain or loss. An Explanation in Gain or Loss with time unknown. Lib. 2. Parag. 10. A Tradesman bought Nutmegs at 8 ss the l, and sold them all again at 9 ss the l, what day of payment must he allow, whereby to gain after the rate of 20 l per cent. per ann. in all such like Cases, as 100 l to 12 Months, so 20 l. secondly, as 8 ss the price, unto 1 ss the gain made: the first rule is reversed, the other direct: in the third row of the Table stand the products, and under that again (in the fourth Table) is placed the operation in a reversed proportion, but may be made direct if you please: the answer to the question here in this, is 7 Months and 15 days, the time of payment; which makes the gains proportionable unto 20 l per cent. per annum, the thing required: had the Proposition been of loss, the operation is the same, so it needs no example but this. PROPOSITION XII. If 124 l of Cinnamon cost 20 l sterling, and that sold again for 23 l what day of payment was there given, when the Merchant made after the rate of 16 l per cent. per ann. of his money so laid out? this question will be solved by the last Proposition, and found that the day of payment was at the term of 11 Months and ¼ of a Month. PROPOSITION XIII. With the price and quantity of any Wares on Merchandise, to find how they must be sold, upon several days of payment, either in Gain or Loss, at any rate given per cent. per ann. The RULES. Rule 1 As 100 l sterling shall be to the gain so the price of wares so shall 12 Month's Rule 2. to the prod. of times An Explanation of Gain or Loss in several payments, Lib. 2. Parag. 10. A Merchant had certain Wares which stood him in 60 l: these goods he sold unto another, who paid him so much money for earnest, as that the Merchant made of his money laid out 12 l per cent. per an. yet was to receive it at two several payments, viz. 40 l or ⅔ of his money 2 Months after the Wares were delivered; and the other 20 l, or ⅔ at the term of 3 Months after the contract: now to find what money the buyer paid down, observe this table, where in the first row stands this proportion, as 100 L is to 12 L gains, so 60 L the price of the wares; in the first place of the second rule the term of a year (in months) is inserted; lastly ⅓ or 2 ●/3 month's, which number is composed by the sum produced of the terms for payment and the money, viz. as 2 months, by ●/3 of the money to be paid, and ●/3 of the money by 3 months, whose sum is ●/3. these terms multiplied according to the double rule direct, will produce, as in the third table; these three numbers, viz. 1200. 12. 140. which in the fourth table are reduced to 120.12. 14. and maybe made (retaining the same proportion) 10. 1. 14. or 5. 1. 7. the quotient will be found by any of these, or reduced unto 1 ⅖ L. that is 1 L. 8 ss. the earnest given, and the rate at 12 L. per cent. per ann. as was required; if there had been more times of payment given, the proportional sums to be paid, multiplied by their peculiar terms or respite of time given, and those sums collected into one total, the operation will be as in this last example. PROPOSITION XIV. By the gain made, or loss sustained per cent. per ann. to find what any other s●●me must gain or ●●se in any part of a year. The RULES. Rule 1. As 100 L in money Is to the gains or loss, So any other sum; If for 12 months' Rule 2. What for 3 Mo. An explanation of gainer loss proportionable to a stock and time. Lib. 2. Parag. 10. Admit a man employs his money in the way of trade, by which he gains 16 L per cent. how much does he gain with 80 L in the same employment for 3 month's time? The state of the question, or any of this kind by the first rule, is thus: As 100 L to 16 L the gain or loss, what 80 L in the second rule there is 12 months and 3 months: their products (according to this double rule of Three) are 1200 and 240, as in the third table, and in the fourth table as 120 to 16 L gains, so 24; or reduced, it will be as 5 to 16, so 1 unto 3 ⅕ L, or 3 L 4 ss, the proportional gains made by 80 L in 3 month's time: if it had been employed to loss, then 3 L 4 ss must have been subtracted from the stock, which in this proposition was 80 L, the remainder will be 76 L 16 ss; the question solved. PROPOSITION XV. By the profit of a small stock in money, and a short time to find what gain is made in the hundred per ann. or the contrary in loss. To solve this, and prove the last, I will reverse the former proposition; and suppose a Tradesman employed 80 L for the space of 3 months, in which time he gained 3 L 4 ss. and here it is required to find after what rate it was per cent, per ann. the state of the question will be as 80 L to 3 ⅓ L the gains, so 100 L, and if in 3 months, what in 12 months, or a year? The answer to this will be as in the last Prop. 16 L per cent. per an. PROPOSITION XVI. Divers tradesmen join their stocks together, with which they buy a Commodity, whose price and quantity is known, from whence they take unequal shares, what part then must every one pay? The RULE. As the whole quantity of any Merchandise: Is in proportion unto the whole price thereof, So shall each particular man's part or share be proportionable to the money he must pay. An explanation in equation of payments in gain or loss. Lib. 2. Parag. 11. Four Grocers did lb L ss A 240 30 00 lb L B 300 37 10 1656 207. C 516 64 10 D 600 75 00 The totals 1656 207 ●0 join their stocks together, and bought with it 1656 lb of Pepper, which cost them ●07 L, whreeof A did take for his part 240 lb B 300 lb, C 516 lb, and D 600 lb; the total sum of the wares must be the first number, the whole price the second, and each particular the third number, and then A must pay 30 L. secondly, B 37 L 10 ss. thirdly, C must disburse 64 L 10 ss. and lastly, D must pay 75 L. and in the same manner it will be proportioned with gain or loss, to each respective part as in the table, PROPOSITION XVII. In wholesale, or by retail, if the price of any two commodities be known with the price and quantity of the one to find what quantity of the other shall be equivalent to it. The RULE. As the price of any Wares (the quantity unknown) Shall be in proportion unto 1, or a unity of it, So the price of that whose quantity is known, Will be to a parcel of the first, and equal to the other. An explanation of Barter, or vending one commodity for another. Lib. 2. Parag. 8. A Tradesman exchanged Salt at 20 d. the bushel for Sugar at 15 d the lb, of which commodity he desired 1 C weight gross; how many bushels of salt will there be required to be equal in value unto 1 C weight of sugar: by the first table here you will find that 112 lb of sugar at 15 d the pound comes unto 1680 d. then according to this rule and in the second table, you may see the proportion, viz. if 20 d bought 1 bushel of salt, than 1680 d will buy 84 bushels of salt, being equal to 112 lb of sugar. This trading with one merchandise for another is called Barter, derived from Barato, implying an exchange of commodities, the most ancient way of commerce. PROPOSITION XVIII. The worth of any wares in ready money, if valued at a greater price in barter, how to set a rate upon the price of any other commodity to be bartered for it, that shall be proportionable to the first. The RULE. As the price of any Merchandise in ready money Is unto the value of the same Wares in barter, So the second Wares (in the first denomination) Shall be proportionable to the price of it in barter. An explanation upon the rate Wares made proportionable in barter. Lib. 2. Parag. 8. If 8 Bolts of Holland cost 37 L 10 ss, and valued in barter at 40 L, and this to be exchanged for Wool at 6 L 5 ss the hundred, what must it be rated at in the barter? according to the rule and state of the question, a fourth number will be found (as ●n the table) 6 L 13 ss 4 D, the rite put upon the Wool in barter proportionably answering the increase, as the Holland is prized, which is the thing required. PROPOSITION XIX. In any gross weight, as Sugar, Currens, etc. upon the allowance of Tare, to find the Neat weight, and the price of it; and if there be Cloffe allowed to find also the Neat and the price, by knowing what the gross cozened was valued or bought at. The RULE. As 100 gross weight of any Wares or Merchandizes Shall be in proportion to the pounds allowed for Tares, So will any gross wares that are known or weighed Be unto the allowance given for the Tear thereof. An explanation of Tare, Neat and Cloffe in gross weights. Lib. 2. Parag. 8. Tare is an allowance betwixt the Merchant and the Grocer, for all such commodities as are weighed in boxes, chests, casks, etc. for which the allowance is agreed upon when they bargain, as from 6 lb to 18 lb in the C, and is called the Tare; which subtracted from the gross weight (that is, the chest, or cask and wares together) the remainder is the Neat; as suppose here are weighed four chests of Sugar, the one in gross weight 6 C and 14 pound, the second 7 ¾ C and 10 pound, the third 8 ½ C and 4 pound, the greatest 9 ¼ C, the sum or total of them in all 31 ¾ C gross: then the proportion is as 1 C is to the Tear given (which admit 14 lb) so 31 ¾ C unto 444 ½ lb. or by the rule of Three without fractions (as in the table) and then to the fourth number found, viz. 434 lb. add in the Tare for the parts, as ¾: which is for the ½ 7 lb & for ¼ add 3 ½, the sum is 10 ½, the total 444 ½ lb. as before, which divided by 112 l is 3 ¾ C. 24 ½ lb, the true Tare of the gross weight 31 ¾ C: from whence subtract 3 ¾ C 24 ½ lb, the remainder will be 27 ¾ C 3 3/2 lb, the Neat weight, as in the table appears, and so for all questions of this kind. As for Cloffe, it is only an allowance for the refuse of the commodity, which hangs upon the chest or cask, for which is usually allowed but 3 or 4 pound in every parcel, with the parts of pounds, if there be any; but yet in this I strike off ⅛ C 3 ½ lb, and there will remain 27 ½ C 14 lb Near: now to find what it is a pound by the whole weight, or the contrary, see Prop. 1. PROPOSITION XX. Upon any Indian Spices sold by wholesale, with the allowance for Trett, to find the subtle weight Neat, and how much it is a pound; or by the pound subtle of one sort, to find what any other quantity did cost. The first RULE. As 26 pound weight of any Indian Spices Is in proportion unto 1 pound for the Trett, So shall any quantity of the same Wares Be proportionable unto the Trett thereof. The second RULE. If 50 lb weight subtle of any Spice propounded Did cost a certain and known sum of money, What is the price of another quantity of the same, When 4 pound per cent. shall be allowed for Trett. An explanation of Trett, with reduction of the pound subtle. Lib. 2. Parag. 8. Here first you are to know, that the custom betwixt the Merchant and Grocer is to weigh all sorts of his wares by the C gross, which contains 112 pound, for which in some he allows Tare, as was said before; but in Tobacco and sundry other commodities, with all Indian and Arabian Spices, their gross weight is reduced into simple pounds, usually called the pound subtle, and in every such hundred there is commonly four pound allowed, which is called the Trett; that subtracted, is by many called the Neat weight: this is readily found by dividing the whole quantity by 26, according to the first rule (which produceth the Trett) and that quotient subtracted from the total of the pounds subtle giveth the Neat of the refined weight. As for example: A Grocer bought 50 pound of Mace subtle, which cost him 18 L 15 ss, what shall the subtle weight of 4 ¾ C 1 lb gross of the same commodity stand him in, with abating 4 lb per cent. for Tret? first the gross weight of it reduced will be 533 lb subtle, which divided by 26, giveth 20 ½ lb for Trett, and subtracted from 533 lb, the remainder is 512 ½ lb subtle Neat, as in the first and second table; and in the third table the question is stated, if 50 pound of Mace cost 18 ¾ L, then 512 ½ lb will stand him in 192 3/16 L, that is 192 L 3 ss 9 D, as by the operation in the table will appear; and to find what it is a pound, the first Proposition will resolve the query. PROPOSITION XXI. If 1 pound of Cloves cost 3 ss 6 D, what is the price of 676 pound, deducting 4 pound in every hundred for Trett? The subtle weight here propounded is 676 lb, and abating 4 lb per cent. for Trett, which comes unto, by the last Proposition, 26 lb, which deducted from 676 pound, there will remain 650 pound, then by the first Proposition, if 1 pound cost 3 ½ ss, what shall buy 650 pound? the answer to this question will be 113 L 15 S, and so for any other of these kinds. PROPOSITION XXII. A man delivers unto a Merchant a certain sum of money to be received of his Factor upon Bills of Exchange in a foreign Country and Coin, the rate and proportion of the moneys in both places being known. The RULE. As a unite, or any one piece of a known Coin, Is in proportion unto a Coin of another value, So any sum of money delivered in the first Coin Unto the quantity, to be received of the second. An explanation of Exchanges betwixt Foreign Coins. Exchanges of all Coins, Weights and Measures of foreign places with one another are easily performed by the common rule of Three (if first reduced unto any certain proportion) by which means any one thing may be converted into the species of another, in respect of value or quantity, as by some few propositions with examples shall be illustrated, and first for this; suppose that sixty pound received at a place where one pound is 13 ss 4 D sterling or English, how much must the man receive in London at 20 s the pound Sterling, whereby to make them equal in value: The answer to this proposition will be 800 s or 40 l, as in the table, equal to 60 Marks, the value of one being 13 s 4 d. had the question been stated in any other denomination, the solution would have proved the same with the second term, as 1 l Scotl. to 40 groats, so 60 l Scotl. unto 2400 groats, or in the fraction of a pound Sterling, thus, as 1 l Scotl. to ⅔ l Sterl. so 60 l Scotl. to 40 l. Sterl. Thus sometimes you may ease yourself by changing the denominations (all being true) depending upon the seventh Paragraph of my second Book. PROPOSITION XXIII. Any pieces of Coin, if equal unto some one piece of another, and that equivalent to a third, to be exchanged with the first; how much money of the one will discharge a bill of the other. This differs not essentially from the last; as admit 10 Rials were equal to one Ducat, and one of them worth 5 s 6 d, how much money Sterling will discharge a bill of Exchange for 4500 Rials? the proportion will be: As 10 Ryals is to 5 ½ ss. what 4500 Rials? the answer 2475 s, that is 123 l 15 s sterling, the question solved: for if A be made equal to B, and B = to C, then A and C are equal, as by the second Axiom. Lib. 2 par. 7. PROPOSITION XXIV. If upon return of money a certain rare per cent. shall be required, to find in any sum of money how much must be abated, at the rate propounded, upon such exchange or return of money. The RULE. As the sum of 100 l with the allowance per cent. Shall be in proportion to 100 l where it is to be paid, So will any sum of money received of a Merchant Be proportionable to the money that shall be delivered. An explanation upon return of money after any rate per cent. Lib. 2. Parag. 8. A Merchant of London was to return money to be delivered at Durham, as admit 616 l received by a Carrier, which to secure and deliver at the place appointed what was to be returned, the Merchant did allow 2 l 13 s 4 d per cent. upon this abatement, how much was the sum paid at Durham? first add unto 100 l the money to be abated per cent. the total in this proposition will be 102 l 13 s 4 d, which must be the first number in the rule of Three direct, and will be in proportion thus, as 102 ⅔ L, or made an improper fraction, viz. ●0●/3 to every 100 L returns, so 616 L received will be proportionable unto 600 L, the Proposition answered. PROPOSITION XXV. Upon assurance and return of money at any rate in the pound sterl. to find what a greater or lesser sum will be worth, assured at the rate propounded. Observe in any proposition made, the true state of the question, and whether it be customary, or of that predicament; if customary, it is something tolerable in small sums, although a little erroneous; this caveat concerns other propositions, only note well the difference of these, in the last, the Assurancer was to have so much money out of the sum delivered to him, as should but discharge the money he returned, which the last rule does solve, where the Assurancer had 16 L out of the 616 L, so answers the question in the rate required, which admit imposed upon every pound sterl. the proportion will be: as 1 L is to the rate given, so will the sum to be returned, unto the money due upon it for the assurance. And for the probat of this, suppose (as in the last proposition) 600 L were to be returned from Durham to London, allowing the assurancer 6 ⅖ D upon every pound sterl. the rule is as 1 L to 32/5 D, so 600 L shall be in proportion to 3840 D, that is 320 ss, or 16 L, as before, due for the securing of 600 L, and not 616, as in the last proposition, which is erroneous, though allowed of by many. PROPOSITION XXVI. The rate or proportion for the exchange of any money betwixt two places being known, to find how much money of the one place will discharge a bill of exchange in the other city or town. The RULE. As any one pound sterl. or other piece of money, Is in proportion to the difference of Exchanges, So will any sum propounded of the first money Be proportionable to the coin where it is payable. An explanation of two ways concurring in one production. The rate for exchange here in this example is of a foreign coin, whereof 1 L 3 ss 4 D is equal to 1 L sterling, how much of that foreign money will discharge a bill of exchange for 240 L 13 ss 4 D sterling? in this case 1 L or 20 ss is the first number in the rule, the difference in exchange is 3 ss 4 D, the sum to be exchanged is 240 L 13 ss 4 D. with these 3 numbers you may find 40 L 2 ss 2 ⅔ D. which added to the money paid makes 280 L 15 ss 6 ⅔ D, the total to be received upon exchange: but the more usual way is according to the table and prescribed rule, viz. as 20 ss is to 23 ⅓ or 20/3 ss. so 1●●●●/3 unto 5615 9/● ss. which reduced is 280 L 1586 ⅔ D, as before. PROPOSITION XXVII. By knowing the money paid unto a Merchant; and likewise the sum received upon bills of exchange in a foreign coin, to find how the exchange went between those places. This proposition is but the former reversed, and so requires no rule (but that of proportion) nor example but the last, where the first money paid is 240 l 13 s 4 d; the foreign money received upon bills of exchange was 280 l 15 s 6 ⅔ d, the middle number here in the rule of Three must be 1 l sterling, or 20 s, if you please: the former numbers reduced into improper fractions will stand in the rule of Proportion thus, viz. As 14440/3 ss is to 20 ss, so will 50540/9 ss be to 1 l 3 s 4 d, the rate which the Exchange went at, according to the former Proposition, enucleated in this. In all these questions, or any others (appertaining only to the exchange of money) there is nothing more required, from the value or estimate of any known Coins, to find what sum of the one, shall be equal or in proportion unto the same quantity of the other, as if 1 l sterling were equal in value unto 25 s of some other coin, the proportion of equality would be, viz. As 1 l sterling, or 20 s is to 25 s, so any sum of the first coin to an equal quantity of the second, or which is all one (●ib. 2. parag. 1. Aziom 13.) as 4 to 5, so any quantity of the first to an equal sum of the second, and likewise the contrary to these, viz. as 5 is to 4, so any known sum received of the first coin, will be equal in quantity to the sum of the second due to be repaid in exchange, which is the sole scope of this rule, or the mark that is ●imed at in the exchange of money, as for the profit, experience in trading will discover it. PROPOSITION XXVIII. A Merchant delivers so much money, with this condition, to be repaid in a foreign Coin and Country, within any limited time, as a year, and at any rate per cent. per an. for interest allowed of there. The RULES. Rule 1 As 1 L English or 20 ss Sterling To a sum of that money, So 1 L Sterl. in another coin. If 100 L Sterling, Rule 2. What 8 L Sterling interest? An explanation of Exchanges. Lib. 2. Parag. 7. Axiom 13. and Parag. 10. As in this example, suppose 350 L of English money was delivered in London to be repaid upon bills of exchange a year after the receipt thereof, and to allow 8 per cent. per an. in that country where 24 ss was equal unto 1 L sterl. from hence the proportion is, as 20 ss is to 350 L, so 24 ss: in the second row of the table it is reduced unto 2. 35. 24. and in the third row to 1. 35. 12. by this or any of them you may find the fourth proportional number to be 420 L, the sum to be paid in the foreign coin; and in the fourth row of the table you will find 100 L of that money under 1. and beneath 12 stands 8 L for a years interest; these will make 3 numbers, viz. 100 35. 96. from whence a fourth proportional number will be produced, as in the fifth table, viz. 33 L 12 ss, the interest due upon 453 L, so the total to be received is 453 L 12 S, according to the condition and state of the question. PROPOSITION XXIX. To reduce weights that are customary in one, or divers Countries, to an equality from one denomination into another, or the weight of any ponderous body being known, to find the quantity of a greater or lesser weight. The RULE. As the proportional parts of 1 ounce, 1 pound, 1 stone, 1 C weight, etc. Shall be to any quantity propounded in that weight, So will the weight of any other place, town, or country Be proportionable to the weight thereof demanded. An explanation in reduction of weights, Lib. 1. Parag. 8. The question here propounded is of a commodity whose gross weight is 2 C or 16 stone, at 14 pound to the stone, and it is required to find how many stone there are, where custom admits but of 8 pound: the proportion of a stone weight in these two places is as 14 to 8, or as 7 to 4: in this rule the third term is the least, and yet requires a greater number; from whence it is evident the rule must be reversed, and the fourth proportional found in the table, will be 28 stone, equal to 16 stone at 14 pound to the stone, the thing required. PROPOSITION XXX. How many hundred or pounds of Troy weight will there be found in 5 C Aver de pois, when as 1 pound 2 ounces 12 penny Troy, is equal unto 1 pound or 16 ounces of the Civil, or Merchant's weight. An Explanation. This depends upon the last Proposition, and so requires no other rule, but only to reduce the gross weight into pounds subtle, which are 560 pound, and since 1 pound Aver de pois is equal to 14 ⅗ ounces, what 560 pound: by the rule of Three direct you will find 8176 ounces, which divided by 12, the quotient will be 681 ⅓ pound Troy; and so for all other questions of this kind. PROPOSITION XXXI. The customary measure of any place being known, with the quantity of one propounded, to find how much it will make by a greater or lesser measure of another place. An explanation in reduction of Measures. Lib. 2. Parag. 8. An Innkeeper bought 20 quarters of corn, to be delivered where the custom of the place required, 8 ¾ gallons to every bushel, how much must the Farmer send in, according to the Statute measure, containing 8 gallons (commonly called Winchester, where the Act was made) for to fulfil the condition as the bargain was agreed upon: the state and operation of this question, or the like, differs not in the form from the 29, as in the margin is evident, where 8 ¾ gallons, or 35/4 multiplied by 20 quarters, the quantity propounded in that measure; which divided by the third number, viz. 8 gallons, the quotient will be 21 ⅛, that is, 21 quarters and 7 bushels of the lesser measure, equal to 20 quarters of the greater, the thing required. PROPOSITION XXXII. How many yards or els of any one place propounded will be equal, or make a given number in some other, which hath proportion to the measures of a third place, etc. and that in any known quantity unto the first. The RULES. Rule 1. As 20 els or aulness of Lions To 25 yards of London, So will be 60 els of Lions. 100 els Antwerp. Rule 2. 47 els of Antw. A plural proportion. I TWO III IV V Antwerp Antwerp = Lion's: Lion's = London = 47 els: 100 els = 60 els: 20 els = 25 yards An explanation in reduction of measures from plurality of proportion. Lib. 2. Parag. 10. In this Proposition there is an equality or proportion derived from divers descents and collateral lines, and may be continued like a British pedigree: the equality here re-required is betwixt 47 els of Antwerp and the yards of London that shall be equal to them, if their measures were not known (in any certain proportion) but as derived from some other, and that from a third, and so continuing a proportion until you arrive at one that runs directly from the first. As for example: here is required how much 47 els of Antwerp will be of London measure; if the proportion were known that four els of Antwerp were but equal to 3 yards of London measure, there would be no more in it then to multiply the els propounded, viz. 47 by 3, which product 141 divided by 4, the quotient would have been 35 ¼ yards: but suppose this proporrion not known, but by derivation, to be collected from others, as in this plurality of measures you will find that the city of London, according to the English standard for measures, hath proportion to the els of Lions in France, and those again to Antwerp, in the Low countries, from whence the proportion will arrive (according to the first table) as 20 to 25, so 60; then in the second table, as 100 els of Antwerp, to so many yards of London (supposed to be found) what will 47 els of Antwerp require, to have an equality in their measures: in the third row or table they are both reduced into a single rule, and in the fourth table unto their least denominations, viz. as 4 is to 1, so 141 in proportion to 35 ¼ yards, as it was before, the thing required, and I hope explained, from whence I will proceed to the customary rules used in Factorage. PROPOSITION XXXIII In the first place you must consider what the Merchant allows his Factor in lieu of his pains, and the adventure of his person; as whether ½, ⅓, ¼, ⅕, etc. that proportional part taken from an integer, the remainder is the Merchants, the other shows the value of the Factor's person. The RULE. As the proportional part of the Merchant's adventure Shall be to the whole stock adventured in his charge, So will the proportional part allowed to the Factor Be to the estimate of his person in the employment. An explanation of Factorage. Lib. 2. Parag. 8. If a Merchant entrusts his Factor with a sum of money, upon condition he should have half the gains; in this case the Factor's person was valued equal to the adventure: but admit ⅓ part of the gains were to be allowed the Factor, and 1000 L committed to his charge, the Merchant's share will be but ⅔, which is in proportion to 1000 L, as ⅓ is unto 500 L, the estimate of the Factor's person as by the rule and table appears: and if in this employment 2000 L were gained by the adventure (with all charges defrayed) the Factor's share would be 666 L 13 ss 4 D, and the Merchants 1333 L 6 ss 8 D, the one but half the other: if the Factor had been allowed but ¼ of the gains (in this adventure) his person had been valued at 333 L 6 ss 8 D, and his gains would have amounted to 500 L; if ⅕ had been his proportional part, than the Merchants had been; and his gains 1600 L, the Factors 400 L, and the estimate of his person in this employment 250 I, etc. PROPOSITION XXXIV. If a Merchant shall deliver unto his Factor any sum of money, and does agree for to allow him 2/7 parts of his gain, with this proviso, that he employs such a stock of his own as shall be mentioned in the contract between them, what shall the Factor's person be valued at, and how much will his gains amount unto? find by the last Proposition what the proportional parts are unto the Merchant's adventure, and from the Factor's part subtract his stock adventured, the remainder will be the estimate of the Factor's person, and the 2/7 parts of the whole gain will produce his profit. As for example, Suppose a Merchant delivers to his Factors charge 2000 L, conditionally that he employs 300 L of his own in the same adventure, the proportion will be, viz. as 5/7 is to 200 L, so will 2/7 be unto 800 L, from whence subtract 200 L, the remainder is 600 for the estimate of the Factor's person in the employment; and admit the gains at his return were 3675 L 8 ss 9 D, the 2/7 parts of it will be found 1050 L 2 ss 6 D, and the Merchant's share will be 2625 L 6 S 3 D. both Propositions answered. PROPOSITION XXXV. Of Factorage. A Merchant did condition with his Factor, to allow him for the adventure of his person a part of his stock, and according to that proportion of the whole adventure, he should share in the gains, from hence to discover what the Factor's person was valued at, and the proportion of his profit is the thing required. To explain this, suppose a Merchant entrusts his Factor with 1680 L, and with this condition, that if he gained so much money he should have 240 L for his pains, and so proportionally for a less or greater increase: in all questions of this kind reduce the two sums (like fractions) into their least denominations, viz. 240/1680 which will be 1/7, then by the 33 Proposition, as 6/7 is to 1680, so 1/7 to 280 L, the estimate of the Factor's person in this employment; and suppose he gained (all charges defrayed) 1481 L 7 ss 6 D, what must he have for his pains? The answer will be 211 L 12 ss 6 D (lib. 2. parag. 9 quest. 6.) that is ●/7, according to the Articles of Agreement made. PROPOSITION XXXVI. A Merchant conditions with his Factor to allow him, out of his gains, a certain profit in the pound sterling, by which it is required to find what the factorage will amount to in any sum propounded. The RULE. As 1 L sterling, or any other sum of money given Shall be in proportion to what is allowed for factorage, So the gains of the adventure in the first denomination Will be proportional to the gains for the Factor's share. An explanation of Factorage, Lib. 3. sect. 1. cap. 7. table 1. A Merchant had a due, but doubtful debt owing him in another Country, where he was to employ a Factor, who had letters of credence to demand his money due, and with this condition, to have 13 ⅓ D in every pound sterling, that he should procure of it: the Factor by his industry recovered 1200 L of the debt, what does his salary amount unto? by the rule of Three you will find 67 L 10 S, and so like wise in the table, according to the rule of Decimals 〈◊〉 by this you may state other questions of Factorage, and in this form. PROPOSITION XXXVII. A Merchant takes up money to freight his ship, with condition to allow 26 L per cent. and that to be paid where the goods should be landed, with this proviso, that the Creditor shall stand to such hazards as belongs to sea, viz. Shipwreck, or Pirates, etc. The RULE. If the sum of 100 L sterl. or any other money Shall require upon adventure 26 L for interest, Then any greater or lesser sum of the first money Will be in the same proportion to the required gain. An explanation of Sea-hazard, or Bottomree. Lib. 2. Parag. 8. Cambio maritimo, some call this rule, wherein the Creditor stands to the hazard with the Merchant at Sea, that if the ship be lost he loses the money adventured: the operation of this rule is facile, the interest just, and the explanation short: the money here contracted for (according to the conditions of 26 L for interest per cent.) is 640 L, and the fourth proportional found will be 166 ⅖ L, or 166 L 8 S, as in the table of the margin does appear, which with the principle makes 806 L 8 S, to be received where the money was payable, or should be due, the ship being arrived with the adventure at the appointed Port of the Country or Kingdom, as the voyage was intended. THE SECOND PART. Definitions and Etymologies of Usury and Interest Money, with the several operation compendiously displayed. USury is derived particularly from Usura, ab usu aeris, or generally from the use and occupation of any thing, as Money, the worth or estimate of it, upon some mutual contract, wherein the Debtor allows the Creditor a Loan in lieu of the money or goods borrowed; which in times past hath been at liberty as men could agree; but when the unsatiable avariciousness of rich miser's attracted extortion from the indigency of borrowers, the corroding use of money was by provident Laws confined to a certain rate, as to 10 L per cent. per ann. afterwards to 8 L, and now of late to 6 L, that is, if a Creditor lends 100 L for a year, he may legally exact 6 L profit for the use of the money lent. These are divided into 3 parts, viz. Principal, Time, and Interest: the first signifies the sum, or value of the money or goods so lent: Time is the forbearance of it, as Days, Weeks, Months, or Years. Thirdly, Interest is the profit that arises from the other two, and is derived from two compound Latin words, viz. inter and est, ab edo, to eat or devour, as it is the property of Use to do; they have these proportions, as the Principal to the Time, the Rate contracted for, and Interest to itself, and generally as commixed with one another. Interest resembles Janus with two faces, one looking upon the time past, the other on that to come: i● this tract the Principal runs, like a snowball rising upon an even superficies, equally moving, but the increase unequal, although proportional to the body, as it is magnified in the motion; and if continued, in time it will gather up all: this is called Forbearance. And Rebate or Discount of money is like the tract in which a snowball moved, and in its descent takes up all until it is stayed, leaving the ground bore from whence it takes a seeming original, where Time hath not arrived, but beholds it, as seamen an object, which seems little at a great distance, and increases to the Optic sense by unsensible approaches. Use or Interest hath in either Predicament two Species, viz. Simple or Compounded, the first is computed from the Principal and Time only, upon a certain rate given or allowed, whether ascending or descending, as in Forbearance, or upon discount, which are thus explained; if 100 L be continued for 2 years, at 6 L per cent. per ann. the creditor at that term of time is to receive but 112 l, that is 12 L for simple interest in loan of the money forborn; here 100 L is the principal lent, the term 2 years, the use 6 L per ann. whereas in compound interest, the first payment attracts a proportional use: as admit in annual disbursements, in the second year there is an use required, or imposed on the 6 L due, if continued, and therefore it is called interest upon interest. Discount or rebate of money is upon a legacy, or sum due to be paid at a time to come, yet satisfied or discharged with so much present money, as immediately put forth at an interest or rate allowed forborn until the legacy should have been due, returns again to its first principal or sum, the money paid being computed at the same rate of interest for the forbearance, as was the discount made, as by following examples shall be illustrated, observing these proportions, viz. The RULE. As the principal and time for which a loan is allowed, Shall be in proportion unto the interest thereof, So will any other sum of money to be borrowed Be proportional to the interest for the same time. PROPOSITION XXXVIII. What comes the interest of 145 L unto, forborn a year at 6 L per cent, per ann? An explanation of simple interest. Lib. 2. Parag. 8. Any question of this nature is with facility performed by the common rule of Three, as if 100 L principal forborn a year requires 6 L interest, the 145 L for the same term of time will exact 8 7/10 L, as in the table, that is, 8 L 14 ss. the question answered. PROPOSITION XXXIX. A Money-Merchant employed at Use 250 L sterling for 5 months, at the rate of 8 L per cent. per ann. and the simple interest of it is here required. An explanation of simple interest, lib. 2. par. 10. All questions of this Prin. In. Prin. 1 Rule 100-8-250 M— M 2 Rule 12-5 Product 1200. 8. 1500 Or as 3 to 1 so 25 Facit 8 L.. 6 ss 8 D kind do consist of 5 terms, viz. 100 L Principal, secondly, the rate for Use, as in this, 8 L. thirdly, the Principal lent, as in the first rule; in the second rule under 100 l is placed the time for which 8 L was due, as at a year, or 12 months: the fifth number is the term for the principal borrowed, viz. 5 months, the products of those are 1200. 8 and 1250, and may be reduced (as in the 13 and 14 Axiom, lib. 1. parag. 7.) to 3.1. 25. the quotient here answers the question, viz. 8 L 6 ss 8 D, the proportional interest, for the sum and time required. PROPOSITION XL. A Banker did lend 650 L. which principal was repaid at the term of 6 months 3 weeks and 3 days; what came the interest of it to at 6 L per cent. per ann. An explanation of simple Interest. lib. 2. parag. 10. In this double Princi. Inter. Princ. 1 Rule 100 ... 6 ... 650 D— D 2 Rule 365 192 Prod. 36500. 6. 124800 Or as- 365 to 6. so 1248 facit 20 l 10 ss 3 d. 2 34/73 Q rule the first term is 100 l. the second its interest for a year, the third is the principal lent: the first term of the second rule 365, the number of days in a vulgar year; the last term is 192, the number of days terminating the time, accounting 4 weeks to the month; the products of these are 36500, and 124800, and reduced will be 365 and 1248. which multiplied by 6 l the interest is 7488, and divided by 365 the quotient will be (in a direct proportion) 20 l 10 ss 3 d 2 34/73, as in the table, the simple interest of 650 l. the time required. PROPOSITION XLI. A man received 8 l 6 s 8 d for 5 month's interest of a sum unknown, but at the rate of 8 l per cent. per ann. and the principal is here the thing demanded. An explanation of this Proposition, lib. 2. parag. 10. quest. 4. In the first rule, as 8 l interest is to 100 l principle, what 8 ⅓ the interest received: in the second rule stands the terms of time, viz. 12 more. and 5 m. which multiplyers and the 2 dividers increased reciprocally by one another, they will produce these 3 numbers, viz. 40. 100 100 all questions of this nature are made either direct or reverse, accordingly as the products are placed; but by either way the quotient will be discovered, 250 l the principal lent, as in the table; the proposition solved according to demand. PROPOSITION XLII. What shall the simple interest of a mixed principal, as 265 l 13 ss 4 d 1 q. amount unto at 6 l per cent. per ann. An explanation of interest money by the rules of practice: Lib 2. Parag. 9 Que●t. 4, 5, & 6. The increase 100 L. 6 In. 265 L. 13 ss. 4 ¼ D. Products L 15 94. 00. 1 ½ 20 100 L divid. 6 L multipl. S 18 80 12 D 9 61/4 Q 2 46 26/100 or 23/50 of any mixed sum will be thus discovered; first place your proposition according to the demand made, as here viz. if 100 L requires 6 L interest, what shall the principal given, as 265 L 13 ss 4 ¼ D in this, 6 is the multiplier, with which increase the principal, as in lib. 1. sect. 1. parag. 4. exam. 9 and the product will prove 1594 L 0 ss 1 ½ D. here draw a line as in the table, cutting off two places on the right hand, a● 94, because 100 is the divisor, the quotient 15 L secondly, the remainder 94 or 94/100 reduce into. shillings, by the multiplication of 20 the product will be 1880, and being no shilling left in the last operation, sever 2 places with the line, the quotient is 18 ss, the remainder 80, which by 12 reduce into pence, and add to it 1 D remaining in the first product, cut off 2 figures on the right hand, viz. 61, so on the left hand there will be 9 D. the 61 reduced into farthings by 4, produceth 244, to which add the ½ D remaining yet in the first product, the sum is 246, that is 2 Q. and 46/100. so the total interest for a year of 265 L 13 ss 4 ¼ D at 6 L per cent. comes unto 15 L 18 ss 9 D 2 2●/50 Q, as by the operation in the table is conspicuous. PROPOSITION XLIII. To find the Use-money of any sum, whose principal and interest are of several denominations; as 956 L 7 ss 6 D, at 5 L 17 ss 6 D per cent. per ann. An Explanation. To solve any question Prin. Inter. L ss D 100 5. 17. 6. 956 7 6 5 The products 4781 17 6 478 3 9 239 1 10 ½ 119 10 11 ¼ The interest money L 56 18 14 0 ¾ ss 3 74 17 6 D 8 88 Q 3 55/●00 of this kind, state the proposition as in the head of the table, then take the greatest denomination of the interest allowed, as in this example, 5 L. which multiply through all the denominations of the principal here stated, according to sect. 1. parag. 4. examp. 9 the product will prove 4781 L 17 ss 6 D. had the interest been 6 L, the product would have contained the principal once more; but being 17 ss 6 d, divide it into proportional parts, lib. 2. parag. 9 quest. 4. as in the table at A, viz. 10 ss. 5 ss. and 2 ss 6 d. for the 10 ss take ½ the principal, that is half of 956 l 7 ss 6 d, which will be 478 l 3 ss 9 d. next for 5 s take ¼ part of the principal, or ½ the last, which is 239 l 1 s 10 ½ d, and the half of that again is 119 l 10 s 11 ¼ d. the total of all these is 5618 l 14 s 0 ¾ d. this done, divide the greatest denomination by 100 l, or cut off two places on the right hand, and you will find 56 l and 81 remaining, with which proceed as before, and you will discover (as in the table) the interest to prove 56 l 3 s 8 d 3 5●/100 q. the simple use for a year, as was required: and thus may all other propositions be expeditely solved by this operation, and the rules of practice. PROPOSITION XLIV. What amounts the interest money unto upon a mixed principal, for a time less or greater than a year, as 645 l 6 s 8 d, lent for 11 months, at 5 l per cent. per ann. An Explanation. State the question Prin. Inter. L S D 100 L 5 645 6 8 Product 3226 13 4 Products 1613 6 8 806 13 4 537 15 6 ⅔ The interest L 29 57 15 6 ⅔ 20 S 11 55 Months 11 D 6 66 6 Q 2 66 A 3 2 as in the head of the table; that multiplied by the interest allowed in this 5 l, the product is 3226 l 13 s 4 d. now being the money is to be continued but 11 months, take proportional parts, as in the table at A, 6 M. 3 M. and 2 M. for the 6 M. take half the first product, than ¼ and ⅙. or thus, ½ of 3226 l 13 s 4 d. is 1613 l 6 s 8 d for 3 Months, the ½ of that is 806 l 13 s 4 d. lastly. 2 month's ⅓ part of the former, which is 537 l 15 s 6 ⅔ d. this done, proceed as in the last, or by 42 proposition, and you will discover the interest money to be 29 l 11 s 6 d 2 66/100 q. the demand performed, and if it had been required for any longer forbearance, find the interest for the term of years by the former propositions, and the parts of a year by this. PROPOSITION XLV. What shall the Use-money come unto of any sum, whose Principal, Interest, and Time, are all compounded numbers, viz. 543 L 13 ss 4 D, to be continued for 9 months, at 5 L 12 ss 6 D per cent. per ann. An Explanation. The proposition Princip. Interest L S D 100 5. 12. 6 543 13 4 Products of interests per cent. per ann. 2718 6 8 271 16 8 67 19 2 The total is 3058 2 6 Interest in respect of time 1529 1 3 764 10 7 ½ (A) L 22 93 11 10 ½ 5 l 12 s 6 d 20 5 L S 18 71 months 9 ½ 10 ss D 8 62 B 6 ½ ¼ 2 ss 6 D Q 2 ½ B 3 ½ being stated (as in the head of the table) multiply the principal by the greatest denomination of interest, viz. 5 L, as in the table at A, according to the prescribed rules of practice, lib. 2. paragr. 9 quest. 4. for the 10 ss take half the principal, and for the 2 ss 6 D ⅛ or ¼ part of the last, so you will produce these 3 numbers 2718 L 6 ss 8 D. secondly, 271 L 16 ss 8 D. and thirdly, 67 L 19 ss 2 D. the total is 3058 L 2 ss 6 D to be divided by 100 L, if forborn a year; but the term of time here is but 9 months, as in the table at B, which I divide into two parts, viz. 6 M. and 3 M. then take half the total, that is, 1529 L 1 ss 3 D, and for 3 months the half of that 764 L 10 ss 7 ½ D. the total is 2293 L 11 S 10 ½ D, to be divided by 100 L. now proceed according to the former examples, and you will find the interest demanded 22 L 18 S 8 D 2 ½ Q. the proposition solved. If this question had depended on months, weeks, and days, you must have taken proportional parts, and proceed, as before is specified; so there needs no more examples for the ingenious, to whom all other questions (in the rules of practice) will be direct, and indirect for others, to encumber their understandings with multiplicity of ways in this kind, therefore I will only show you the discount of money, and proceed to decimal Tables of compounded Interest. PROPOSITION XLVI. Upon any interest per cent. and the sum of money that shall be due at the term of a year, to find the worth of it in ready money, or present pay, the interest deducted from the sum. The RULE. If 102 L and the annual interest due at a years end Be worth a 100 L at one entire and present payment, What shall any other sum due at that term Be worth in ready money, the interest deducted? An explanation of this rule in discount, or rebate of money, lib 2. parag. 8. Admit the interest were 6 l per cent. per ann. which added to 100 l makes the sum 106 l to be rebated: for if this principal were due at the term of 12 months, it were worth 100 l present pay, because at 6 l per cent. it would increase in a years space unto 106 l again: from hence this rule of antepayment is framed, and all questions of this condition (if terminated by a year) are solved, as in this example; suppose 546 l shall be due at the term of a year, and desired presently, rebating at 6 l per cent. per ann. the proportion is evident, viz. as 106 is unto 100, so shall 546 be to 515 09●/1000 l, which decimal fraction is 1 s 10 ½ d. so 546 l, which should have been due at a years end, is worth upon discount present pay 515 l 1 s 10 ½ d, as in the table is demonstrated by a decimal, and in a natural fraction, it will be 515 5/53 l. PROPOSITION XLVII. A Legacy, or any sum of money not presently due, and to be discharged at two several payments, as the first at 6 month's end. the other at the term of a year, how much money will discharge it at one payment, discounting 10 l interest per cent. per ann. The RULE. As 12 months, or the true term of a year Is in proportion to the annual interest allowed, So the months, or term for day of payment Shall be proportional to the interest for the time. An explanation of discount or rebate of money, for days, weeks, months, etc. In all questions upon rebate of money at simple interest for the space of a year, observe the last proposition: but if for a greater or shorter term of time, viz. months, weeks, days, etc. or days, weeks and months; in all such cases you ought to find a proportional interest for those annual parts, because the money due at several payments is terminated therein, as the other in a year. As for example: there is a Legacy of 1000 l bequeathed to T. W. at equal payments, viz. 500 l to be paid at the end of 6 months, and the other 500 l at the period of a year: what is it worth present pay, rebating at 10 L interest per cent. per ann. as for the last payment, form the rule of Three according to the last proposition, viz. as 110 L to 100 L, so 500 L will be in proportion to 454 L 6/11 L. now for that due at 6 month's end, find a proportional intere t for the time (according to the rate allowed) as by the first of these tables in the margin, which is 5 L. and by the second table's operation you will find for the fourth proportional number 476 4/21 L, the value of the first 500 L present pay: the sum of both payments 930 170/231 L, 930 L 14 sh 8 48/77 D. the Legacy as valued present pay upon rebate at 10 L per cent. per ann. simple interest, which in the first table required no rule to discover it, but many other propositions may, therefore it was inserted by me, and made plain as it is general; if the respite of time had been for more or fewer months, with weeks and days, the manner of operation is the same in effect, therefore to write ane more of this would prove superfluous unto the ingenious Arithmetition; so of simple interest I will here conclude. Many rules do seem originally derived from Truth, or extracted from Art, but if well observed, they are easily detected, being erroneously reered upon fallacious grounds, and their derivations from imaginary principles, viz. as in interest, discount of money, Equation in payments relating to Time and Interest, etc. These I have inserted to please some, but not to delude any, the ways being common in general received customs, plain, easy, and of good use, the errors not being great or considerable in small sums, or in a short forbearance, as in Bonds, they rarely pleading prescription of years, or exceeding an annual revolution, upon forfeiture in transgressing a penal Law; whereas rebate or discount of Money often depends upon long terms, involved with multiplicity of antedated years, as in purchasing of Leases, Pensions, Annuities, Reversions, etc. therefore these were composed so compendious as I could, since there be better rules extracted from Artificial numbers, and Decimal Arithmetic in compound Interest, as those following. A Decimal Table of Interest money forborn any number of days, weeks, months, or years, to 25 inclusive, accurately calculated at 6 l per cent. per ann. Day's decimal numbers Years decimal numbers I 1.000160 I 1.060000 TWO 1.000319 TWO 1.123600 III 1.000479 III 1.191016 IV 1.000639 IV 1.261477 V 1.000798 V 1.338226 VI 1.000958 VI 1.418519 VII 1.503630 Weeks VIII 1.593848 IX 1.689479 I 1.001118 X 1.790848 TWO 1.002237 XI 1.898298 III 1.003358 XII 2.012196 Months XIII 2.132928 XIV 2.260904 I 1.004867 XV 2.396558 TWO 1.009760 XVI 2.540352 III 1.014674 XVII 2.692773 IV 1.019613 XVIII 2.854339 V 1.024576 XIX 3.025599 VI 1.029563 XX 3.207136 VII 1.034574 XXI 3.399564 VIII 1.039610 XXII 3.603537 IX 1.044671 XXIII 3.819750 X 1.049756 XXIV 4.048936 XI 1.054865 XXV 4.291871 The second Table of compound Interest money discounted for days, weeks, months, or years, unto 25 inclusive, exactly calculated at 6 l per cent. per ann. Day's decimal numbers Years decimal numbers I .999840 I .943396 TWO .999681 TWO .889996 III .999521 III .839619 IV .999361 IV .792093 V .999202 V .747258 VI .999042 VI .704960 VII .665057 Weeks VIII .627412 IX .591898 I .998883 X .558394 TWO .997767 XI .526787 III .996653 XII .496969 Months XIII .468839 XIV .442301 I .995156 XV .417265 TWO .990335 XVI .393646 III .985538 XVII .371364 IV .980764 XVIII .350343 V .976013 XIX .330513 VI .971286 XX .311804 VII .966581 XXI .294155 VIII .961859 XXII .277505 IX .957239 XXIII .261797 X .952643 XXIV .246978 XI .947988 XXV .232998 Table 3. Forbearance of annuities, rents, or pensions from 1 year to 25, at 6 l per cent. per ann. Table 4. Discount of annuities, rents, or pensions, from 1 year unto 25, a● 6 l per cent. per ann. Table 5. Purchase of annuities, rents, or pensions from 1 year unto 25, at 6 l per cent. per ann. Years Deci. N. Years Deci. N. Years Deci. N. I 1.00000 I .94340 I 1.06000 TWO 2.06000 TWO 1.83339 TWO .54544 III 3.18360 III 2.67301 III .37411 IV 4.37462 IV 3.46510 IV .28859 V 5.63709 V 4.21236 V .23740 VI 6.97532 VI 4.91732 VI .20336 VII 8.39384 VII 5.58238 VII .17914 VIII 9.89747 VIII 6.20979 VIII .16104 IX 11. 4913● IX 6.80169 IX . 1470● X 13.18079 X 7.36009 X .13587 XI 14.97164 XI 7.88687 XI .12679 XII 16.86994 XII 8.38384 XII .11928 XIII 18.88214 XIII 8.85268 XIII .11296 XIV 21.01506 XIV 9.29498 XIV .10758 XV 23.27597 XV 9.71225 XV .10296 XVI 25.67253 XVI 10.10589 XVI .09895 XVII 28. 21●88 XVII 10. 477●6 XVII .09544 XVIII 30.90565 XVIII 10.82760 XVIII .09236 XIX 33.75999 XIX 11.15811 XIX .08962 XX 36.78559 XX 11.46992 XX .08718 XXI 39.99272 XXI 11.76407 XXI .08500 XXII 43.39229 XXII 12.04158 XXII .08305 XXIII 46.99583 XXIII 12.30338 XXIII .08128 XXIV 50.81557 XXIV 12.55036 XXIV .07968 XXV 54.86451 XXV 12.78335 XXV .07823 The construction of Decimal Tables for Use money, made proportional for any interest, time, or term of years required, with the applications of these illustrated with useful and compendious examples. All questions of compound interest money may be comprehended and solved by one of the 5 precedent Tables, calculated (without sensible error) and extracted from the former prescribed rules, as in the 38 and following Propositions of simple interest: And first, here observe that these Tables are composed upon the worth or interest of 1 L principal per ann. after the rate of 6 L per cent. yearly payments, as the Law now commands, with a prohibition of any greater interest, upon the penalty of a forfeiture, excepting such cases wherein there depends some apparent hazard, as in Exchanges, Cambio Maritimo, or in Bonds and Obligations not satisfied, that antedate the Act, etc. and upon this foundation are erected these tables, whose calculations I will first exhibit to your view, both for farther satisfaction, & the composing of any others upon other rates or proportions; and then proceed unto the use and explication of these. The proportions of compound Interest Money, in decimal numbers, due upon the loan of 1 L Principal. RULE 1. As 100 L sterling lent for term of a year To the sum of the Principal and interest allowed, So shall 1 or an unite of the first denomination Be unto the principal and interest in a Decimal. RULE 2. As 100 L sterling forborn the term of a year Shall be in proportion to the last years Decimal, So will the sum of the principal and interest Be unto a Decimal for the succeeding year. An illustration. Prin. Decimal Principal Decim. I 100 l is to 106 so will 1 l be to 1.060000 TWO 100 l is to 1060000 so 106 l is to 1.123600 III as 100 l is to 1123600 so 106 l unto 1.191016 IV 100 l is to 1191016 so 106 l unto 1.262477 V 100 l is to 1262477 so 106 l unto 1.338226 In the construction of these tables, take an unite with what number of cyphers you please for the Radius, one place more than you intent to inscribe, but of no necessity, as here 1,000,000, the seventh place being an integer, all the other towards the right hand are fractions, or representing their places according to my third book of Artificial Arithmetic: now for framing the first table (wherein the principal is forborn) add the principal and interest together; the sum is here 106, annex 4 cyphers or points to it, equal to the Radius, and it is a decimal fraction for the 1 L forborn a year; or by the first rule, as 100 L is to 106, the principal and interest allowed, so shall 1 L principal be in proportion to 1.060000, which is a mixed decimal fraction, and comprehends the 1 L principal, with the interest 06/100, and reduced is 1 ss 2 d 1 ⅗ q. but leaving it involved, as the increase of the first year, all the rest will be discovered by the second rule successively, viz. as 100 L to 1.060000, so 106 the principal and interest to 1.123600, the decimal for the second year; then for the third year, as 100 L is to 1123600, so 106 in proportion to 1.191016 in the same manner of operation, you will find 1,262477 for the fourth year, and 1,338226, a proportional decimal fraction comprehending 1 l principal forborn 5 years, with the compound interest of it, which is 6 ss 9 d, and not one farthing more: thus you may proceed to what number of years' occasion shall require. To find Decimal Numbers for any parts of a year, as months, weeks, days, or for half years and quarterly payments. Take the decimal for a years interest, viz. 1.060000, whose Quadrat Root extracted as in lib. 2. parag. 1. examp. 5. you will find 102956, a proportional decimal, for the interest of 1 l forborn 6 months, in money 7 0●44/10000 d. the mean proportional betwixt 1.02956 and 1.06000 will be the decimal for 9 months, viz. 1.04467, and between an unite or 1, and 102956 will be .1.01467, a decimal mixed number (according to compound interest for 3 months) equal in value to 1 L 0 ss 3 ½ d. and thus you may with facility discover all the other numbers: if the annual table had been for half yearly or quarterly payments, you must find 1 or 3 mediums between every continued and succeeding year; which to effect, I refer the Reader to lib. 2. parag. 6. prop. 1. and 4. and observe to annex cyphers to the Decimal (whose Root is required) in such a number as to equal the places in the table, and point them from the left hand to the right, so that the first prick of your pen may be over the radius or integer. In other things observe lib. 2. parag. 1. exam. 5. The framing of the second Table for discount or rebate of money. RULE 1. As 100 L with the interest due at a years end Is unto the principal 100 L present payment, So an unite or 1 L endue at the same term of time Shall be to a Decimal fraction interest deducted. RULE 2. As 106 L upon discount for the term of a year Shall be in proportion to the decimal last found, So will 100 L Sterling, or its value, present payment Be proportional to a Decimal for the time required. An illustration. Prin. Decimal Principal Decim. I 106 l is to 100 L so will 1 l be to .9433962 TWO as 106 l is to .9433962 so 100 l is to .8899964 III 106 l is to .8899964 so 100 l unto .8396192 The second Table for discount or rebate of money at 6 L per cent. per ann. is thus composed for the first year and Rule 1. as 106 L due at a years end shall be worth 100 L present pay, so will 1 L principal (according to the rule of Three in Decimals) be in proportion unto 9.433962; which fraction is in money 18 ss 10 d 1 66/100 q. the value of 1 L endue at a years end, and presently paid upon discount: but if not due until 2 years shall be expired: say as 10 6 L in proportion to the first years decimal fraction, viz. 9.433962, what shall 100 L produce? 8.899964. then for the third year, as 106 L to the last decimal found, viz. 8.899964, so will 100 L present pay be in proportion unto 8396192. which fraction of 1 L reduced is 16 sh 9 ½ d. and thus you may continue it to what number of years you please, and inscribe what places of decimals you think fit, but make them all to one Radius, and one place less upon discount, being but fractions or parts of 1 L principal, viz. 943396/1000000 as in the first year. How to find Decimal numbers for parts of a year upon discount, or for half years and quarterly payments. These are composed after the same manner as the table of money forborn, excepting only in pointing the numbers for the Roots extraction, the first Decimals being all mixed numbers, and those of discount are every one proper fractions, having only a point prefixed for the Radius or integer; therefore in these make the first point under the second figure on the left hand. As for example, 943396 is the decimal for the years rebate of 1 L, put the first point under the figure of 4, and so in order to the right hand, the root thus extracted will be .971286 in 6 places for the discount of 1 L, 6 months as 6 L percent. the square root of that again will be .985538 for 3 months; and thus proceed with mean proportionals until the places are all complete between the radius and the decimal last found; as for half yearly and quarterly payments, they are discovered as were those before in the forbearance of money, to which I refer you, and Lib. 2. Paragr. 6. Proposition 1. and 4. observe the 2 Tables, for out of these grounds the other 3 are framed and erected as followeth. The invention of decimal Fractions, or proportional numbers for the third Table. Here are two tables Forbearance of money at 6 l per cen. (1) Forbearance of Rents at 6 l per cen. (2) 1 1. 060000 1 1.000000 2 1. 123600 2 2.060000 3 1. 191016 3 3.183600 4 1.262477 4 4.374616 5 1.338226 5 5.637093 6 1.418519 6 6.975318 7 1.503630 7 8.393837 inscribed for 7 years, whereof the one is the transcription of the first breviate, out of which the third table is composed, & thus: an Annuity, Rent, or Pension of 1 L per ann. is but so much money due at the term of a year, therefore on the head of the table I place the Radius, against the interest and principal of 1 L forborn a year, viz. in the first table 1.060000, in the second 1.000000. the sum of these two numbers is 2.060000, the rent which will be due at the two years' end: in which time there will be 2 L in arrears, and the annual interest of 1 L, to which add the second years forbearance, viz. 1.123600, the sum will be 3.183600 for the rends 3 years forborn, and thus in order, the 6 years added together will make the seventh as 8.393837, and the seven years the 8, viz. 9.897467, which reduced is in money 9 L 17 ss 11 d 1 ½ q. and so much 1 L yearly rent or annuity forborn 8 years does amount unto at 6 L per cont. per ann. annual payments and compound interest: in this manner you may proceed, according to what number of years the first table comprehends. The construction of Decimal Fractions, or proportional numbers for the fourth table. The first of these 2 Discount of money at 6 l per cen. (1) Discount of rents at 6 l ●er cent. (2) 1 .943396 1 .943396 2 .889996 2 1833392 3 .839619 3 2673012 4 .792093 4 3465105 5 .747258 5 4212363 6 .704960 6 4917323 7 .665057 7 5582380 tables is transcribed out of the second breviate, from whence the the 4th is framed after the manner of the last, for 1 L Pension, Rent or Annuity due at a years end is worth but so much upon discount as the interest rebated, which at 6 L per cent. is included by this Decimal 943396. and the second years number must be increased by the Annuity, Rent, or Pension discounted for: therefore add 889996 unto 943396, the sum will be 1833392 the decimal fraction for the second year, and so proceed to the seventh year of the first table, by adding that number, 665057 unto 4917323, the sixth years discount in this second table, the sum will be 5582380. the Decimal for 7 years rebate of rent at 6 L per cent, and in this manner continue on the tables to what number of years you please. Here 1 L Annuity discounted for 7 years, is worth in ready money 5 L 11 ss 7 d 3 q. compound interest rebated. How to find the Decimal Fractions, or proportional Numbers for the fifth table. RULE 1. As the Decimal for 2 years rend rebated Is equal in value to 1 L annuity for 2 years, So is 1 L of annual annuity the same term of time In proportion to the Decimal purchased by 1 L: Or, As 1.833392- to 1 L- so 1.000000 unto .54544. Or thus, As 1 L 16 Purchase sh 8 d- is to Annuity 1 L, so will 1 Purchase L be to 10 sh Annuity 10 9/10 d. An Illustration. The Annuity, Rent, or Pension, which 1 l will purchase for a year, lies involved in the decimal of 1.060000, according to the first table, there being only one years' forbearance of 1 L, then for the second year take the decimal fraction of 1 L rend discounted for the term of 2 years, which is 1.833392, in money 1 L 16 sh 8 d. and it is evident how this sum is equal unto 1 L annuity purchased for 2 years, and consequently the proportion will be as in the rule before: if 1 L 16 sh 8 d, or 1.833392 (the decimal for two years rend rebated) be equal to 1 L annuity to continue 2 years; what annual rent or pension will 1 L purchase for the same term of 2 years? which fourth proportional number discovered, will prove .54544, as in the second year of the fifth table, and reduced, is in money 10 sh 10 9/10 d. the annuity or annual pension purchased by 1 L for 2 years, which at 6 L per cent. in the term of 2 years' returns to its first principal, the interest considered: now for the third years decimal, as 2.673012 is unto an unite with cyphers, so will 1 L for a purchase be in proportion to 37411, in money 7 sh 5 d 3 q. an annuity to continue 3 years; and thus proceed to what number of years you please, since by the third, fourth and fifth etc. are framed out of the 3, 4, 5 years, etc. respectively answering the years of discount or rebate of rent, as in the fourth table; the fifth being thus finished, if the half years and quarterly payments be required, they may be extracted by finding mean proportional numbers, as hath been declared in the calculation of the former table of Decimals; and as for parts of a pound sterling, I refer you to lib. 3. sect. 1. cap. 7. but here note that the precedent tables be continued to one or two places more, otherwise errors will creep in at the root or end of these numbers by annexing of cyphers; for which cause (as it is the common custom) these 3 last tables were framed on a lesser Radius (as by one place or degree of figures and cyphers) then are the two first. The description, use and explantion of these Decimal Tables, accommodated to the compound interest allowed, at 6 L per cent. per ann. calculated without sensible error in the forbearance or discount of Money, Annuities, Rents, Pensions, and Reversions, with the purchase of them due upon yearly payments. The first Table is divided into 4 columns, and gins with one comprehending the parts of a year, ascending by days, weeks and months; the third row contains the years from 1 to 25 inclusive, both noted with numeral letters; upon the right hand of these are placed (in Arithmetical characters) the Decimal numbers made proportional for 1 L forborn, respectively answering the times included, calculated u●on the Radius of a million 1000000. the second table is for discount of 1 L principal, after the rate of 6 L per cent. per ann. made by the former Radius, viz. 1000000, for the same parts and term of time as was the last. This discount or rebate of money is by some termed Interest damageable, by reason it is ever less than the Principal, although upon a day's discount, or any shorter time, as you may see in the head of the table, with a title to each column; the first hath the Radius prefixed to the Decimals, the second of Discount have only points to denote their places, as Primes, they being all proper fractions, and parts of 1 L. the third, fourth and fifth Tables have each 2 columns only, the first numbering the years from 1 to 25 in numeral letters, and against those annual computations are placed the Decimal numbers depending on those years, as by their titles do appear: the third and fourth increases, the fifth declines the Radius; which prolix numbers in these the three last Tables, extends to 100000, and so much for the model and form of them. The first Tables use illustrated. QUESTION I. If 1000 L be forborn 1 day, what shall be the interest of it, after the rate of 6 L per centum per annum? Look in the first Table for 1 day; against which (under the title of Decimal numbers I find 1000160, which multiplied by the principal 1000 l, or annex 3 cyphers to it, the product is 1000, 160000, from whence sever the Radius by cutting off 6 places from the right hand, you will find 1000 L for integers struck off, and the remaining fraction .160000, which reduce, as in lib. 1. sect. 2. parag. 1. paradig. 10. or multiply it by 20, and from the product cut off 6 places, and so proceed; you will find the loan of 1000 L to be 3 ss 2 d 1 ⅗ q. QUESTION II. What will the interest of 300 L amount unto, if forborn 3 weeks, after the rate of 6 L per centum per annum? The decimal for 3 weeks is 1.003358. which multiplied by the principal 300 L produceth 301, 007, 400. from whence cut off 6 places and reduce them, you will find 301 L 0 ss 1 D 3 q. the increase of 300 L for the time required, viz. 1 L 0 ss 1 d 3 q. QUEST. III. What will the interest of 200 L rise unto, if forborn 6 months, after the rate of 6 L per centum per annum? The Decimal for 6 month's forbearance of 1 L is 1.029563, so the rule is in this and all the rest; as 1 L, or the decimal 1,000,000 is in proportion unto 200 L, the principal propounded; so the decimal of 1 L forborn the same term of time shall be proportional to 205.912.600. from which compound fraction sever 6 places numbered from the right hand, the integers are 205 L, the fraction 912600 reduced, will be in money 18 ss 3 d, so here the interest of 200 L for 6 months proves but 5 L 18 sh 3 D, whereas, according to custom, you may discover amongst the vulgar errors the loan of this principal forbo●n half a year comes to 6 L, that is, 1 sh 9 D too much. QUESTION IU. How much comes the interest of 150 L unto, if forborn 7 years, at the rate of 6 L per centum per annum? The Decimal against 7 years is 1503630. which multiplied by 150 L produceth 225, 544, 500, and reduced, is in money 225 L 10 sh 10 68/100 D, so the increase of 150 L, all interest forborn 7 years, swel● to the sum in clear profit 75 L 10 sh 10 68/100 D, which by the common current of simple interest does multiply in the seven years' apprenticeship (when the Principal shall be discharged the Indenture) but 63 L, which is less (I conceive) then the intention of the English Laws allow by 12 L 10 sh 10 d. for if any loan upon a principal can be legally exacted in equity, use upon the interest (so often as due) may be as justly claimed by the same prerogative, according to Humane institutions, not warrantable by the Divine Law. QUESTION V If 210 L be forborn the term of 3 years, 3 months, 3 weeks, and 3 days, what will be the increase at 6 L per centum per annum? The Products I Decimals 1191016 2382032 TWO 25011336 Year: III 25378356 Months IV 25463576 Weeks V 25475773 Days The total — 254 L 15 ss 1 ¼ d. In all questions of this kind, seek the Decimal for the longest term of time allowed, as here 3 years, whose artificial number is 1191016, which multiplied by 210 L (the principal lent) or by 21, lib. 1. sect. 1. parag, 4. exam. 7. as in this table and first row: in the second stands the product, viz. 25011336. to which you may annex the cipher in 210 L. it is not material, the number being one place greater than is the Radius, & yet the product one cipher defective; therefore strike off but 5 places from the right hand, and the fraction reduced, the sum would prove at 3 years' end 250 L 2 ss 3 d. But to proceed, the second row for the term of years multiplied by 10.14674 (the Decimal for 3 months) produceth in the third row of the table 25378356, the number for 3 years and 3 months, as noted on the right hand of the Table; which multiplied by the Decimal 1003358 for 3 weeks, the product will be in the fourth row 25463576, the artificial number for 3 years, 3 months, and 3 week; and lastly, multiplied by 1000479 (the decimal for 3 days) the fifth row will specify in the product 25475773, the artificial number for the whole time, viz. 3 years, 3 months, 3 weeks, and 3 days; from whence sever the integers, and reduce the fraction, the total appears (as in the table) 254 L 15 ss 1 ¾ d. the true compound interest for the sum and time required. The second Table of compound Interest illustrated by Examples. QUESTION VI At the term of 6 months A is to pay unto B 500 l, but do agree in receiving it presently upon discount, after the rate of 6 L per cent. per ann. what sum of money will discharge it? In the second table (for discount of money) I find the decimal for 6 months .971286, which fraction of 1 L Sterling multiplied by 500 L, or 5 the product will be 4856430, to which annex 2 cyphers, the number will be 485,643,000; from the right hand cut off 6 places, and reduce the fraction, there will appear 485 L 12 sh 10 ¼ d, the true sum upon rebate, that will discharge 500 L 6 months before 'tis due, which according to the best vulgar custom comes near the truth, as by Pro. 47 of this book (the discount being but for a short time) viz. 485 L 8 sh 8 d 3 q. QUESTION VII. A had a Lease in reversion, which at the expiration of 7 years was valued worth 1200 L. which Lease B would purchase present pay, rebating at 6 L per cent. per ann. what will be the value of i●? This differs not essentially from the last, for it is no more but to find the present value of 1200 L not due until 7 annual revolutions be completed. Look in the second Table for discount of money, and in the column against 7 years you will discover 665057, which Decimal multiplied by 1200 L, produceth 798,068,400, from the right hand sever 6 places and reduce the fraction, the sum will appear in money 798 L 1 sh 4 ½ d very near; and so much money present pay B must disburse to A for his Lease in reversion, commencing at 7 years' expiration, the thing required. QUESTION VIII. A is to pay unto B a Legacy of 1800 L, which is to be discharged at 3 several and equal payments, viz. at the end of 6 months 600 L, at the term of a year 600 more, and the last payment 6 months after that: B desire, it presently, and A is willing upon discount at 6 L per cent. per ann. what sum will discharge it at one present and entire payment? The sum hear The decimals L S d 1 582771600 1 582. 15. 5 2 566037600 2 566. 0. 9 3 549784200 3 549. 15. 8 1698. 593400 1698. 11. 10 propounded is 1800 L at 3 equal payments: the Decimal for discount of 6 months is 971286. which multiplied by 600 L (the first payment to be due at the half year's end) the product is 582771600, which reduced does prove 582 l 15 s 5 d, then is there 600 l upon a years rebate: the decimal for that term of time is 943396. which multiplied by 600 L will produce 566,037,600. which reduced into money is 566 L oh should 9 d due upon the years rebate, as in the second row of the table: now the last payment is 600 L upon a year and a halfs discount, to find an artificial number for this; the Decimal for a years discount is 943396, and for 2 years 889996. the product of these will be 83.96.18.66.64.16. the Quadrat extracted as it is pointed will be 916307, a mean proportional number betwixt the first and second year, according unto the construction of these tables before delivered, and if multiplied by 600 L the last payment (due at that time) the product will be 549784200, as in the third row of the table, and is in money 549 L 15 ss 8 d. the total 1698 L 11 ss 10 d. which sum will discharge all the 3 payments at one time, and present upon discount; and the 3 several Decimals (whose total is 1698593400, and reduced, will prove the same total sum: the money deducted is 101 L 8 ss 2 d. The third Tables use of compound interest demonstrated by examples. QUESTION IX. If an Annuity of 60 L per ann. be all forborn 7 years, how much will it amount unto when that term expires. Look in the third table for Annuities forborn the time specified, where against 7 years you will find the Decimals 8.39384. which multiplied by 60 L (the annual rent) the product proves 503.63040. cut off 5 places, whereby to sever the integers from the fractions, which reduce into money, and you shall find 503 L 12 ss 7 d. the true value of the 60 L annuity forborn 7 years; the question solved. QUESTION X. A did owe unto B 186 L, and upon covenant to pay unto the said B a rent of 20 L 13 ss 4 d per an. until the debt should be discharged; yet after this contract, they both agreed to respite the payments, until the last were due, with this proviso, to pay it all in then, allowing interest for the forbearance, at 6 L per cent. per ann. Find what number of years would have terminated the Annuity first agreed upon betwixt A and B, for the payment of 186 L by 20 L 13 ss 4 d annual rent, which will be performed by the example in the Table, viz. as 62/3 L is to 1 year, so will 186 L be unto 9 years: which rent is to be respited during the aforesaid term. Look in the Table of Rents forborn, where against 9 years you will find this 11.49132 to be multiplied by the decimal of 20 L 13 ss 4. the Decimal of 13 ss 4 d is (as in lib. 3. sect. 1. chap. 7.) 66667, to which prefix the integer 20 L, the total is 2066667. this multiplied by 11.49132, the decimal for the term of years, the product will be 237-4873183044, according to the rules of Multiplication in Decimals, lib. 3. sect. 1. cap. 4. sever off 10 places for the fraction, the integer will be 237 L, reduce 5 or 6 places of the fraction, making the Radius one place more, you will find 9 ss 9 d very near: so A must be responsable to B, or their heirs at 9 years' end for 237 L 9 ss 8 d 3 ●/10 q. This exactness was not required, nor yet so great a number taken for the fraction of 13 sh 4 d. but these if understood, the ingenuous will ease themselves by my labours, to which end I will proceed. QUESTION XI. A was to pay unto B 200 L at the full term of 5 years, for which debt A was contented to make B a Lease of a Farm to continue in force the same time, whose annual rent was 35 L. which of them gained by this contract, interest allowed at 6 L per cent. per annum? In the Table of Rents forborn, under years look 5, the decimal number against it is 5.63709. which multiplied by 35 L (the Rent respited the term of 5 years) the product will be 197.29815, and reduced into money is 197 L 5 sh 11 ½ d. which subtracted from 200 L, the remainder is 2 L 14 sh 0 ½ d. and so much A did gain by the bargain or contract made with B. The fourth Table exemplified in discount of Annuities, Rents, Pensions, or Reversions, at 6 L per cent. per ann. compound interest. QUESTION XII. What is the present worth of 80 L Rent or Annuity, to continue 25 years, rebating at 6 L per centum per annum? Look in the fourth Table for 25 years, against which I find 12.78335. This compound Decimal multiplied by 80 L (the Annuity propounded) the product proves 1022.66800. which reduced into money will be 1022 L 13 sh 4 ¼ d, the true value of 80 L per annum yearly payments, rebated for 25 years according to demand. QUESTION XIII. A man hath a Lease of Lands or Tenements worth 15 L per ann. more than the rent, and hath a Lease yet 4 years in being; the Tenant desires to take another in reversion for 21 years at the same rent, what must the Lessee pay for a Fine, interest allowed at 6 L per centum per annum? for 4 years for 25 years 346510 1278335 1 1732550 5 6391675 2 51.97650 6 191.75025 3 51 L 19 sh 6 D 7 191 L 15 sh 0 D 4 139.77375 8 139 L 15 sh 6 D First seek the Decimal for the term of four years 346510. which multiplied by 15 L, or by 5, as in the first Table in the margin, according to lib. 1. sect. 1. parag. 4. exam. 5. the product in the a row will be 51.97650, in money 51 L 19 sh 6 d. and so much the old lease in being is worth, when the new for 21 years enters possession: now admit the term of the old Lease and the new added together, the sum of years is 25, the profit or overplus of Rent is to continue all the time, therefore 1278335, the Decimal for 25 years, multiplied by 15 L, as in the fifth row of this table, produceth in the 6.119.75025, equal in value to 191 L 15 ss. the difference of the first Lease and the total time in the 8 row is 139 L 15 ss 6 d. and so the difference of decimals in 4 row reduced is very near, without a material error, being 139 L 15 ss 5 7/10 d. QUESTION XIV. A Tenant hath a Lease of 21 years, the present thereof is 41 L per ann. during the term of 7 years, and after that time shall be expired, the Lessee is to pay 50 L rent per ann. for the residue of the term, what is the value of this Lease in ready money, interest discounted at 6 L per cent. per annum? for 21 years for 7 years 1 11.76407 5 5.58238 5 9 2 588.20350 6 50.24142 3 588 L 4 ss 0 ¼ d 7 50 L 4 sh 10 d 4 537 L 19 ss 2 ¾ d 8 537.96208 In the fourth table (of Rents rebated) the Decimal of 21 years is 11.76407. which multiplied by 50 L (the rent of 21 years) the product is 588.20350, as in the second row of this Table; which reduced is 588 L 4 ss 0 2/4 d, as in the third row, which had been the true value of it, at L per ann. for the whole term of time; but the first 7 years of this Lease was but 41 L annual rent, therefore the first Decimal was too great, by the difference of rent, which was 9 L per annum; then look into the fourth Table for 7 years, and against it you will find 5.58238. which multiplied by 9 L, as in the first row of this Table, the product in the sixth, is 50.24142, and reduced is 50 L 4 ss 10 d very near; which subtracted from the third row, the remainder, is 537 L 19 sh 2 ¾ d, as in the fourth row; or subtract the Decimals found in the sixth, from the second row; the difference will be 537.96208. which artificial number reduced would be 537 L 19 sh 2 ¼ d, as before; the true value of the Lease required. QUESTION XV. There is a Lease to be taken for 21 years at 30 L per ann. and 100 L Fine: the Lessee likes the bargain, but not the condition, desiring the annual rent to be but 10 L yearly payments, and is willing to give such a Fine as shall be proportionable to the rent abated, during the aforesaid term of 21 years, and here the Fine is demanded. In all questions of this kind The Decimal 1 11.76407 2 L 235.28140 3 S— 5.628 1256 4 D— 7.536 take the rent abated, which is here 20 L per ann. for 21 years, whose decimal (in the 4th Table of Rents rebated) is 11. 76-407, as in the margin; which multiplied by 20 produceth 235.2814, that is, 235 L. reduce the fraction (neglecting the cyphers) the value of 20 L per ann. (the difference of Rent) for 21 years, is as in the 2, 3, and 4 row, in all 235 L 5 sh 7 ½ D. this added unto the former Fine, 100 L, makes in all 335 L 5 sh 7 ½ D, the true sum to be paid for a Fine, in lieu of 20 L Rent per ann. abated during the Lease of 21 years; the thing required. QUESTION XVI. A had a Lease of 130 L per ann. to continue 24 years; B had another of 210 L per ann. and 11 years to come; these 2 men mutually exchanged Leases; A (upon the contract) paid unto B 20 L in ready money, which of these had the better bargain, and how much? A B 12.55036 788687 3765108 1 1577374 1631.54680 2 1656.24270 1631 L 10 sh 11 D. 3 1656 L 4 sh 10 D. 1651 L 10 sh 11 D. 4 4 L 13 sh 11 D. Against the 24 year of the fourth Table, look and you will find the Decimal of it 1255036, for A. secondly, the lease of B 11 years, hath this decimal 7.88687. these 2 numbers multiplied by their respective rents, as in the first row of this table, according to lib. 1. sect. 1. par. 4. exam. 6 & 7. or by the vulgar way. In the second row of the margin A does produce 1631.5464, and B 1656.2427, neglect the cyphers, and reduce the numbers: in the third row you may find the Lease which A exchanged is worth in present money 1631 L 10 ss 11 d. and the lease which B was owner of being 210 L per ann. for the term of a 11 years, proves in currant coin the sum of 1656 L 4 sh 10 D. and A mended his in the barter or exchange 20 L, which makes the value of his lease, as in the fourth row, 1651 L 10 sh 11 D. which still is less worth by 4 L 13 sh 11 D, as in the fourth row (by subtraction) is evident, and that B lost so much money by the bargain. The fifth Table does demonstrate in its use the purchasing of Annuities, Rents, Pensions, or Reversions, at 6 L per centum per annum compound Interest. QUESTION XVII. What Annuity, Rent, or Pension, will 250 L in ready money purchase for a Lease of 7 years; interest allowed at 6 L per cent. per ann. Seek the seventh year in the fifth Table (which is the term of years that the Lease continues) whose Decimal number is .17914, and if multipled by 250 L, the product will be 44.78500, and reduced, is in money 44 L 15 sh 8 ¼ d. And this Annuity or Rent to continue the full term of seven years, which the former sum of money will purchase as a yearly revenue during that time. QUESTION XVIII. There was a man who purchased a Lease to continue 25 years, at 10 L per ann. for which the Lessee paid a Fine of 150 L. how much was the annual rent of this Lease valued at, when interest was rated at 6 L per cent. per annum? This differs little from Decimals 1 07823 39115 2 11.73450 3 11 L 14 ss 8 D the last; for here you are to find what Annuity or Rent 150 L in ready money will purchase for the term, as in the fifth Table against 25 years stands this Decimal 07823. which multiplied by 15, as in the margin, in the first row of numbers, whose product in the second row with the cipher annexed, is 11.73450. that reduced, is in money 11 L 14 ss 8 D (the farthing neglected as not material) and this annual Annuity 150 L will purchase for 25 years: therefore add this unto the Rent paid, viz. 10 L per ann. the total is 21 L 14 ss 8 D. the question answered. QUESTION XIX. There is a Lease of 25 years to come, set at 10 L rent per ann. and the Fine demanded is 150 L. the Tenant is willing to give 100 L, and a proportional annual revenue during the whole term, what will be the rent required, the loan for money allowed at 6 L per centum per annum? This does not vary essentially Decimals 1 07823 5 2 3.91150 3 3 L 18 ss 2 ¾ D from the former: for the Fine being diminished, the annual rent must be increased: take the difference betwixt the two Fines, viz. 100 L, and 150 L, as 50 L the Decimal for the term of years 25 is .07823. which multiplied by 50, or by 5 (as in the first table of the margin) the product in the second is 3. 91150. which reduced in the third row is 3 L 18 sh 2 ¾ D. the rent which 50 L will purchase for 25 year; which added to the former Annuity of 10 L per ann. makes the whole rent 13 L 18 sh 2 ¾ D, according to demand. QUESTION XX. A Citizen giveth over his Trade unto a faithful servant, leaving him his shop ready furnished, the Wares prized at 1408 L, the Lease of his house valued at 250 L, so in all 1638 L, which the Master was to receive by equal and annual payments in the space of 7 years, the interest agreed upon at 6 L per centum per annum, what annuity will discharge this debt. To discover this annual Rent, look in the first Table for the term of years specified, and against 7 you will find .17914. This multiplied by 1658 L produceth 297.01412. the Decimal● reduced will prove in money 297 L oh ss 3 ¼ D. Which Annuity or Rent, for 7 years annual payments, discharges the whole debt with interest, at 6 L per centum. QUESTION XXI. A Tenant took a Lease of a House and Land for a term of 21 years, paying 160 L Fine, and 16 L Rent per ann. at 7 year's end the Lessee was resolved to put it off: What annual Rent or Annuity must he set the Tenement at, to withdraw his former Fine, or reserving the same Rent, impose another proportional for the years to come? Interest at the rate of 6 L per cent. per ann. Rent Fine 1 .08500 5 9. 2949● 51000 13.6 2 13.60000 6 5576988 2788494 3 13 L 12 ss 929498 4 29 L 12 ss 7 126.411728 First to impose a proportional Rent, find by the first Table (of Annuities to be purchased) what 160 L will buy for the full term of 21 years, whose Decimal is .08500, which multiplied by 160 L, or 16, as in the first row of this marginal table, the product in the second is 13.60000. in the third is reduced to 13 L 12 ss. & this annual Pension 160 L will purchase for 21 years; which added to 16 L per ann. (the Rent of the Tenement) does evidently show the nature of the Lease, as in the fourth row 29 L 12 sh. and setting of it at that rate the remaining years, the Tenant saves himself. To discover what Fine must be imposed, the old Rent reserved, and yet a proportional part for the first Fine. The term of years remaining are 14, whose Decimal in the fourth table of Discount is 9.29498, which multiplied by the Decimal of 13 L 12 sh last found, viz. 13.6, as in the fifth row, in the s●xth stand their several products, and in the seventh row the total sum, as 1●6. 411728, from whence strike off 6 places, which are fractions (according to the Rules of Multiplication in Decimals) and reduce the test, the Fine will be discovered 126 L 8 sh 2 ¾ D, which saves the Tenant harmless, the old Rent still reserved, without gain or loss; the thing required. As for the Decimal of 12 sh. find the fraction, or see lib. 3. ca●. 7. table 1. Rules I have here delivered, equally balanced betwixt the Buyer and Seller, Debtor and Creditor, whereby neither side might deceive, none yet be deceived by fallacious or ambiguous contracts. As for Interest Money, here are composed rules both according to Custom, prescriptions of Art, and the precepts of humane Institutions, which tolerates Usury, confined to a Loan of 6 L per centum per annum. I cordially wish the frugality of the people would lessen the trade of money, and sink the Impost to a Land rate; yet there would be many Money-corm●rants, and their pro●it great, because such Estates lie dormant in Banks, obscured from the inquisition of a sax; and rarely appea o● wake but with the noise of a Forfeiture o● the Owners Land, or the liberty of his person. The Interest, like a Monster, by an unlawful conception, and a prodigious birth (grown greater than the Principal) makes appeal to the rigour of the Laws, against those who bore too prodigal a Sail, and now like to suffer wr●ck betwixt Scylla and Charybdis, or swallowed by those yawning waves. Usury is like a Cancer, which by an unperceptible Consumption ingratefully wastes that body where by Corruption it took a being; I wish none to adore the Golden Calf, nor yet slight the materials, their use being good and laudable, where Virtue is Treasurer, Discretion Controller, and Charity Purse-bearer: but if abused by being cast in another mould, or the three adverse parties in office, it will as e● lie catch those (who make worldly wealth their Mammon) as lime does Birds; so the danger is great, and the more, when usually the love of Money multiplies, as their Stocks and Magazines increase; and those who have most are often most miserable in want, ignorant in the use of temporal blessings, and glutted with excess, become immedicable by those surfeits; like men in Dropsies, the more waterish they grow, the more they desire drink, with an unsatiable thirst, so feeds the humours, and that the disease. And thus I will conclude with the ingenuous Poet, Ovid. Sic quibus in●umui suff●sa venter ab unda, Quò plus s●n potae, plus sitiuntur aquae. In English thus, Men s●ell'd with Dropsies grow excessive dry, And drinking, covet more until they die. THE THIRD PART. General Rules of Practice, by the Art of Memory. MErchandizes and all Commodities are sold either by number, weight, or measure, and those by gross o● retail, viz, as in Tons o● sized Loads, by the Thousand, by the Weigh, Tod, Clove, by the Hundred, whereof there are accounted 4 sorts, as 6 Score, 5 Score and 12 lb, 5 Score, and also 60 Warp, and sundry sorts by measure, as the Tun, Cauldron, Quarter, Barrel; the Gross, Dozen, etc. as by the following Tables are delineated, whereby their values in the least species or denomination may be computed in the greater, without obstruse rules or encumbering the memory with reservations, but by vulgar notions and natures common dictates only, having imprinted in your memory the gross sums, and what those amount unto in coin of the least denominations, as in shil. g●oats, pence & forth. but for those, to whom these accounts are unknown, or where for want of practice they have been obliterated, or the recollection troublesome, I shall present you here with a Table ready calculated, for the Numbers, Weights, and Measures, most frequently used in England, and generally received either from former Statutes, or customary Laws, ratified by the undeniable prescription of Time, and intermixed, are these: A TABLE of Numbers, Weights, Measures, and what their several gross sums amount unto, in Shillings, Groats, Pence, and Farthings, registered in Arithmetical Characters, and by Numeral Letters. lb Subtle Shill. Groats Pence Farthings Nu. Gross L S L S d L S d L S d 2240 or XX. C g. ●12 0 37 6 8 9 6 8 2 6 8 2000 M.M 100 0 33 6 8 8 6 8 2 1 8 2016 or XVIII Cg. 100 16 33 12 0 8 8 0 2 2 0 1800 MDCCC 90 0 30 0 0 7 10 0 1 17 6 1680 or XV. C g. 84 0 28 0 0 7 0 0 1 15 0 1500 M D 75 0 25 0 0 6 5 0 1 11 3 1344 or XII. C g. 67 4 2● 8 0 5 12 0 1 8 0 1200 MCC. 60 0 20 0 0 5 0 0 1 5 0 1120 X C g. 56 0 18 13 4 4 13 4 1 3 4 1000 M 50 0 16 13 4 4 3 4 1 0 10 896 VIII. C g. 44 16 14 18 8 3 14 8 0 18 8 800 VIII C 40 0 13 6 8 3 6 8 0 16 8 672 VI C g. 33 12 11 4 0 2 16 0 0 14 0 600 VI C 30 0 10 0 0 ●0 0 0 12 6 560 V C g. 28 0 9 6 8 2 6 8 0 11 8 500 V C 25 0 8 6 8 1 8 0 10 5 448 IV C g. 22 8 7 9 4 1 17 4 0 9 4 400 IV C 20 0 6 13 4 1 13 4 0 8 4 365 CCC LXV 18 5 6 1 8 10 ● 0 7 7 ¼ 336 III C g. 16 16 5 12 0 8 0 0 7 0 300 CCC 15 0 5 0 0 1 5 0 0 6 3 256 CCLVI 12 16 4 5 4 1 1 4 0 5 4 252 CCLII 12 12 4 4 0 1 1 1 0 5 3 224 TWO C g. 11 4 3 14 8 ● 18 8 0 4 8 200 CC 10 0 6 8 0 16 8 0 4 2 166 CLX 8 0 2 13 4 0 13 4 0 3 4 144 CXLIV 7 4 ● 8 0 0 12 0 0 3 0 120 CXX 6 0 2 0 0 0 10 0 0 2 6 112 I. C. g. 5 12 ● 17 4 0 9 4 0 2 4 100 C 5 0 1 13 4 0 8 4 0 2 1 90 XC 4 10 1 10 0 0 7 6 0 1 9 80 LXXX 4 0 1 6 8 0 6 8 0 1 8 70 LXX 3 10 1 3 4 0 5 10 0 1 5 ½ 63 LXIII 3 3 ● 1 0 0 5 3 0 1 3 ¾ 60 LX 3 0 1 0 0 0 5 0 0 1 3 56 LVI. ½ C g. 2 16 0 18 8 0 4 8 0 1 2 50 L 2 10 0 16 8 0 4 2 0 1 0 ½ 48 XLVIII 2 8 0 16 0 0 4 0 0 1 0 42 XLII 2 2 0 14 0 0 3 6 0 0 10 ½ 40 XL 2 0 0 13 4 0 3 4 0 0 10 36 XXXVI 1 16 0 12 0 0 3 0 0 0 9 32 XXXII 1 12 0 10 8 0 2 8 0 0 8 30 XXX 1 10 0 10 0 0 2 6 0 0 7 ½ 28 XXVIII. ¼ C 1 8 0 9 4 0 2 4 0 0 7 25 XXV 1 5 0 8 4 0 2 1 0 0 6 ¼ 24 XXIV 1 4 0 8 0 0 2 0 0 0 6 21 XXI 1 1 0 7 0 0 1 9 0 0 5 ¼ 20 XX 1 0 0 6 8 0 1 8 0 0 5 18 XVIII 0 18 0 6 0 0 1 6 0 0 4 ½ 16 XVI 0 16 0 5 4 0 1 4 0 0 4 14 XIV 0 14 0 4 8 0 1 2 0 0 3 ½ 12 XII 0 12 0 4 0 0 1 0 0 0 3 10 X 0 10 0 3 4 0 0 10 0 0 2 ½ 8 VIII 0 8 0 2 8 0 0 8 0 0 2 6 VI 0 6 0 2 0 0 0 6 0 0 1 ½ 4 IV 0 4 0 1 4 0 0 4 0 0 1 2 TWO 0 2 0 0 8 0 0 2 0 0 ½ 1 I 0 1 0 0 4 0 0 1 0 0 0 ¼ A description of this Table. Be pleased to observe here are 13 Columns, whereof the first contains (in Arithmetical Characters) the principal Numbers, Weights and Measure, noted in the head of the table with lb for pound weights, or Num. for numbers, beginning with 2240, a sized load, tun, or 20 hundred gross; from hence continued down to an unite, in such an order or series of numbers most frequently used in the Commerce and Trade of this our British Island: in the next stands their subtle and gross weights, noted with numeral letters, distinguished in the postscript by the letter g, denoting gross: the third and fourth Column shows what sum of money they do make in Shillings: in the fifth, sixth and seventh Column, what so many groats amount unto: the three next Columns what those gross or subtle sums do make in pence: the three last (in the least denomination of money) & what the value riseth to Pounds Sterling, Shillings and Pence, as by their titles in the head of these tables do evidently appear. The benefit of this Table, by sundry Examples illustrated to ease the Art of Memory. EXAMPLE I. It is required (without calculation) what 2240, or 20 C gross amounts unto in shillings, look under its title and you will find 112 L 0 ss. in groats 37 L 6 ss 8 D. so many pence comes to 9 L 6 ss 8 d. and in farthings 2 L 6 ss 8 d. EXAMPLE II. In things sold by Retail. Admit a Commodity vended for 3 half pence the pound, and it is required at that rate how much it comes unto by the Tun; I take it in the least denomination of Coin, (that is 6 farthings) and at one farthing the pound under that title I find 2 L 6 ss 8 D. for which you must impose so much on every farthing contained in the price, which was 6. then consequently the sum must be 6 times 2 L. secondly, 6 times 6 ss. and also 6 times 8 D. or for brevity, 6 times 2 L is 12 L, and 6 Nobles is 2 L. in all 14 L, at 1 ½ D the lb, as was required. EXAMPLE III. At 3 D the pound, what comes a ton unto? under the title of Pence I find 9 L 6 sh 8 D, at a penny the lb. than 3 times that is 28 L. admit a commodity at 8 D the lb. you may work this as before, but being the price is 2 groats, under that title you will discover 37 L 6 sh 8 D. and since 2 groats is the price of one lb, that doubled is 74 L 13 sh 4 D. if the price had been 3 sh the lb, the ton would have come unto 336 L, as by the table is evident. EXAMPLE IU. How much comes 10 D a day unto by the year? I look down in the table for 365 (the number of days in a vulgar year) and under the title of pence I find 1 L 10 sh 5 D. now 10 times that is 15 L 4 sh 2 D, or sum it up in your memory thus, viz. 10 L then 10 Angels, 10 groats and 10 D. or take, it in groats and pence, as 6 L 1 sh 8 D, and 1 L 10 sh 5 D maketh 7 L 12 sh 1 D. this doubled (the question depending on 2 groats and 2 pence) the sum will be as before, 15 L 4 sh 2 d. EXAMPLE V. If one pound of Cheese cost 3 ¾ D, what comes the Weigh unto? This properly belongs to a certain quantity of Wool and Cheese, cons●sting of 32 Cloves, whereof one contains 8 lb, so the Weigh is 256 lb. which having found in the table, I seek it in the column of pence, and find 1 L 1 sh 4 D. and in the row of farthings 5 sh 4 D. the sum 1 L 6 sh 8 D. and being that there was 3 times so much to be imposed in either denominaation, the sum is 4 L for the Weigh at the rate propounded. EXAMPLE VI. Wine sold at 2 sh 5 D the gallon, how much is that a tun, containing 252 gallons? Having found the number, look against it (the column of shillings) and you will discover 12 L 12 sh (at 12 D the gallon) which doubled is 25 L 4 sh. to which add 5 L 5 sh (for the 5 D) the sum is 30 L 9 S. the column of pence answering that number being but 1 L 1 sh, so it is easily multiplied by 5. Or take it as mixed in their several columns of groats, and pence, 'twill be all one in the total. EXAMPLE VII. Currants sold at 3 ¾ D the pound, how much comes 1 C weight gross unto? The farthings contained in the price are 15, and against 112 in the last column I find 2 sh 4 D to be imposed on every farthing, that is 15 groats & twice 15. in all 1 L 15 S. or take it in both columns of pence and farthings, as 3 times 2 sh 4 D. and thrice 9 sh 4 D. if this question had been propounded on 100 lb subtle, the answer will be 15 pence and twice 15 shillings, that is 1 L 11 sh 3 D. EXAMPLE VIII. Fish sold by the warp or couple, at 2 sh 10 ½ D the warp, what co●es 4 C unto? 60 in this commodity is 120 Fi hes to the C. look 60 in the table, against which I find (in the column of shillings) 3 L, then for the 2 should I set down 6 L, or keep it in my memory; in the next column I observe 1 L, and for the 2 groats in the price 2 L, then for 2 D (in the column of pence) I impose 2 Crowns, and for the 2 farthings in the price 2 sh 6 D, in all 8 L 12 sh 6 D, the price of 60 couple: now 4 times that is 34 L 10 sh. according to the demand. EXAMPLE IX. If 1 pound of Indigo cost 8 sh 7 D 3 q. what comes a quarter of 100 unto subtle? The answer will be 10 L 16 sh 1 D 3 q. look 25, and for the 8 sh in that column impose 10 L. for the 7 D, 14 sh 7 D. and lastly, for ¾ D take 1 sh 6 ¼ D, the sum will be as it was before, and according to my rules of Practice, lib. 2. parag. 9 By these 9 Examples all obscurities in this kind are cleared, difficulties made easy, and burdens to the memory removed, made facile even to common capacities, without tedious rules of Art, the Numbers, Weights and Measures of Commerce and Trade-being known to those who are conversant, or Masters in their own occupations; and if otherwise, this will be a guide to conduct them, without deviation, to the end of each gross sum, and may be accommodated unto the Numbers, Weights and Measures of any foreign or transmarine place; if occasion requires, or necessity urges, which I refer to the ingenious. Any day of the year assigned for the receipt or payment of money, or other business, to find what day of the week 'twill fall upon for any time to come. The Julian CALENDAR. Bis New-year's day Month's Days 1659. Saturd. I January 1, 8, 15, 22, 29 ✚ 1660 Sunday TWO February 5, 12, 19, 26 1661. Tuesd. III March 5, 12, 19, 26 1662. Wednesd. IV April 2, 9, 16, 23, 30 1663. Thursd. V May 7, 14. 21, 28 ✚ 1664 Friday VI June 4, 11, 18, 25 1665 Sunday VII July 2, 9, 16, 23, 30 1666 Monday VIII August 6, 13, 20, 27 1667 Tuesday IX Septemb. 3, 10, 17, 24 ✚ 1668 Wednesd. X October 1, 8, 15, 22, 29 1669 Friday XI Novem. 5, 12, 19, 26 1670 Saturday XII Decemb. 3, 10, 17, 24, 31 The Tables use explained. This Table contains 6 Columes; the first hath only 3 Crosses, to signify those years against them to be greater than the rest, being Bissextiles, or Leapyears: in the next are the years that shall be elapsed since the birth of Ch●ist, from 1659. unto the year 1670. In the third Column are placed the week days, which gins each year respectively, or the first day of January: in the fourth and fifth stands the 13 months: the last column shows the weekdaye, in every month, on which New-year's day did fall upon in any of these years. EXAMPLE I. It is required to know what day of the week shall be the fourth of December in the year 1659. against which I find Saturd. for the first day of the year, and likewise the third of Decemb. the next is Sunday the thing desired. The saturdays in this month 1659. are upon 3, 10, 17, 24, 31. Saturday concluding both month and year, and Sunday beginning the year 1660, as in the Table. EXAMPLE II. Admit it were required in a Leap-year, to know what days of any month shall be Sunday: here you are to observe that in Bisextiles, or Intercalary years, there is one day added to February, which then hath 29. so after that month take one from the day found, as in the year 1660. the first Sunday in March, in October, and the last day of December is required. New-year's day I find to be upon a Sunday, and in the Columns of Months against March stands 5, which should have been the same day of the week; but being February had 29 days this year, the 4, 11, 18, 25, are the Sundays in March this year. Secondly, against October I find 1, which should have been the same with New-year's day in a common year, but now the last of September, so the 7 day of October shall be the ●●rst Sunday, likewise 14, 21 28, and from any other number subtract 1. and then for December, the last Lord's day shall be 30, and the 31 to conclude the year shall be Monday. A Gregorian CALENDAR. Bis New-year's day Month's Days 1659. feria 4 ☿ I January 1, 8, 15, 22, 29 ✚ 1660 feria 5 ♃ TWO February 5, 12, 19, 26 1661. feria 7 ♄ III March 5, 12, 19, 26 1662. feria 1 ☉ IV April 2, 9, 16, 23, 30 1663. feria 2 ☽ V May 7, 14, 21, 28 ✚ 1664 feria 3 ♂ VI June 4, 11, 18, 25 1665 feria 5 ♃ VII July 2, 9, 16, 23, 30 1666 feria 6 ♀ VIII August 6, 13, ●0, 27 1667 feria 7 ♄ IX Septemb. 3, 10, 17, 24 ✚ 1668 feria 1 ☉ X October 1, 8, 15, 22, 29 1669 feria 3 ♂ XI Novem. 5, 12, 19, 26 1670 ●eria 4 ♀ XII Decemb. 3, 10, 17, 24, 31 This Calendar of 12 years is made for the payment or receipt of Money, or Merchandizes assigned upon a prefixed day of the month in Foreign parts, to find on what day it will fall upon: observe this Table does not essentially differ from the former in construction, but in the days of the months, the Reformed Account being 10 days before ours, so that the 22 day of December, according to the Old Style or computation, is the first day of January in the New, and so all the other month; precedes our 10 days, their Septimana, or Weekdays are diversely reckoned, but most usually thus, viz. Sunday, Feria prima, Dies Dominica, or Dies Solis, ☉. Monday, Feria secunda, or Dies Lunae, ☽. Tuesday, Feria tertia, or Dies Ma 'tis, ♂ Wednesday, Feria quarta, or Dies Mercurii, ☿. Thursday, Feria quinta, o● Dies Jovis, ♃. Friday, Feria sexta, or Dies Veneris, ♀. Saturday, Feria septima, Sabbath, or, D●e● Saturni, ♄. The f●r●t computation is an Arithmetical progression from 1 to 7. the other according to the Planets, denoted by their Characters, as they are appro●riatod unto those peculiar days: in other things this Table differs not from the former, so I refer the Reader to those 2 Examples. THE SECOND BOOK. Demonstrating a Sympathetical affection between Arithmetic and Geometry, by solution of several Problems or Propositions of Magnitude with exactness by the assistance of Art and Numbers. PROBLEM I. In any right-lined Triangle propounded with the Perpendicular and Basis, to find the Area or content of it in square Inches, Feet, Yards, Perches, &c, in whole numbers or fractions. The Theorem. The superficial content of any right-lined Triangle is half the Square produced, in multiplying of the Basis by the Perpendicular. Lib. 1. Prop. 16. Trigon. diagram IN the Triangle A.C.D. from the Angle at A. let fall a Perpendicular, as A. B. upon the Basis or ground-line, C.D. This Perpendicular suppose to be measured in inches or feet, etc. but here in this admit 4 feet, and the basis C. D. 5 feet, the product of these is 20 square feet, the half of this 10 feet superficial content of the Triangle A.C.D. the thing required. All right-lined multiangular and irregular figures may be reduced into Triangles, and thus measured, a Problem of great use to the Surveyer. PROBLEM II. In all plain rightangled Triangles, with either of the two sides known, to find the third side; from whence with any line how to describe or set out a perfect square for any Plate or Building, etc. The Theorem. In any lain right angled Triangle given, the square made of the Hypothenusal (or Subtendant side) is equal to the square made of both the containing sides. Lib. 1. Prop. 23. Trigon. In the last Triangle A.C.D. having let fall a Perpendicular from the Angle at A, as the line A. B. making 2 right angled triangles, viz. A.B.C. and A. B. D. whereof A.B. is 4. and b. C. is 3. their squares 9 and 16, the sum of them 25. whose quadrat root is 5, as by the demonstration may be explained in the second book, pag. 122. the true length of A.C. the Hypothenusal required, and the squares of A. B. 16. and B.D. 4. will be 20. wh●●e root will be A.D. as 4 4/9 or 4 47/100. but neither of them exactly true, as lib. 2. par. 1. examp. 5. but to return, if the Subtendant side A. C. were known, and one of the other two containing sides, the third side will be discovered; as admit A.C. 5, and A.B. 4. their squares 25. and 16. the difference 9 whose quadrat root is 3. for the side B.C. or if the square of 3 (that is 9) were taken from 25. the remainder will be 16. the root 4. for the Perpendicular A.B. In all plain right angled triangles, these numbers are only rational, to be found without fractions, or their products and quotients increased or diminished by some common number, from whence divers mechanical men do use and acknowledge it as a maxim in their trades in setting out Structures and regulating their works in perfect squares, after this manner: Take a long line (as your occasion requires) of which take 3 equal parts at pleasure, than 4 such succeeding parts, and from thence 5, so the line is now divided into 12 equal parts, by 3, 4, & 5. these parts extended will enclose a right angled triangle, as A. B. C rectangled at B. and proportional in all the parts, as by the first Book, 19 Prop. Trigon. This right angle found, you may describe a Parallelagram, or a Quadrangle if you please, as C.D.E.F. and A.B.D.E. or A.B.C.F. etc. PROBLEM III. The three sides of any right lined Triangle being given to find the superficial content thereof, without knowing the Perpendicular. The Theorem. From half the sum of the 3 sides subtract each particular side, the total of their mutual products increased by half the sum of the 3 sides, the quadrate root of that product will produce the superficial content. Suppose a Triangle with all the three sides known or found by any true measure, as admit in Feet, and the dimensions these, viz. 15 F. 20 F and 25 Feet, the sum of them is 60 F, the half 30 F. from whence subtract the particular sides, the differences will be 5.10.15. these by multiplication continued will produce 750, that product again increased by the sum of half the sides (which here is 30 F.) will produce 22500. the Quadrat root of it is 150. the number of square feet contained in the superficies of that Triangle required: having here the superficial content of this Triangle, by the first Problem before you may easily find the Perpendicular, for 150 feet is but half the long square made of the Basis and Perpendicular, then 300 the whole square divided by 25 the Basis of this Triangle, the quotient will be 12 feet for the Perpendiculars height, and so in any long square the superficial content divided by the longest side will produce the shortest, and divided by the lesser side will discover the greater. If in multiplying or dividing any square figures or numbers (that happen in fractions) you must consider their sides, for ½ multiplied by an unite will produce but ½, and ½ by ½ is but ¼ of that Square, as by the first and second little Quadrats made of the line A. B. in the Scheme, pag. 109. to which Book and Parag. I refer you, and to my first Book of Trigon. Prop. 31. The 3 sides of any plain Triangle given, to find the Perpendicular, and in what part of the Basis 'twill fall. The Theorem. Square the 3 given sides, add the 2 greater squares together, and from that sum subtract the less, h lf the remainder divide by the Basis or greater side, the quotient will be the greater Segment. As for example, admit the 3 sides of a plain Triangle given, 30 40. and 50. the Basis, which the Perpendicular will divide into two Segments, in this 32 and 18. making 2 right angled Triangles; now with either of the two sides, find the third as before, which according to the state of the question will prove 24. the thing required. PROBLEM IU. The dimension of any plain right angled Superficies, and first of Board, by square measure, as a foot, or 12 inches, whose Quadrature contains 144 inches. The Theorem. The superficial content of all rectangled figures are found by the multiplication of any two sides by one another that encloseth the right angle. A foot is here allowed the integer, by which board, glass, etc. is usually measured, every one of these dimensions is divided into 12 equal parts, called inches, and are the next immediate fractions to that integer, as by the Scheme pag. 103. Arith. does appear. Now suppose a stock of board to be measured in number 20. each board is in length 18 feet, & in breadth 10 inches, the length is in inches 216. which multiplied by 10 shows the superficial content to be 2160 inches, that divided by 144, the number of square inches in one foot, there will be found in each board 15 square feet, and consequently in the 20 boards 300 feet, the superficial content of the whole stock required: if the boards be tapering (as most stocks are) the common custom is to take the breadth in the middle, or the Arithmetical mean, that is half the breadth from the sum of both ends; as admit the breadth of the last stock had been 9 inches at the one end, and 11 at the greater, or 8 inches and 12. the sum had been 20 in either, the half 10 inches as before. Admit there were 24 panes of glass propounded to be measured, each pane containing in length 22 ½ inches, in breadth 14 ½ inches, and the superficial content is required in feet; the breadth and length here given converted into half inches produceth 29 and 45, the square of them is 1305 half-squared inches in each pane, which multiplied by 24 (the number of panes) the whole product is 31320. and since half the root or side of any insquared is bu● ¼, as by the demonstration of fractions pag. 109. divide 31320 by 4, the quotient will be 7830 square inches: which divided by 144, the quotient will be then 54 ⅜ feet, the true superficial content of all the glass required. By Decimals. Divers questions that fall in fractions, may be readily performed by artificial number, as thus, the length of one pane here propounded is 1 foot 10 ½ inches, the breadth 1 foot 2 ½ inches: for these fractions see the first Section of the third Book, and fifth Table Chapter 7. where you may find the decimal for 10 inches to 5 places .83333, and for the ½ inch or 5/10 this 04167, the sum of them .875; and for 2 ½ inches these .16667 and .04167 the total .20834, before these prefix their integers, and then their numbers will stand thus 1.20834 and 1.875, the products of these are 22656375, which multiplied by the number of panes, viz. 24 produceth 54.3753000, which is 54 3753/10000 feet, and exceeds the former not 4/10000, and that by reason of the irrational fractions which cannot be exactly true, yet the greater number will have the lesser error. PROBLEM V For boarding a Room. There is a Gallery containing in length 271 feet, in breadth 35 ½ feet; how many feet of board will floor this room? To find how many superficial feet this room contains will be discovered by the last Theorem, for 271 feet multiplied by 35 ½ feet, that is by 71, or more compendiously by increasing 271 by 7, according to my former rule, as in pag. 38. which will produce 19241 half feet, and that divided by 2 (as by the demonstration in fractions, pag. 109.) the true superficial content will be 9620 ½ feet. And here you are to consider in all such cases there will be loss in their breadths by seasoning and jointing them, and in their lengths to fit them on a joyce; some will prove faulty, as shaken or maim, and sundry other casualties, for which you must allow, especially in good work, and rebated 1/10, in square joints 1/15 or 1/16 lost in well shooting of the boards, although seasoned. PROBLEM VI. To measure Hang, Wainscot, Pavements, Land, etc. The last Theorem is an undoubted speculation to all these, so I will show the practice of it compendiously with examples; and first, there is a Room to be hanged, containg of Flemish yards, in height 4 of those measures, and in compass 25, the product of them is 100 the superficial content; in this room there is a chimney-piece containing 9 ½ square yards, and the Window 10 ½ yards, the sum 20, which deducted from 100 yards, the remainder will be 80 Flemish yards to furnish that room. And as for Wainscot, the operation's the same, but differing in yards, and sometimes by custom, in takin those measures, as in the height and compass of the room wainscoted, some using a small line extended strait upon each panel, and then rising over each stile and quarter. Thus Joiner's will make their work both of a greater height and compass then a line extended over all can do, the reason the workman gives, they must be paid where their Plane goes; but their measures admitted of, find the superficial content in the same manner as it was before, yet the Wainscot of that room, by the same measure, may exceed the other 5 or 6 yards, yet more or less according to the Joiner's work. Pavements are usually measured by the foot, or yard square, as Board and Hang are: the longest lineal measure used in England is the Rod, Pole, or Perch, whose lengths are various for Land, as custom hath introduced and continued them in particular Countries, and those from 15 to 25 feet in length: the most equal and generally received Perch is 16 ½ feet long commanded by Statute, yet 160 square Pole is one Acre of ground, according to the Rod by which it was measured, and in that Province where it is allowed. But as for our present purpose, the Survey being taken (though the field be never so irregular) it may be reduced into Triangles, and then measured, as was said before in the first Problem. diagram Example. The Area here surveyed is represented by the figure A. B. C. D. E. whose superficial content by natural Arithmetic will be thus discovered: Draw a strait line from E to C. now A C in this proves a subtendant side to the right angled triangle A.B.C. whereof A.B. was measured by the chain, and found equal to C. D. 45 ½ Perches: from E let fall a Perpendicular on C. D. as E.F. measured with the scale (by which the Plate was taken) 39 Poles: the work thus prepared, by the first Prob. B. C. 60 2●/33 or 60 2/11 P. multiplied by 45 ½ P. according to the rules of fractions (as in lib. 1. sect. 2. Parag. 4. Parad. 4.) will produce 60692/22 take ½ of it, 'twill be 60697/44 which is 1379 21/44 square Perches. Again, by the first Problem, in the Triangle E.C.D. the line C.D. 45 ½, or 91/2 Poles, multiplied by the Perpendicular E.F. 39 P. produceth 3549/2 square Perches, ½ of it is ●549/4 that is 887 ¼ P. the true content of the Triangle C.E.D. the sum of these two Triangles is 2266 ●/11 P. which divided by 160, the square Perches contained in an Acre, the quotient will be 14 Acres, 0 Rood, and 26 square Pole, the superficial quantity of the Field, as was desired. And thus the Triangle A.E.D. proves 3 A. 12 Pole. Any Parallelogram or long square propounded, whose dimension is required, multiply the length by the breadth, the product answers your desire: As for example in Decimals, the figure to be measured is A.B.C.D. in length B.C. or A.D. 60 P. 10 ½ F. in breadth 45 perches 8 feet and 3 inches, what is the Area or superficial content of this ground? 160 square perches makes one Acre, which contains 4 Roods, and one of them 40 Pole; now from the dimension of this field, in the seventh table of Decimals look 10 ½ feet, that is 21 half feet, whose Decimal is 6364, to this prefix the integers given as 60 Pole, which number will stand thus, 60.6364. and 45 perches 8 feet and 3 inches, that is ½ a rod, will be 5 for the half pole, so the multiplier is 45.5, the product of these is 2758.95620, which decimal fraction being very near an unite, the integer I make 2759 square perches, which divided by 160 pole, the quotient will be 17 acres and 39 square perches, the Area or superficial measure of the Field required. This Proposition in Decimals is useful for Surveyors. PROBLEM VII. Reduction of any squared Superficies from a greater unto a less, and the contrary, where the custom of several countries' allows of various measures. The Thorem. The Area or superficial content of any figure is in proportion unto a greater or lesser quantity, as are the squares made of those measures by which the figure was measured. In the last Problem there was 80 square yards of Arras hang according to the Flemish measure, which hath proportion to the English Standard, as 3 is to 4, whose squares are 9 and 16, and being they were in the lesser measure, multiply 80 by 9, the product is 720, which divided by 16, the quotient will be 45, the true number of yards according to London measure, as was required. In the Land measure there was last found 17 acres 39 pole, which admit according to the Statute, the length of a perch was to be 16 ½ feet, and it is required to know the content of that field where the pole is but 15 feet long: these measures in half feet will be as 33 is to 30, which reduced is as 11 to 10, the squares of these are 121 100, the Area found was 17 A. 39 P. and is in the least denomination 2759 perches, which multiplied by 121 produceth 333839, and divided by 100 is 3338, rejecting the fraction, being less than half a pole, and divide 3338 by 160, the quotient will prove 20, the remainder 138 P. divided by 40 will be 3 R. and 18 P. remaining; so the field of 17 A. 39 P. will prove in the lesser measure 20 acres, 3 rood and 18 pole, the proposition solved in the Parallelogram A B C D, including the woody and marish ground. PROBLEM VIII. The making and dividing of Rules in proportional parts, whereby the superficies of any right angled figure may be conveniently measured with more brevity by instrument, yet with less exactness then by Arithmetic. The Theorem. With the breadth of any rectangled figure given, divide the square inches contained in a foot, the quotient and fraction will show the inches and parts of the figures length which shall be equal to a square foot. The Carpenter's Rule for measuring of board and timber is commonly in length 2 feet, a thing necessary, but of no necessity whether longer or shorter, for this length will contain a foot of board, although but 6 in. broad, and what the length of the Ruler cannot comprehend, is usually termed under measure, with which I will begin. The side of a foot square is divided into 12 equal parts, called inches, the quadrat of 12 is 144, the number of square inches contained in a foot, and if it were demanded what length shall be required at 1 inch broad to be equal unto it (being an unite is divider) the answer will be 144 inches, that is in length 12 feet; if the breadth were 1 ½ inch, with which (according to the last Theorem) divide 144 the quotient will be 96 inches, that is in length 8 foot, at 2 inches broad 72 inches or 6 feet in length, at 2 ½ broad 4 F. 9 ●/● inches, at 3 inches in breadth 4 feet in length, and so proceed in the under measure by half inches if you please, until you come at 6 inches, with which divide 144, the quotient will be 24 inches, or 2 f. in length: the board measure being now upon the two foot Rule, containing 24 inches, and each inch usually divided into 4 equal parts, the board measure commonly proceeds also by quarters of an inch, so 144 divided by 6 ¼, in the quotient will be 23 1/25 inches in length; then for 6 ½ inches take in length upon the Ruler 22 2/13 inches; at 6 ¾ take 21 ⅓ inches, at 7 you will find 30 4/7 inches, and thus proceeding in quarters to 8 inches, which will require 1 ½ foot or 18 inches in length, and 9 broad 16 inches, at 10 broad 14 ⅖ inches, at 11 inches broad 13 1/11 inches, and 12 inches is the side of a foot square; from hence ascending, the square exceeding 12 inches the length will lessen, but thus proceed by quarters to 24 inches, from thence to 3 feet broad by half inches, and after that by whole inches only, the difference growing scarcely sensible, and howsoever not considerable in things of this nature, for if they should be continued to 4 feet, the difference betwixt 47 and 48 inches in this square measure will be but 3/47 parts of an inch; these proportionals found, you may inscribe them upon a Ruler, with figures to them, and so made ready and apt for common use, if exactness be required, make use of the Problems delivered you before. Yards are divided after the same manner, in their proportional squares to any breadth assigned, but usually such measures are taken in feet, one F. being the least in breadth that is commonly measured, 9 feet making a yard square, 3 being the side, & frequently without any under measure, beginning at 3 feet for the breadth for any such superficies to be measured, from thence proceeding by inches with their quarters to 10 feet in breadth, and more if need require: the side of this square contains 3 feet, that is 36 inches, whose quadrat is 1296 square inches, that divided by 48 (which is 4 feet) the quotient will be 27 inches, that is 2 feet 3 inches in length equal to a yard square; 5 feet broad requires 21 ⅗ inches; 6 feet or 72 inches must have 18 inches in length, 7 feet broad 15 ●/7 8 feet broad 13 ●/2 inches in length, 9 feet will have 12 inches in length and a long square 10 feet in breadth, every 10 ⅘ in length will be equal unto a yard square; and according to these dimensions, having found the parts in length answering to the feet in breadth from 3 unto 10 F. by the parts found you may inscribe them upon a Rule 36 in. in length, then find the quarters in the same manner to place between the feet and inches. The pole for Land-measures is only divided into equal parts, is quarters, etc. and so likewise the chains, distinguished usually with brass rings, and those again by tenns, both ready, exact, and of excellent use, especially in Decimal Arithmetic. PROBLEM IX. To find the Area of the least Quadrangle, or square figure that can comprehend the circumference of any Circle propounded. The Theorem. The Diameter of any Circle squared makes each side a Tangent to the Peripheria, or circumference thereof. A Demonstration. diagram Definitions and terms. A Circle is a Geometrical Figure comprehended with one line, as E F G H E, termed the Peripheria or Circumference, the part contained (as in other figures) is usually called the Area, in the middle of which there is a point, denominated the centre, as at 14; from whence all right lines drawn to the Circumference are equal, and infinite, and if any one be terminated at both ends with the Circumference, it is called the Diameter, as the line E G, 14; whose square is 196, for the content of the Quadrangle A B C D, each side being a Tangent, so named from touching, and not intersecting the circle, as in E F G H, being the least square that can be made containing the circle: which in this Problem is the thing required. PROBLEM X. The diameter or circumference of any Circle being known, to find the greatest circumscribed Quadrangle, or square made within that Circle propounded. The Theorem. The quadrat root extracted from half the square made of the Diameter shall be the side unto the greatest Quadrangle that can be circumscribed by that circle. An exact proportion betwixt the Diameter and the Circle was never yet discovered unto man, his knowledge therein confined within a small circumference of his own imagination; but as for circles within our capacities accommodated to humane use, the proportions are usually these, viz. as 7 is to 22, or as 71 unto 223, so will the Diameter of any circle be in proportion to the cicumference of it, the last is most exact, but the first most in use, as in this, the Diameter E G being 14, the circumference of the circle will contain 44 of those equal parts, so the one being known the other mey be found wonderful near the truth; now as for this Problem draw a line, or suppose one drawn from E and G to F, these sides are equal by construction, and by the 11 and 28 Proposit. Trigon. lib. 1. will enclose a triangle, right angled at F. and by this second Problem, the subtendant side E G squared will in quantity contain the squares of the two equal sides, and consequently half the square of E G 196, that is 98, must be equal to the square of F G, whose quadrat root extracted in tenths, lib. 2. parag. 1. examp. 5. Arith. there will appear 9 399/1000, for the side of the greatest inscribed Quadrangle, the content of the whole square is 98 as before. PROBLEM XI. To find the nearest quadrature of a Circle, that is such a square, whose superficial content shall without sensible error represent the Area of the Circles Peripheria. Three Theorems. A squared Diameter multiplied by 11, and the product divided by 14, the quotient is the vulgar Area: or thus, the semidiameter multiplied by half the circumference is the supposed quadrature, or the circumference squared and divided by 12 4/7 will be equal unto the superficial content of the given circle: these 3 do err, and yet agree in one. 1. In the last demonstration E F G H is a circle propounded, whose superficies is required, or the nearest square in content unto it; the Diameter E G 14 squared is 196, which multiplied by 11 produceth 2156, and divided by 14 the quotient will be 154, the superficial content required of the Quadrangle I K L M, the side of which square is the root of 154, that is 12 ⅖ I K. II. For the second Theorem I take again the same circle, the half of which circumference is 22 of such equal parts in which the semidiameter was found 7, which if multiplied by 22 will produce 154 for the superficial square of that circle, not exactly true but wonderfully near, as by the former rule, and the root extracted in centesms is 12 41/100 very near. III. According to the third Theorem of this Problem, in the former figure admit the circumference only known in this 44, w●ich squared is 1936, and if divided by 12 4/7 that is ●8/7, the quotient will be 13552/88 or 154, as before; thus diversity of ways confirms your work upon a good foundation, although an exact proportion betwixt the Diameter and Circumference is inextricable to Art, but real in Nature, and conspicuous to Man, although he cannot find it out, but leaving those that seek it, while I show you the use of these last Problems; and that you may find the Diameter or Circumference mutually by the other proportion, if required, observe as 71 is unto 223, then if 14 were the Diameter multiplied by 223 and that product divided by 71, the quotient will be 43 69/71 for the Circumference; which if it had been 44, the Diameter would be 14 2/223. A Treatise of Solids. All solid bodies usually measured, are divided into 5 Species, viz. Cylinders, Squares, Pyramids, Cones, and Segments of the two last, the forms of which figures I here present unto your view, as an ocular dedemonstration. diagram Definition of these figures, which in their several species are here propounded, and their dimensions required. Solid bodies to be measured are considered in general, whether they be Circular, Right or Obliqne angled; and these particularly in respect of each superficies one to another, or in, relation to their sides and bases, and most of them comprehended by some one of these 5 species. I. Cylinders are bodies long and round, equal at either end, right angled at their bases with their sides, whose length and circumference is contained under one superficies, as is the figure A B C D. II. Squared or rectangled Solids are all such bodies, whose sides are parallels one to another, and right angled with the basis at either end, as the figure E F. G H. I K L. III. A Pyramid hath but one basis, which is usually right angled in the 4 sides, the foot always acuteangled with each superficies, and terminated at the other end in punto, as the figure M R S T. iv A Cone hath but one base, and that round, the other end is terminated in a point, as was the former, and the length is contained under one superficies, as is the figure V W X Y Z. V Segments are parts of either of the two last, the lesser end being cut off, so making 2 square or 2 round unequal bases, one acute angled with the side, the other obtuse, as the figures N O R S T, or P R, or W X Y Z, etc. PROBLEM XII. To find the solid content of bodies that have two bases, and those equal at either end, the distance betwixt them or the length of which body is measured in a right line. The Theorem. All bodies whose length is a strait line, right angled with either basis, the superficial content multiplied by the length (in the same denomination) will produce the solid content in inches, feet, yards, etc. as it shall be required. First for the Cylinder A B C, whose length A C is supposed 30 feet, the Diameter A B or C D 3 ½ feet, that is 7 half feet; by the last Problem you will find 38 ½ which divided by 4 (as in lib. 1. parag. 5.) the quotient will be 9 ⅝ or 77/8 square feet for the basis, which multiplied into the length 30 feet, the product will be 288 ¾ feet, the solid content of that Cylinder, as in the figure does appear. If this round and long body were a tree with the bark taken off, and intended for board, slit work or building timber; in all such cases (according to common custom) a fourth part of the circumference is taken for the square of all such timber, called Girt Measure, which square multiplied by itself, and that product in the length (of the same denomination) will show the total content: As for example, admit the Cylinder were the trunk or body of a tree thus to be measured be the former Problem, the circumference will be found 22 half feet, that is 132 inches, the fourth part of which girt is 33 inches, whose square 1089 multiplied into the length, viz. 30 feet, that is in inches 360, the product will be then 392040, which divided by the cube of 12, the number of inches in a solid foot, viz. 1728, the quotient will be 226 1512/1728, or reduced to 226 ⅞ feet, the true content required according unto customary measure for timber to be squared. If the bark be on, as in Ash, Elm, or any timber felled in the winter season, it is as the buyer and seller can agree, or referred to custom, which in some places abating one inch in the square found, and for old Elms 1 ½ inch, in other places abating a tenth part of the solid content so measured, that is, allowing 1 foot in 10. of these two ways both are indifferent in timber, that is 17, 18, 19, or 20 inches square, but in small timber an inch abated in the girt is too great an allowance, and too little when the timber is very great; but here I will not prescribe you either way, for a foot allowance in 10 hath as great an error as the other, but contrary to the former, when one is too little, the other is too much; so I will write no more of this but caveat emptor. The measuring of square Timber. Timber or stone cut four square, the sides are parallels one to another, as the figure E F and I K, which here suppose 3 feet, the other two sides as H I or G K 2 feet, their square or basis is 6 feet, as H I G K, which multiplied into E H the length that product will be 180, and so many cubical feet this squared piece contains; but here observe, that according to girt measure the 4 sides make 10 feet ¼ of that is 2 ½ feet for the common square, which is apparently false, custom herein exceeding the truth, and will prove 187 ½ feet, which is too much 7 ½ F in this squared piece of timber. PROBLEM XIII. The dimension of Pyramids and Cones, either in Timber or Stone, and to find the solid contents of either species in inches, feet, or yards, etc. The Theorem. The magnitude or solid content of these figures is found by multiplying the superficial basis of either, in a third part of the length. The Pyramid (whose dimension is here required) admit represented by the figure M R S T, the side of the basis 3, whose square is 9, the Pyramid in length 30 of the same measure, the third part of it is 10, which multiplied by 9 (the superficial basis) the product will be 90, for the solid content required. If the solid content of M P Q were desired, the square at P or Q the basis is 4, the side being 2, which multiplied by 6 ⅔ being a third part of the length M P or M Q 20, the product of 4 and 20/3 is 26 ⅔ the solid content required: if these dimensions were in yards, it would contain 26 such cubes and 18 feet; or if found in feet, than 26 F. and 1152 inches, as in lib. 2. pag. 154. Arithmetic may be proved. The little Pyramid M N or M O is but 1 yard or 1 foot at base, 10 times that is the length, one third part of it, or 10/3, the product shows the content to be 3 ⅓, that is in cubical yards 3 and 9 feet; if the dimensions were in feet, the solid content would have been 3 feet and 576 cubical inches, the thing required. All Cones are measured as the Pyramids are: as for example, in the figure V Y or V Z the Diameter at the base Y Z is 3, the superficial content of that circle, by the 11 Problem will be found 7 1/14 or 99/14 which multiplied by 10 (one third part of the length) the product proves 990/14 or 70 5/7 the solid content of the Cone in cubical inches, feet, or yards, according to the parts, by which it was measured. If the solid content of the lesser Cone V X had been required, whose length is 20, the diam. at X the base is 2, and by the 11 Problem the superficies of it is 3 1/7 or 22/7 which multiply by 20 the height, produceth 440/7 or 62 6/7 the content in cubical parts of 3 times the Cone; the third part of 62 6/7 is 20 20/21 the true dimension of the figure V X. To find the lesser Cone V W, the diameter at the base is 1, the square 11/14, which multiplied by 10/3 the length, the product will be 110/42 or 55/21, that is 2 13/21 the content of the Cone in cubical pars required. PROBLEM XIV. The dimension of all Segments in tapering Timber or Stone, etc. as they are the parts of solid Pyramids and Cones. The Theorem. Unto the squares of the two extremes or bases, add the Geometrical mean square, which sum multiply by ⅓ part of the height, or ⅓ of the total by the height, the product will be the Segments solid content required. Admit the solid content of the Segment P Q R S T were required (which is part of the Pyramid M T as in the figure) the squares at the 2 ends are 4 and 9, their products 36, the square root of it is 6, for the Geometrical mean square, the sum of these 3 squares (viz. 4, 6, 9,) is 19 which multiplied by ⅓ part of the length, that is by 1●/3, the product will be 290/3 or 63 ⅓, the solid content of that Segment. To find the Segment N O P Q, the superficial squares of the 2 bases are 4 and 1, their products 4, the square root 2, which 3 number, viz. 4, 1, and 2, added together makes 7, and multiplied by 10/3 (viz. N O P Q by construction) the product will be ●0/3 or 23 ⅓, the true content of that Segment, and the sum of these two, viz. 63 ⅓ and 23 ½ is 86 ●/●, to which if you add the little Pyramid found by the last Problem 3 ⅓ the total of this with the 2 Segments completes the solid content of the whole Pyramid M R S T 90, as before, which demonstrates all the several dimensions to be true, in the same manner the Segments of Cones are measured, having first found the squares at either end as in Prob. 11. so to the ingenious Artist no more examples will be required, yet being a thing in controversy, and not well understood by mechanical men, for ampler satisfaction I will explain it with one demonstration more, Segments being the most frequent form of all, and so more diligently to be observed. An ocular Demonstration in measuring of all tapering timber, whether round or square. diagram The Pyramidal Segment here proposed for to be measured is A. whose length is 15 feet, the square at the greater basis is 9, and the lesser end 1 foot, as at A, and according to the last Problem the mean square 3, the sum is 13, which multiply by ●/● part of the length in this by 5, there will be produced 65 feet, the true solid content required; which to prove, take the Segment A into pieces as B C and D, there would be 9 of them, and all of one length, but several forms, viz. B a foot square taken from the middle of the Segment, then will there be 4 pieces like wedges a foot square at the base, and ending in a line at A, having no thickness, as the four corner pieces are perfect Pyramids, containing one foot square at the base, the other ending in a point, as D, each of whose dimensions, according to the last Problem, must be 5 cubical feet, being the length is 15, and consequently 4 of these will contain 20 feet, and as for C, 2 such pieces turned end for end, will be equal to the figure B, containing 15 feet; then one of them is 7 ½ feet, and 4 of those will make 30, the total of the 9 pieces in content is 65 cubical feet, equal as they are all together in the figure A, which is evidently proved, as was required. But this Segment (according to common practice) is measured in the middle) by taking the Arithmetical mean, that is by adding the sides of the two squares together, and taking the half of that for the common square, as in this, 1 and 3 maketh 4, the half 2, whose square 4 multiplied into the length, 15, the product will be 60 feet, for the content, according to custom, which is apparently erroneous, and 5 feet too little in this piece, as before was demonstrated; divers other errors (in measuring of solid bodies) are crept in for want of Art, and having got possession of ignorant people, they plead prescription and custom of the place, whereas Custom cannot establish a Law upon a bad Title, and a false ground nor Error prevail against Truth, nor Ignorance convince Reason, supported by Art upon Demonstration; but leaving the roughhewn and cross-grained people to their own imaginations, although themselves confess a profit by some trees, and a disadvantage by others, but know not from whence, as in flat timber, which some call ill weighed; as for Cants, and multiangled figures, their bases may be measured by triangles according to the first Problem of this Section: which found, their contents will be discovered by some one species in the dimension of the 5 former figures, according to the precepts of Art. Yet I would not have any man for to exact upon the buyer, but wish him some advantage or allowance in every load of green timber, as in every 40 or 50 cubical feet; and my reason is, because no green timber will hold measure when the bark is taken off, and some trees will shrink more than others, as I have found by experience, in a month's time 2 feet in a load and more. Here I have briefly delivered you the manner and custom in measuring most kinds of solid bodies, whereby to understand what you do, yet such exactness is not always requisite in rough timber, especially where there is much to be measured, and therein to avoid confusion, mark the trees as you measure them with 1, 2, 3, etc. and enter them in a book with the year and day of the month, the owner's name, and the field or wood wherein they grow: this done, make 4 columns, one for the number of trees, secondly for their lengths, in the third column their squares, and in the last their contents; by this means any tree will be quickly found, and if any mistake be you may correct it at your leisure; observing this course many abuses will be avoided betwixt the buyer and seller, as in cutting any tree shorter, or altering their marks, etc. and will be ready for your own justification: and besides, it is necessary in some place of your book to enter the buyers name, with the conditions agreed upon for the price by the foot or load, with articles for measuring, as in girding any tree more than once, and whether in the buyers or the cellar's choice so to do, also what allowance to be abated for bark, as in Elms, Ashes, etc. and sometimes in Oaks the bark will not run; besides a limited time should be agreed upon how long after the fell the trees must be measured, and the ground cleared; such things as these do often make cavils, when not agreed upon before; below the middle of any tree the buyer may gird it where he pleases, and this is general, divers other particular things there are for which I refer you to practise, and how the common Ruler is made for measuring of timber, observe this following Proposition. PROBLEM XV. The framing and dividing of the Carpenter's Rule, whereby the content of squared solid Bodies (as Timber or Stone) may be discovered in cubical feet. The Theorem. Divide the number of Inches contained in a cubical Foot, viz. 1728, by the square inches of any common Basis given, the Quotient shows what length must be required in inches of that body to be equal unto a cubical Foot. All Timber to be measured by this common Ruler, is supposed to have four equal and parallel sides, or reduced unto it by girding the tree with a small line, and taking ¼ part or that for a common square. This done, and the Ruler divided into inches, and every inch into eight or ten equal parts, begin first with the under measure, as one inch square will require 1728 in length, that is 144 feet, to make one cubical foot; 2 inches hath 4 for the square with which divide 1728, the quotient will be 432 inches, or 36 feet in length to make one foot; 3 inches square will require 16 feet in length; 4 inches 9 feet, 5 inches squared is 25, with which divide 1728, the quotient will be 69 3/25 inches, that is 5 feet 9 3/25 inches for the length, at 6 inches the side of the square, that is 36, will have 4 feet in length: 7 inches must have 2 feet 11 13/49 inches in length to make one foot, at 8 inches square 2 feet 3 inches in length will be equal to one cubical foot. Thus you may find the under measure to each quarter of an inch, with decimal fractions, if you please, as at 8 ¼ inches, by my former rules of fractions, the square will be 1089/16 with which divide 17280, the quotient will be 2 feet 1 inch and 4/10 or ⅖ very near: and a tree 8 ½ inches square will require in length to make a foot 23 9/10 inches almost. Now having past the under measure, you may proceed in the same manner to 36 inches, or more if you please, but it is unnecessary, the increase being so little, as half inches not well to be distinguished; so in case a tree shall prove above three feet for the side of the square, I have prescribed rules in number how to measure it, or by the Ruler thus, Take half the square and so measure the tree according to the length, the quantity so found will be ¼ of the whole content, as in the Demonstration, page 109 of my Natural Arthmetick. Thus having found all the parts from an inch unto 36 or 40 inches, you may make a table of them, or inscribe them on a Ruler, as you think good. THE SECOND PART. PROBLEM I. The dimension of round, concave and dry measures, as Pecks, Bushels, Strikes, Cornhoops, etc. diagram THe Measures used in England of this kind are more various and uncertain then are the Weights; few Market Towns, Villages, or Farm Houses, but have bushels of several capacities, whereas the Statute made at Winchester commands them a size of just 8 gallons: the modern Writers of this subject do affirm, that a Wine gallon (according to the Standard) filled with water, and then poured into a square or regular vessel, placed Horizontily, hath been often measured, and found to contain 231 cubical inches very near, yet perhaps intended for 230 ⅖, by reason that ancient Writers do affirm the proportion between the Wine & Beer gallon to be as 5 is unto 4, and so consequently a Beer gallon to hold 288 cubical inches, which is ⅙ part of a solid foot; and it is very probable that the measures instituted by the wise and just precedent race of this Age, extracted their weights and measures as from a grain of Wheat, and thence proportionally derived from one another, as a pint of good Wheat, equal in weight to 1 lb Troy, and a bushel to weigh half a hundred gross, & something more, but leaving this discourse and not measuring their actions by imaginations only, but find the content of this bushel in Wine measure, the depth of it being found 8 ⅛ inches, the diameter 17, whose superficial square by the 11 Problem will be discovered 227 1/14 inches, neglecting the fraction multiply this superficies by 8 ⅛ or 65/8, the product will be 14755/8, that is 1844 cubical inches, which divided by 230 ⅖ or 1152/5, the quotient will be 8 gallons, for the capacity of this vessel in Wine measure, as was required; and with the former fraction 1/14 more, it will prove 8 7/1000 gall. but when a broken number proves considerable it must not be omitted, but in such a case as this reject it if you please. This former Figure represents a Bushel, which was well approved of in the Country, although I cannot; for having measured it (as before) I filled it with a Wine quart, and found it to contain but 8 gallons Wine measure, whereas (according to Winchest. Stat.) it should be 8 Ale gallons, and one of them to contain 288 inches, as Mr. Windgate affirms, M. Outhred, & that excellent Artist M. Briggs allows the content of an Ale gallon but 272 cubical inches, which is generally esteemed on, being sealed by the approbation of these 2 famous Geometricians, and equal to the Standard; yet I will not confine you to their authority, nor persuade you to let another measure your corn by their bushel, since measures are so various every where: the scope I here do aim at is to find the capacity in cubical inches in concave vessels; which found, apply them as you please, or as the place admits. PROBLEM II. Gauging of Vessels, by finding the capacities or quantities of liquid measures, or cubical inches contained in them, from the least Roundlet to a Tunn, either of Wine or Beer. diagram Any Vessel of Wine or Beer may be thus measured; if the Diameter at the Head and Bung be equal, it makes a perfect Cylinder; for the dimension of which figure I refer you to the 12 Prob. and the last: but vessels of this kind are commonly biggest in the middle; and from the bung to the head, or either end, they have circular sides, which if continued would end in a point, as the Cone does, and hath 2 bases, yet is no segment of those which are bounded with right lines, but rather a Sphaerocide or Sphaeroides, which Sphaerall Segment may be thus measured. The length of this Butt or Pipe is 50 inches, the diameter at the bung 30, and at the head or either end 21 inches. This known, by the 11 Problem find the superficial content at the head and bung, which in this will prove 346 ½ and 707 1/7 square inches, take ⅓ part of 364 ⅓, that is 693/6, and ⅔ of 707 1/7 will be 9900/21, both which reduced are 231/2 and 3300/7, add these together, their sum is 3217/14, which multiplied by 50 inches, the length of the vessel, that product will be 410850/14, and reduced to 205435/7, which is 29346 3/7 inches. In this case you may reject the fraction, and divide 29346 by 231, the quotient will be 127 gall. and 9 cubical inches; or make the divisor or dividend a unite more as you see cause; or if more exactness and less trouble be desired, see 22 Axiom lib. 2. parag. 7. as for example, 205425 to be divided by 7 and that quotient by 230 ⅖ or 1152/5 the dividers multiplied will produce 8064/5, that is 1612 or 1613, with which divide 205425, the quotient will be 127 gallons, 1 quart and 1 pint ferè, the content of this pipe in Wine measure, which multiplied by 4 produceth 509 G. 2 Qu. and divided by 5 the quotient will be 101 G. 3 Q. 1 P. and ⅕, the content of this Butt in beer measure. Such Propositions are tightly performed by the decimal Tables. PROBLEM III. With the Diameter of any Circle known to find the Circumference of it in proportional parts unto 100, 1000, or 10000, etc. In all questions of this kind, make choice of some proportion betwixt the Diameter & Circumference, as in the 10 Problem and first, as 7 to 22, unto 22 annex 3 or 4 cyphers, which divided by 7, the quotient will be 31428, or unto 223 annex cyphers at pleasure, 71 must be then divisor, the quotient will be 31408, which proportionals may be made more numerous, if occasion requires; but first suppose 17 were a Diameter propounded, whose circumference is required in a Decimal fraction, the proportion will be as 10000 the supposed Diameter is unto 17 the true Diameter, so will 31428, a supposed circumference, be proportionable to 53. 4·276. which differs but little from the 10 or 11 Prob. PROBLEM IU. With the diameters of two circles known, and circumference of the one to find the circumference of the other, or a diameter with 2 circles given, to find the other diameter. The Theorem. Circumferences of all circles are in proportion one to another, as be their respective diameters. Admit the diameters of two given circles were 17 and 21, the circumference of the first (in a decimal fraction) is in proportion, as 7 to 22.53.43. the second number multiplied by the third will be 1122·03, which divided by 17, the quotient proves 66, the circumference of the second circle, which was required. If the circumferences of any two circles with one given to find the other, the manner of operation is the same, so it requires no example. PROBLEM V. With the diameter and superficial content of one circle, to find the superficies of another, whose diameter is known. The Theorem. All circles have proportion one another, as the superficial squares made of their diameters have. As for example, Suppose the Diameters propounded were 7 and 14, whose squares are 49 and 196 (their proportions as 1 to 4) the superficial content of the first circle is 38 ½, but as a decimal in the table it is 38.5, which multiplied by 196, the product is 7546.0, and divided by 49, the quotient will be 154, the superficies of that circle, whose Diameter was only known, as by the 11 Problem may be also proved. PROBLEM VI. The superficial content of any two circles propounded, with one of their diameters given to find the other. The former squares are here again propounded with one diameter known, whereby to find the other: in this example I take 14 for the diameter, the superficial square of whole circle is 154. the diameter squared is 196 for the second number, and the superficial square of the other circle is 38 ½ or 77/2, which multiplied by 196 produceth 15092/● or 7546, and divided by 154, the quotient will be 49, whose square root is 7 for the diameter required, as in the last Problem. PROBLEM VII. To find the content or convex superficies of any Sphere or Globe, whose diameter or cicumference is found or propounded by 4 several ways, according to Art. 1. By the Diameter or Circumference, find the Area or superficial Content of that Circle, according to the former Problems; which multiplied by 4 produceth the convex superficies required: As for example, admit 7 were the Diameter of a Sphere, the nearest Quadrature of that circle is 38 ½, which suppose inches, that multiplied by 4 will produce 154 square inches for the superficial content required, but not exactly true, Art being in all such cases defective, as was said before. 2. To find the superficial content of any Globe, as thus; multiply the Spheres diameter with the circumference of the same circle, the product will be the thing required, as in the last Example, where 7 is diameter, 22 will be the circumference, the product of these is 154, as before. 3. To measure the superficies of any Sphere, multiply the square made of the circumference by 7, and divide the product by 22, the quotient will be your desire: As for example, admit the circumference propounded be 44, whose square is 1936, which multiplied by 7 produceth 13552, that divided by 22 the quotient will be 616 for the superficies desired in a quadruple proportion to the last. 4. The superficial content of any Globe may be also thus found, multiply the square made of the diameter by 22, and divide the product by 7, the quotient resolves the question: as for example, admit 14 the diameter given, whose square is 196, which multiplied by 22 produceth 4312, that divided by 7, the quotient will be 616, as before: all those 4 agreeing in one. PROBLEM VIII. The dimension of Globes and Spheres by their circumferences and diameters known, and their solid contents found by any measure assigned, and performed 3 several ways. 1. With the diameter or circumference by any one of the former rules, find the convex superficies of the Globe, of which take ⅓ part, or ⅓ of the semidiameter; those multiplied together will produce the solid content of the Globe: as for example, admit a sphere to be measured, whose diameter is 7 inches, the convex superficies will be found as before, 154 inches, the semidiameter of this circle is 3 ½, or 7/2, take ⅓ part of either, as of the fraction here, which will be 7/6; this multiplied by 154 produceth 179 ⅔ inches, the solid content of this Globe required. 2. The diameter of a Globe being given, multiply the cube made of that diameter by 11, and divide the product by 21, the quotient is the solid content. Example, in the last question 7 was the diameter propounded, whose cube is 343, that multiplied by 11 produceth 3773, and divided by 21, the quotient will prove 179 14/21 or ⅔ the solid content of this Globe. 3. By the circumference of any Orb, find the solid content as thus, take half the circumference, and multiply it cubically, and that cube by 49, then divide that product by 363, the quotient gives the solid content required. Example, 44 inches is a circumference given, the half is 22, the cube of it will be 10648, which multiplied by 49 produceth 521752, this divided 363, the quotient will be 1437 ●21/363 or ⅓, and so many solid inches are contained in the Globe. If more exactness, or ampler satisfaction shall be required herein, see Archimedes de dimensione circuli. PROBEME IX. The Diameter, Weight, and Magnitude of a Globe being given, with the Diameter of another, to find the weight or solid content of that Orb. The Theorem. All Globes and Spheres are in proportion one to another, as be the cubical bodies, composed of their Diameters. Suppose 7 inches were the Diameter of a Globe propounded, and the solid content of it 179 ⅔ or 539/3, and the Diameter of another Orb 14 inches, whose solid content is required in the same parts: the Cube of 7 is 343, and the Cube of 14 inches is 2744, the cubical content of the first is given 539/3, which multiplied by 2744 produceth 1479016/3, and this divided by 343, the quotient will be 1437 343/1029, or ⅓, as in the Table does appear, and also in the former Problem: if the weight of the first had been known, and both their Diameters, the weight of the second would have been discovered in the same manner as was the magnitude: and if the Diameter of the one had been known, with the weight or solid content of both, the other would have been found in proportion of their Cubes, as shall be illustrated by the following Propositions. THE THIRD PART, Consisting of Military Propositions. PROPOSITION I. By the Diameter and weight of any Bullet known, with the Diameter of another to find the second Bullets weight. IT is a common received opinion, that an iron bullet of 4 inches diameter will weigh 9 lb, which if it be true, and that all iron will weigh alike in equal magnitudes then this rule is a positive truth, viz. as the cube of 4 is to 9 lb weight, so shall the cube of any iron bullet's diameter, be proportionable to the weight thereof, according to the last Theorem: as for example,, an iron bullet, whose diameter is 6 inches, the cube of it 216; so the proportion is as 64 is to 9 lb. so will 216 be to 30 ⅜ lb, as in the table is evident, which ⅜ is 6 ounces. PROPOSITION II. By knowing the weight of two bullets, and diameter of one to find the other diameter. For illustration of this Proposition, I will reverse the last question, viz. if a bullet of 9 lb weight shall contain 64 inches, in the diameters cube, than a bullet 30 ⅜ lb or 243/8 will require 216 inches, whose cubique root is 6 inches for the bullets diameter; these 2 examples are sufficient for any question of this kind: but observe, if by the diameter of the guns concave you would find what the bullet belonging unto it will weigh, the diameter of it must be ¼ of an inch less than the diameter within the muzzle, although it be not a taper bored gun. To find what thickness they are in metal, their Cylindars, concaves with their bullets diameters; Galaper Compasses are held the best for expediteness, especially those that open with a quadrant divided proportionally in inches, and to 1/10, commonly known to every Engineer: as for proposals of this art, there be divers books extant, to which I refer you, they belonging more to the practice then any Theory; besides doubtful queries are made by them, viz. as whether the quantity of powder can be proportioned by Arithmetic to the weight of bullets, or whether they move in a right line, or circular; or if a Canon be more fortified in metal upon one side then the other, wherefore the gun discharged shall convey the bullet wide from the mark, and the concaves cylindar incline to that side on which the metal is thickest, because most resisted, or wherefore a piece of great Artillery mounted at 18 or 20 degrees of the quadrant shall convey a shot the farthest, and almost twice the level range, also how good powder is known; all which must be referred to experience. This we know, that the sulphur makes it quick to fire, the Charcoal maintains it, and the Saltpetre turning into a windy exhalation by repercussion of the air, causeth such violent effects to amaze the world, as if ambitious to imitate the thunder and lightning, from which good Lord deliver us. This tract I will leave, and return to such Propositions as may be exactly performed by Arithmetic, and founded upon Demonstration. Compendious Rules for marshalling of Soldiers in any rectangular form of battle, either in one body, or in several Squadrons or Regiments. Definitions. Battles are considered in two several respects, one depending upon the number of men to be put into a Military array, the other reflects upon the ground, on which the Battalios are to be ordered. A square battle of men hath an equal number both in rank and file, yet the ground in such cases longer on the file then upon the rank. A square battle in respect of the ground hath the rank and file equal in length, yet the number of men in rank exceeding those in file. In respect of the men to be drawn forth in battalio, it is either termed a square battle, or in proportion, as the men in rank to the number of those in file. PROPOSITION III. If a square battle of men be required of any number whatsoever, the Quadrat root extracted from the list, or number of Soldiers delivered in, shows the number to be marshaled either in rank or file: As for example, a Sergeant Major delivers in a list of 22500 soldiers to be ordered in a square battle of men, the quadrat root of that number is 150, and so many must there be placed in rank, and so many likewise in file, lib. 2. parag 1. examp. 2. Arith. PROPOSITION IU. If the difference of the men in rank to those in file should be in any proportion required, observe these Rules. RULE 1. As the term which is given for men in file Shall be to the term propounded for the rank, So will the number marshaled in this Array Be in proportion to the root, or men in rank. RULE 2. As the term propounded for men in rank Shall be unto the term which is for the file, So will the whole number of soldiers marshaled Be in proportion to the square root for those in file. As for example, 20184 soldiers are to be ordered in battle of array, & in such proportion between the rank and file, as 8 to 3, that is as 8 men in rank for 3 in file therefore 20184 (the number of soldiers) multiplied by 8 and divided by 3, the quotient will be 53824, the quadrat root of it will be 232, as in the first table, for the number of men to be placed in rank, the number for the file is found, if you divide 20184; by 232 the quotient will be 87, or by the second rule to find the men in file; as 8 the term for the rank unto 20184, the number of soldiers, so will 3 the term for the file be in a direct proportion unto 7569, the quadrat root will be 87 for the number of men in file, according to the second table in the margin. PROPOSITION V. To marshal in battalio any number of Soldiers, when there is a double proportion stated, as in respect of the men and ground both for the rank and file. The RULE. As the product of the two terms for the Rank Shall be in proportion to the number of Soldiers, So will the product of the two for the File Be to a fourth number, whose square root is the File. For the illustration of this Proposition, suppose the number of soldiers to be marshaled near 41160 in this order and proportion, viz. as for 3 in rank, 7 in file, and in respect of the ground as 2 is in proportion to 5; the products of these terms are 6 and 35. then say as 6 is to 35, so will 41160 be unto 240100, whose square root will be 490, as in the first table, with which divide the list of soldiers given, viz, 41160, the quotient will be 84, the true number of men both in rank and file, or by the second table, as 35 is to 6, so will 41160 be in proportion to 7056, the root 84; or with this divide the list of soldiers, the quotient will be 490, as before, and so in all questions of this kind, by the rank found you may find the file and the contrary. PROPOSITION VI. If an Army were drawn out in their particular Regiments, and those again divided into several squadrons with their depth and proportion both in rank and file. This Proposition (although of most use) depends upon the former; for having the number of Regiments, or lift of the army, they may be reduced into little squadrons, as the Maj. Gen. shall think fit, and then marshal those according to order in what proportion shall be required, betwixt the rank and file, by one of the 3 last Propositions, of which I have given you examples, not according to custom, or the Military Discipline practised in any place, but whereby you may solve any question of this kind, and not as precedents, but rules only; for the Foot squadrons 10 deep is the most that I have heard of, the usual custom in Europe is 6 deep for the Foot and 3 for the Horse, when they charge the Enemy. PROPOSITION VII. For the encamping of Soldiers in their several quarters. For a quartering of Soldiers in the field, it is performed by the common rule of Three; as for example, suppose a Regiment of 1000 men may be quartered in a square of ground containing 20 perches, what shall the side of a square be to lodge a greater or lesser number; the proportion will be, as 1000 to 400 perches, so shall any number of soldiers be to a proportional square of ground, whose quadrat root is the side required: and for example, admit the number of soldiers were 24000, then say as 1000 men is to 400 the square of 20 pole, so 24000 men will be in proportion to 9600, the quad. root of it in a decimal fraction is 97 ●/10 perches, the side of a square that will incamp those men according to the proportion given: but here are sundry occurrences to be consulted on, which must be referred to the experienced Master de Campo to marshal up together, as in respect of the enemy, the Campanio, the advantage of ground, the securing of passages; and multitudes of other things to be considered, in preserving the Army so well as when engaged in fight, by reserves, or how to draw off and make retreats, etc. depending more upon the practice then any Theory, or prescription of Rules. PROPOSITION VIII. The perpendicular height of any Tower or other place being given, to find at any distance (appointed from the basis thereof) how long any scaling ladder or rope extended must be to reach the top or summity of it. According to the state of this question, the rope or ladder will include a rightangled triangle, and by the second Problem of this book, the quadrat root extracted from the sum of the squares made of the two containing sides will be equal to the Hypothenusal, which is the ladder or length of the rope: as for example, admit there were a Turret in height 45 feet, there was a Moat before it in breadth 22 feet, the square of 45 is 2025, and the square of 22 is 484, the sum of these squares is 2509, the quadrat root of it is 50 feet for the length of the ladder or rope that will reach unto the summity or top of it, if the remainder had been considerable, you might have extracted the root with a fraction, as in lib. 2. parag. 1. examp. 4 or 5. Arith. but some (where ignorance hath got the upper hand of their reason) will say (peradventure) what care they for this; give them rope enough; and so say I with all my heart. PROPOSITION IX. To find the height of an accessible Fort, Turret, or any other place, by a common square, or with two sticks of equal length artificial joined together at right angles. diagram Admit the height required were the Tower C D, I move my station from F towards D, holding the triangle or square parallel with the ground-line and perpendicular by help of a plummet, as at K, where by both ends of the little square I behold the Tower's summity, as at C. Now by the 19 Proposition of my Geometry, A B must be equal to B C, and A B or L D is found by measure 48 feet, the true height of B C, to which add B D or A L (the height of the square above ground) viz. 3 feet, the sum 51 feet, for the altitude of C above the Horizontal plain F D, the proposition answered. PROPOSITION X. To find the distance unto any Fort or place, although not accessible, yet discovered by this square or triangle. Erect a staff perpendicular, whose height is exactly known, as in the last Scheme E G, which admit 6 feet, or 72 inches; upon the top of it cut a notch, so that the square may fall down in it something strait, yet so as to turn at E. suppose the distance required were G D, place your eye at E, then turn the square upwards or downwards, until by the edge of it you see the basis of the Tower, or place at D. the square being fixed, look down from E to F. at which place a mark upon the ground, and measure the distance F G, which is he●e 8 inches: here you have 2 equiangled triangles, viz. G E F, and G E D, and by the 19 pr●position of my Geometry the sides are proportional: now admit this little triangle were delineated in the greater, as G H L, then is G L equal to G E, and G H to G F. thus are they in the rule of proportion, as G H 8 inches is to G L 72 inches, so will E G 72 be in proportion unto G D 648 inches or 54 feet, the true distance required. PROPOSITION IX. To find the height of any place approachable by the shadow which it makes, with the help of a Pike erected perpendicular to the horizontal plane, or by any Turret, whose height is directly known, or by the height of any Tower, to find the distance, though not approachable. diagram The height of the Tower A B is required, to be found by the shadow which it makes upon the Horizontal place, as in this figure; suppose B D by measure found to be 12 per 6 3/10 in. upon the same horizontal plane, I measure the shadow of some other body, or erect a Pike perpendicular, as C D, whose height above ground is 7 feet, and the length of the shadow which it makes extends itself from D to E, by measure 12 feet 3 ⅓ inches: this known, the proportion is, as E D 12 feet 3 ⅗ inches is to C D 7 feet, so the shadow B D 198 feet 6 ●/10 inches unto 112. 98 for the height of the Tower A B, which caused the shadow, that is 113 feet 3 inches and more, much exactness is required (in questions of this nature) or else little truth to be expected, and shadows commonly falling in broken parts, which made me herein use the Decimals, yet with more exactness performed by Natural Arithmetic, and vulgar fractions, and so found 112 F. 11 inches 28/100, if the height A B had been known 113 feet, and the distance or extent of the shadow B D required, the proportion would have been, viz. as C D 7 feet to E D 12.3, so will A B 113 feet be unto 198 feet 5 4/7, that is 198 feet 6 inches and ●/7 for B D, the distance required, which is very near the truth. PROPOSITION XII. To discover the altitude of an accessible place by a mirror or looking-glass, or by a Tower's height known to find the distance unto it. Let the position of your Glass or Mirror be horizontally placed at some convenient distance; from thence go backward into a direct line, until you can descry in the glass the top of the Tower, or object whose height is required; then will the distance from your body to that part of the glass (where the summity of the Turret was represented) be in proportion to the distance from the glass to the perpendicular basis of the Tower or Sconce, as the height of your eye is to the perpendicular height required; for by the Optic Science it is an apparent Maxim, that the angles of Incedence and Reflection are equal, as A D B, and F D E, and your body being parallel with the Tower, the Radius of your sight encloseth a triangle, equiangled with that of the Turret's shadow, as by the 8, 9, or 10 Proposition of my Geometry, and consequently by the 19 proposition of the same book those triangles are proportional in all their sides; this is so visible that it needs no explanation, if they can see themselves from their shadows, or shall ever behold my Trigonometry, to which I refer them for 2 more ample satisfaction; and this to their impartial and judicious censures, yet wishing a legal trial to answer unto my charge (if there shall be any fomented) in the mean time, hopes of a candid construction from a serene verdict free from all obstructions of malice to obtenebrate my intentions, bids me with comfort to proceed. PROPOSITION XIII. A Captain of a Castle expecting to be beleaguered, makes good his outworks, and having fortified those best where he conceived most danger of being stormed; he overlooks the inventory of his Magazine, and takes a list of his Soldiers, with the supernumerary persons, in all 800. by which he finds his provision of victuals good but for 3 months & 3 weeks, that is 105 days: having more men than were necessary, and expecting no relief under 6 months, or 168 days, the question is ' how many men must be dismissed this fort (before the enemy's approach) whereby the same victuals might last the just time required? The Solution. The rule thus stated, according to lib. 2. parag. 8. Can. 9 Arith. in a reverssed proportion, viz. as the provision of victuals for 105 days allowed for 800 men, what number will 168 days require, allowed in the same proportion, which according to the rule, will prove 500 men, as in the operation of the margin is made evident, and consequently there must be 300 meant dismissed of the supernumeraries, ●nd Soldiers not able to perform their duties, or of those least serviceable to defend the works. PROPOSITION XIV. The Castellain commanded the Master of the great Artillery (or chief Gunner) to render a strict account of all the guns (mounted upon this Fort-royal) whether offensive or defensive, with the diameters of each bullet belonging to every piece of Ordnance, with the weight of the said bullet, and quantity of powder; also the distances in Geometrical paces, that each piece will convey the shot, so laden, both at point-blank, and at the utmost random; which accordingly was thus delivered in. The Table or Inventory. The number of Guns The names of this Artillery Each bullet's diameter Every bullet's weight The due charge of powder. The distances in paces at pointblank and random 10 Canon 7 48 15/64 26 lb 340 & 1600 10 Dem. can. 6 30 ⅜ 18 lb 350 & 1700 8 Culverin 5 17 37/64 15 lb 420 & 2100 8 Dem. culv. 4 inch. 9 lb 8 lb 320 & 1600 6 Sakers 3 ½ 6 15/512 5 lb 300 & 1500 6 Minions 3 3 51/64 3 ½ lb 280 & 1400 4 Falcons 2 ½ 2 101/512 2 ¼ lb 260 & 1200 4 Faulconets 2 1 ⅛ 1 ½ lb 220 & 1000 PROPOSITION XV. There are in this Fort 56 pieces of great Artillery, as are specified in the Table, viz, the whole Canon hath a bullet of 7 inches diameter, in weight 48 lb and 4 ounces almost; to which there is allowed for every shot 26 lb of powder: this Gun discharged will carry on the level-range 340 paces, and shot at random 1600. and note that all pieces for battery ought to be planted within ½ or ⅔ parts at most of the paces they carry point-blank; but as for our present purpose, the magazine of powder was found here 10 T. 11 C. 4 st. and 12 lb weight, and the question propounded is, whether this quantity of corn-powder will discharge the 10 whole Canons 20 times round, the 10 Demicanons 30 times, the 8 Culverins 40 times, the 8 Demi-culv. 50 times, the 6 Sakers 60 times, the 6 Minions 70 times, the 4 Falcons 80 times, and the 4 Faulconets 100 times. The Solution. Reduce first the gross weight of powder delivered into pounds subtle, and you will find 21100 gross to be 23632, to which add 4 saint. and 12 lb, that is 68 lb, the total is 23700 lb. (1) (2) (3) (4) (5) (6) 20 10 Cannon 26 260 5200 30 10 Dem. Can. 18 180 5400 40 8 Culverin 15 120 4800 50 8 Dem. culv. 8 64 3200 60 6 Sakers. 5 30 1800 70 6 Minions 3 ½ 21 1470 80 4 Falcons 2 ¼ 9 720 100 4 Faulconets 1 ½ 6 600 450 56 Totals 79 ¼ 690 23190 In the first column stands the number of charges imposed upon every Gun, in the second the number of each piece, in the third are inscribed the Ordnance, in the fourth each particular charge, in the fifth is placed the whole quantity of powder that charges all the guns of each sort, the sixth and last column contains the whole quantity of powder according to the number of their several charges, whose total is 23190 lb. PROPOSITION XVI. The last Proposition was not judged convenient, being but 510 lb of powder remaining: upon which, by order there was deducted from the Magazine 3000 lb, viz. for small shot, for Granades, for murdering shot (in case there should be any breaches made) for waste and priming powder, the query next stated was how many shot about the 20700 lb will make, which according to lib. 2. parag. 11. Arith. may be thus stated: The fifth column (in the last table) contains the quantity of powder that charges all this great Artillery once round about, whose total at the bottom of the same column is 690 lb for the first number, the magazine or whole stock of powder is the second number, viz. 20700, which in this may be totally divided by the first, that is by 690. so the first and second numbers are now 1 and 30. each particular for every species is comprehended in the fifth column, to a proportional allowance, that each piece shall spend, being once discharged, and then the fourth proportional number found shall be the quantity that every kind shall spend at an equal number of shot to be made, whose total (if the operation be true) shall be equal to the second number in the rule, or the Ammunition delivered in for this purpose, as by the following rule is made conspicuous. 260 7800 180 5400 As 690 lb is to 20700, so 120 3600 64 unto 1920 or reduced 30 900 21 630 As 1 in proportion to 30, so 9 270 6 180 690 totals 20700 By this it is made aparent the Canon must be allowed 7800 weight of powder, whereof there are 10 in number, so each whole Canon must have 780 lb, which divided by its proper charge, viz. 26 lb, as in the fourth column and former table, the quotient will be 30 shot, again, for the 4 Falcons there is allowed 270 lb, so for one of those guns 67 ½ lb, which divided by its allowance of powder the quotient will show 30 charges; and so many shot every piece of Ordnance will make round with the allowance according to the last table. A Castle that is fortified both by is Nature and Art, provided with Ammunition, Manned, and victualled well, and all things necessary for a defensive War closely beleaguered; if the men stand sound, and yet surrender it to the enemy before 6 months being expired, it will be conceived the soldiers are better fed then taught. Ladles for guns are proportioned according to the bullets, the plate made plain at first, and in breadth ⅗ parts of the bullets circumference, the ⅖ abated, whereby to empty the Ladle in the Guns chamber. General Rules and Observation, or a Contexture of various questions. Iron bullets are proportioned to Led, made in equal moulds, as 5 to 7, and iron to equal bullets of marble, 7 to 3, the proportioning of their weights is uncertain, when bullets of iron will differ, though cast in the same mould, one metal being more pory then another. The Demi-culvering bullet 4 inches diameter, is generally received as a gage for the rest, whereby to find their weights or magnitudes: this bullet made of some iron, will be just 9 lb weight, and it is a medium almost betwixt the least and greatest sort of Guns upon carriages usually made, yet I have seen and measured one, the diameter of whose concave Cylinder was above 20 ¼ inches, the cube of the bullet 20 inches is 8000; then say as 64 is to 9 lb, so will 8000 be unto 1125 lb for the weight of such a bullet. Gun-founders of brass pieces use an allay of copper and tin, proportioned to every 100 lb weight of brass, but the mixture various; which you may find in any piece of Ordnance, having the true weight of the gun and the allay, as thus; suppose a Cannon to weigh 7000 lb subtle, and the allay for every 100 lb of brass 40 lb copper, and 10 lb tin, than state your question in form, according to the rules of society, as in lib. 2. parag. 11. Arith. as thus: lb lb As 150 is to 7000, so 100 Brass 4666 ⅔ or reduced by 50 40 Copper unto 1866 ⅔ As 3 shall be to 140, so 10 Tin 466 ⅔ 150 totals 7000 All Guns more fortified with metal on the one side then on the other, if discharged at a mark, the bullet will fall wide from the object, inclining to that side which is most fortified or thickest in metal: the reason (I conceive) is, that the thinnest part is soon hot (by the agility of the fire) and so from thence dismisses the bullet with the greater force, or else in imitation of sulphurous Meteors fired in the wombs of clouds, break forth in their deliverance with amazement to mortals, and strikes most at that which is strongest, or most fortified to resist. Two pieces equal in all things, but their length, and if charged and leveled alike, the longest will convey its bullet farthest; yet if discharged together at a mark within distance of either gun, the bullet from the shortest piece will be at the place first, or the object aimed at. Three Cannons discharged G lb G 3 156 10 2 30 6 156 300 Facit 7800 twice spent 156 lb of powder, how much will 10 equal guns to them spend? if 30 times discharged with the same allowance, which stated, as in the margin, according to lib. 2. parag. 10. quest. 3. Arith. or may be reduced to 1. 26. 300, or thus, 1. 156. 50. the fourth proportional found will be 7800 lb. Most guns will shoot at random 4 times so far and more than their level-range, and some of the great Artillery 5 times; the best random of a piece is held when elevated 22 or 23 degrees of the quadrant above the horizontal plain. Always observe to keep your link, stock, match or fire, to lee-ward of the gun or powder. An iron bullet will fly farther than one of lead, but the greatest batters most at 80 or 90 paces; and either of them with most force from a gun a little elevated, then on the level range, although within distance, and the heavier bullet will raze a work the soon. No bullet from a gun, that is leveled and discharged, does move in a direct strait line, but circular, ascending first with the violence of there, and overshoots the mark within the level-range; and as the heat lessens it tends towards the seat of gravity, and at point-blank crosses the line of level, protracted from the centre of its concave cylindar, which arch is greater accordingly as the gun is elevated from 1 to 45 degrees of the quadrant, and lesser if discharged below the level-range. All guns, if over heated with often shooting, are apt to break; those perpetrated with cold and frosty weather are most subject to an eruption at the first shot; the reason is, that in all metal there is a radical humour, which connexes and keeps the parts together, and is made weak by being dilated with overmuch heat, or contracted with too much cold, leaves the parts enervated, and each member of that body dissoluble, or easily discorporated, and the sooner by opposition of its contrary, the agile and penetrating fire invading the condensed cold. Any bullet discharged from a gun does strike most violently against that which is hard, firm and strong to resist, and soon deadened where it wants opposition, as being shot against wool, sand, any soft earth or movable object, and hath more violence at reasonable distance then near the Cannon's mouth which delivered it unto a convoy of the subtle Air; the greatest force of any bullet for battery, is generally conceived from ¼ to ½ of the level-range of that Piece which made the shot, and from ●/2, the force of the bullet lessens in its raptile or violent motion. The weight and content of Casks for Powder. A Barrel or empty Firken aught to weigh 12 lb, and should contain 100 lb of powder neat; so the weight of a Firken thus filled is 112 lb, that is 100 gross, and 24 of such Firkens makes one last containing 2400 lb neat. The common ingredients and quantities in making of Powder. The ordinary Powder is composed of these 3 Simples, viz. to every lb of Saltpetre add ¼ of Charcoal & ⅙ part of Sulphur: To these take any 3 proportional numbers at pleasure, according to the 15 Axiom, lib. 2. parag. 7. or 14 Arith. As for example, 60, 15, 10, or 12, 3, 2, the least without fractions is usually best, as the least trouble, and admit one Last of Powder were the quantity to be made, as 2400 lb, the proportion will be thus: lb 12 Peter 1614 3/17 As 17 is to 2400— so 3 Coal 423 ●/17 2 Sulphur 282 6/17 17 Totals 2400 As for the quality and goodness of Powder, experience is the best tutor; yet as a common and general observation, good powder will be bright in colour, tart upon the tongue, and very salt in taste, apt to burn, and quick in being fired: the Brimstone makes it apt to kindle, the coal continues the inflammation, by which the Saltpetre is resolved into a windy exhalation, and strives to dilate itself restrained in the concave of the gun, vents itself at the mouth of the Piece, being the easiest passage; but if the bullet be rusted in, or over charged, and cannot get out, it will force a passage through the weakest part, as subterranean Meteors do when much rarified, and restrained in the concaves of the solid earth. PROPOSITION XVII. If a barrel of powder will charge a Demiculverin 12 times, burning 8 lb at a discharge, how many shot will a Last of powder make for a Canon that spends 26 lb at every charge? This question is solved, as in lib. 2. parag. 10. quest 4. Arith. in the first rule; stand the quantities of Powder propounded, and the Shot made in the Demiculverin in a proportion direct; the second is each charge of powder reversed, and by their products made direct, as against III. in the last it is reduced to 13. 6. and 192. these by the common rule of 3 will produce 88 8/13. so in a Last of powder there will be 88 shot good for discharging the Cannon, and the 8/13 is 16 lb of powder over, for pri●●ing and waste, etc. the question answered, the thing required. PROPOSITION XVIII. There is a Rope 3 inches in compass, and one 4 ●imes so big is required: the greatness of these is according to the squares made of their diameters, as in the Problems of this book, their circumferences being also in the same proportion; so the square of 3 is 9, which multiplied by 4 produceth 36, the quadrat root of it is 6 inches, the circumference of the Rope required. PROPOSITION XIX. By knowing the weight of a fathom of any Rope, to find the weight of another either greater or lesser. A Rope in compass 4 inches, and every fathom of it admit does weigh 3 lb, how much shall a Coyler Rope weigh that is 6 inches in circumference; which two circumferences, if multiplied by 7, they would retain the same proportion: and so likewise if those products were divided by 22, as in lib. 2. parag. 7. axiom 13. Arith. then institute the rule of Three, with the circumferences given and squared, viz. as 16 shall be to 3 lb weight for a fathom of that Rope, so will the greater square 36 be in proportion unto 6 ¾ lb, that is 6 lb and 12 ounces for the weight of each fathom of that Coyler rope, whose circumference was 6 inches, the thing required. Many such propositions the Gunners do us●, which for brevity I here omit, supposing these m●y suffice young practitioners, so with strong hopes, and a slight fortification, I will conclude this work. PROPOSITION XX. A Mount or Platform is to be raised for battery, on which the great guns are to be mounted; the General commands the Captain of the Pioneers to draw a trench about it, as he and the Engineers should conceive convenient; which according to order was thus designed: the Platform set out 4 square, 70 ●aces on every side: at the line or verge of this Trench (where the labourers first break ground) 16 feet over, to be 10 feet broad at the bottom, and 8 feet deep; the turf to be orderly laid at the brim, and the earth digged out of the Trench disposed of within that, for a wall to raise the Ordnance, and defend the men within the Wo●ks; which wall is ordered to be made 21 feet at bottom, and 18 feet broad at the top. The query is, how high the wall will be made of the earth digged out? and how many cubical yards is in the said Trench? and what the labourer's work may be worth if paid by the great, or task-work. The breadth of this Trench at the brim is 16 feet, at the bottom 10, the sum 26 feet, the half 13, which multiplied by 8 feet (the depth of the said Trench) the product will be 104 superficial square feet; the wall to be made is to be in thickness 18 and 21 feet, the Arithmetical medium 19 ½ or ●●/2, a● in lib. 2. parag. 5. theo●em 1. Arith. with which divide 104, the quotient will be 5 feet 4 inches, the true height of the wall required. The square of this Platform is 72 Geometrical ●aces, that is, 350 feet, and the 4 sides contains in extent 1400 feet, which multiplied by the square made of the Trenches breadth and depth, as before found 104 feet, the product will be 145600 cubical feet; at each corner of these Trenches there will be a Pyramidal Segment reversed with the greater end upwards, whose mean square is 13 feet, as is the Trench, the quadrat of it 169, which multiplied by 8 the depth, produceth 1352 cubical feet, 4 times that (for the 4 corners) will be 5408. this added to the former sum 145600, the total will be 151008 cubical feet, which divided by 27, the quotient proves 5592 cubical yards contained in the whole trench; which at 6 D the yard to dig and carry on to the works comes unto in money 139 L 16 ss, the true manner of measuring a Segment, and likewise the fraction in the last division was neglected, as unnecessary in these gross works. Each of these Segments contains 50 cubical yards of earth, which may raise a Rampire, Sconce, or Bulwark at each angle of the Platform 6 feet higher, 16 feet square at the bottom, an● 14 f●e● a● the top. diagram As for the marshalling and quartering of Soldiers, with sundry other Military Propositions, I have here instated proposals, and delivered Examples for speculation only, and transferred the form to the judgement of experienced Commanders; since most Propositions (depending on this Subject) are undeterminable, but according to the custom of the Country, the advantage of the Place, the number of Horse or Foot, the Enemy's condition, with multitudes of occurences intervening every day, & those circumspectly to be considered by the field-Officers, or Council of war sitting upon this Tragic Scene, as Germany hath learned by sad experience under the Sword's tuition this later age, whose Disciples have been generally Separates, oppugnant in opinion, yet united and armed with factions have commenced War under specious colours to procure Peace, oppressed the Truth to support Religion, suppressed Kings to establsh Monarchy, and by rude Anarchy, pretending to introduce Civility, with divers such zealous Paradoxes by an Hyperbolical Faith. But leaving all to God, whose Decrees are inscrutable, his Wisdom infallible, his Justice certain, his Mer●y without limit; Infinite and Omnipotent in all his works. To whom be all Honour, Praise and Glory world without end. Amen, THE DEBTOR AND CREDITOR: OR A Perfect Method of keeping Merchants Accounts, after the Italian manner. By Thomas Wilsford. LONDON, Printed for Nath: Brook, at the Angel in Cornhill. 1659. PRECEDENTS OF MERCHANTS ACCOUNTS, In form of DEBTOR & CREDITOR, According to the Italian manner, and the most Modern method Epitomised. THe efficient and final cause of keeping Merchants Accounts after this manner, is for sundry respects in Commerce and Trade, by experience proved urgently necessary in steering an ample course of Traffic, yet waving all doubtful reckon, and avoiding confusion in multiplicity of business, and diversity of affairs, of various natures, which a good form will procure, and produce these effects, viz. The Owner or Cashkeeper may at any time, and upon all occasions, readily find out any contract, either by way of Barter or Money, present pay, time limited, or mixed; also any sum of money, Goods embarked, shipped off, returned, or remaining in their Warehouses, with the quantities, qualities, and value of them; also all Receipts and Disbursments, whether charged or drawn upon their Friends, Factors, Correspondents Accounts, their own, or any Company trading by Sea or Land, with the balancing of their Estates betwixt Debtor and Creditor, as what he owes, or is owing him, with all Bills, Bonds, Obligations, Debts or demands contracted on either side. Any man that gins to drive a Trade or adventure a Stock into foreign parts by way of Traffic, or Commerce in multifacious Negotiation, aught to take an Inventory of his present and personal estate, whether it be in ready Money, Goods, Debts, Wares, or Remainers, attracted by any state, Bonds, Bills, Leases, or Reversions transferred unto him, or by transportation of a trade, from some other Merchant or deceased Friend; which Inventory must be entered in form of Debtor and Creditor, according to the engagements of the Factors, Correspondents, Administrators or Assigns, inscribing the Creditors names, the sum, time, and day of payment; and likewise all such debts as are due to him; for all Contracts, whether by Paroll or Obligation in writing (if without limitation of time) are always due upon demand made. An Inventory. The Title entered, the form of it is usually like a common Bill, bearing date the year of our Lord, with the month and day when the Owners estate was surveyed, his name subscribed thereunto. Upon the right hand or margin of each folio or page make 3 columns, to inscribe the pounds Sterling, shillings and pence, both of the Owners Cash or ready money, with the Commodities, as Lands, Houses, Rents, Revenues, Reversions, Bonds, Bills, and Obligations, etc. according to their values or sums due, also the Wares (if any) in his hands unsold, with their quantities, qualities and values in number, weight and measure. These entered and summed up, take a rveiew of the engagements, as whether in Factorage, Company-accounts, or entrusted for the use of others (expecting his own share or part if any) with all his proper debts, as wares of others unsold, ready money in his hands accountable to others; all Bills, Bonds & Obligations by promises of payment, yet not satisfied; all under the notion of debt-demanders, decreasing the stock. This Inventory is best reserved privately in the owner's possession. Thus having balanced his estate, the party may plainly discover what is his own, and so commence a Trade without confusion, employing what stock he shall think convenient for any Adventure: and divers Merchants do continually keep an Inventory, but usually after this manner, viz. A book in a large Folio, every page hath 3 columns ruled in either margin denoting Pounds, Shillings, and Pence; on the lefthand page his debts are inserted, and on the right hand what is due unto him, and from whom, the money and wares with quantities and qualities inscribed between them; and sundry other books they use, the chiefest they use are these following. The number and names of Books usually kept in great Merchants Accounts, are these: I. A Book for petty expenses, and daily disbursments of trivial sums of money, kept like a Compendium of the Cash-book, and these small accounts collected into one sum, each week and month with a general total every year. II. A Book of Letters, or missive Characters, received, or sent upon public or private business into foreign parts, with the dates thereof, and some breviate of the business. III. A Copy-book of charges at home, or Foreign accounts, whether proper or for company, by assignment for others, or Factorage, with abreviate of Receipts or Acquittances. iv A Book in Octavo of Memorandums to help the memory, containing Bargains and Sales, Promises, Engagements by paroll, or designed affairs in Commerce and Trade, with the year, month, and day, the parties names, etc. V A Cash-book, for inscribing the sums of money in the Cashiers possession, with all receipts and payments, whereby to find what remains in bank at any time, and what debts are due, one inscribed against the other. VI A Wast, or Shop-book, wherein are to be inserted all Wares, Goods, and Commodities arrived or shipped off, received in or delivered out, imported, or exported, and to whom, with the year and day of the month, every parcel distinguished by a line drawn betwixt them, describe in the margin the mark of the said parcel, with some note of reference to the Journal page, and also the number, weight and measure of each parcel, with the quantities, colour, charges, value, or price of them. VII. Besides these there are Diurnals and books kept of Ship-accounts, whether outward or homeward bound, viz. daily occurrences, Ship-expences, charges, and disbursements accidental, etc. VIII. A book of Fraighcage, Cargazones, or bills of Lading, Mariners wages and necessaries for them, with divers other supernumerary Accounts, not commonly kept by all Merchants, nor yet convenient for this Treatise, or my intended design. The scope here aimed at is a compendious form in keeping the Journal and Leager-books, by way of the Italian manner, included by Debtor and Creditor, with divers precedents, in posting and entering the Commodities or Merchandises with the description of those books; for to nominate all those, which some particular Merchant's Adventurers do keep, would make a Catalogue in a poor Scholars Library, and herein superfluous, each book of them being but a relative Index unto the two last. The Diary, or Day-book, aught to be in a large folio; upon the front thereof write the year of our Lord and Saviour in numeral letters, Arithmetical characters or both, than the title of Journal noted with a capital letter, as A B, or C, etc. Thus made conspicuous, the title of each page or parcel within the book is dated with the year, month, and day of any Wares, Goods, or Commodities bought, sold, exchanged, received in, or delivered out: every page on the right hand hath 3 columns in the margin, expressing in money the value of the said Goods or Wares inscribed, in pounds Sterling, shillings and pence, etc. and upon the margin on the left hand one column for 1.2.3.4.5, etc. as quotations to the Leaguers, page and folio. Besides all this, 'tis convenient to enter the mark, number, weight and measure of the Commodities or Parcels, the Debtors and Creditors names, with the time place and manner of payment, or what is convenient to be inserted, in explaining the contract, whether imported or exported Goods, without blotting or interlining any thing. X. The Leaguer is a collection of all the merchants books drawn together in a large folio, charged upon some account in this order: as the book is o●ened, ●lace the Creditors upon the right hand page, and all the Debtors on the left, the pages numbered by 1.1.2.2. so as the Dr. and Cr. make but one folio u on either side in both margins, there are also columns (which bond the matter inscribed) in number various as the Merchants please, or the multiplicity of their employments shall require, whereof I will r●nder some precedents hereafter: the words (most frequently used) in transporting or posting of Wares or Commodities from the Journal or Diary into the Leager-book, are these: In the first place (on the Debtors side) inscribe the word [Too] after which let the Account immediately follow: and on the Creditors part, usually the first word is [By] preceding the name of that Account; and note that every parcel is charged and discharged with the same sum: and observe that most Accounts are best written in one line, or so compendious as you can; some men of very great Commerce and trading keep a Calendar, Register, or an Alphabetical Index, of the names of Men, Wares, Ships and Voyages, with a mutual reference of numbers to these and the Journal-pages, where the Goods are e●●red, according to Debtor and Creditor: and this is always annexed before the Leaguer, either side or page of the Leaguer being noted with one and the same numbers. The Definitions of Debtor and Creditor. By Debtor or Debtors in Merchants books, is understood the account that oweth or stands charged, and the word Creditor or Creditors signifies the discharge. So all things received, or the Receiver is always made Debtor; the things delivered, or the Deliverer, is the Creditor: all which are compendiously comprised by some Accountants according to these following breviates. And thus stands all Bills, Bonds, Obligations of things lent, or promises to lend, or cause to be lent or paid, included in the same predicament with the two former (whether simple or mixed) the foundation of this structure. What Debtor and Creditor contains in sum, unfolded in the Merchant's books, viz. All Goods, Men, Money, Voyages, Ships, Cargazones, Bills of Exchanges, Wares, or Commodities, etc. with Profit and Loss, are contracted into two kinds, viz. Debtor and Creditor, and those contained in these Predicaments under 12 Species, whether proper for Company, Factorage, Domestic or Foreign, or mixed with the other two. How these are comprehended under the notion of Debtors or made so by Commerce and Trade. I. Lands, Rents, Revenues, Money, Wares, or Commodities, either in possession or shipped away to foreign parts, with all those who are engaged to pay or deliver, must be inscribed in the Merchant's Books, Debtors to the Owners, to their Cash or Stock. II. Whosoever receiveth, or the Goods received, whether Money or Merchandizes, upon their own proper account, for Factorage or Company, w●ite them, o● the Wares Debtors. III. If a man delivers Wares, or pays money, Bills or Exchanges upon the account of another, that party (upon whose account 'tis paid) becomes Debtor. iv A man who delivers an Assignation in payment, whether his own or not, but for the use of another, than the party (upon whose account 'twas delivered) is made Debtor. V Money received that was taken up at Interest, then is Cash commonly written for the Principal, Debtor; but the Loan, or Profit and Loss, for the Interest money is made Debtor. VI An Adventurer or any other for him, that sends merchandizes unto a foreign place or Region, whether proper or for any Company, consigned to a Factor or Resident there, the Voyage or Ship is written Debtor. VII. A Merchant that insures any wares, and receives the money presently, then is the Insurer or Cash written Debtor; but in case the Insurance be not immediately paid, then is the party (for whom they were insured) Debtor for the Insurance reckoning. VIII. Upon advice that any Goods insured for Proper, or Company-account, and those shipped to sea were cast away in part, or all, the Insurer or Insurance reckoning is Debtor; but if otherwise, than Profit and Loss are subscribed Debtors. IX. Upon Returns or Advice from a Factor, that the shipped wares were received, whether for Proper or Company-account, enter the Factor there resident Debtor. X. A Factor that draws an Echange upon the Merchant, Company, himself, or others; to the party or parties (u●on who●e account it was drawn) inse t Debtor, and the like upon Exchanges remitted, the acceptor (who must discharge it) is Debtor. XI. A Merchant or Company that loses by the sale of Wares, by Bankrupts, Exchanges, Insurances, Interests, Gratuities, or whatsoever proves a detriment in Commerce and Trade, to all or any of these subscribe profit and loss— Debtor. XII. In balancing of Accounts, if there be money or wares in the house unsold, or in the possession of their Partners and Correspondents, who have not rendered satisfaction; then must People and Cash, b●th in the old and new books, be written— Debtors. Creditors in Merchants Accounts are generally but reconversions from Debtors discharged, or the Principal, included briefly in these 12 Species. I. Stock or Cash for wares unsold, Creditor and the people, to whom any one is indebted; whose names are distinctly to be specified under the notion, or by the subscription of Creditor. II. What things soever a Merchant delivers, or engages to be delivered, whether for Proper, Factorage, or Company-account in money or wares, either the goods delivered or the party promised is Creditor. III. Any man that delivers money, Wares Exchanges or Assignations upon another's account, that party or parties (upon whose account 'tis received) mu●t be subscribed Creditor. iv A Merchant borrows money at interest, which being received, the party who delivers it, or the lender, is Creditor for the Principal, and enter the Loan reckoning or profit and loss for the interest money Creditor. V A man having a Principal of another's in his possession, the time of payment expired, and yet detained, the owner of the principal is for the interest of that time only to be written Creditor. VI A Merchant receives advice from his Factor of the sent Goods received and sold, or not, enter then Voyage to such a place, consigned to his correspondent Creditor. VII. Any Wares or Adventures safely arrived, the insurance not paid, but customs and charges defrayed, inscribe one of these, viz. Cash, Charges, the Insurer, Insurance reckoning, or Profit and Loss, Creditors. VIII. Goods or Commodities insured for Proper, or Company-account, shi●t to sea, and by misfortunes cast away, as by letters from the Factor or Resident appears; subscribe Voyage to the place consigned, to such a man for Proper or Company account Creditor. IX. A Me●chant from his Factor or Resident receives Returns in money or merchan●izes, in lieu of the Wares received and sold, whether for Proper or Company account, the Factor or Correspondent that caused these goods to be delivered is Creditor. X. An Exchange drawn by a Merchant upon his Factor for his Proper, Company, or any others account, or remitted; then the party or parties for whose account it is drawn, charged, sent or remitted, make Creditor the correspondent Debtor. XI. A Merchant or Company that gains by Gratuities, the sale of Wares, Exchanges, Insurances, Interests, or what things are beneficial in Commerce and Trade, to all or any these subscribe Profit and Loss— Creditor. XII. In balancing of Books, Factors, Partners, and all others unsatisfied (if Fortune hath favoured the Merchant) Stock and the People will be in the old and new books Creditor. To shun mistakes betwixt Debtor and Creditor, and not to be prolix, I prescribe these rules and instructions, more compendiously than others have delivered them; yet those I have seen in some copies epitomised (like the good works of this Age) easily to be remembered, without over-fraighting the Readers memory, which for the assistance of young men I will render in this breviate a transcription of others presented to your view in two tables, one balancing the other. The Debtors or whatsoever oweth are 1 What Goods we have 2 Who receives any thing 3 All Wares that we buy 4 The men to whom we sell 5 Those for whom we buy 6 Those for whom we pay 7 The men who are to pay 8 Goods that are insured 9 Those for whom we insure 10 Voyages where we send 11 Goods on which is gained 12 Profit and Loss The Creditors or what is to receive are 1 From whence it ariseth 2 Who delivers any thing 3 Those of whom we buy 4 The Goods which are sold 5 The parties which sell 6 Goods wherewith we pay 7 Those who are to receive 8 The party who insureth 9 The insurance reckoning 10 Goods sent or shipped off 11 Goods on which is lost 12 Profit and Loss. The Title of the Journal. Anno first January 1658. In LONDON. The form of regulating a Journal page, with the marginal quotations specified, as thus. L S D Dr 1 The pages prepared according to th● Diaries former description, yet in multiplicity of affairs many do make two columns in the left margin, the first for to insert Debtor or Creditor, the other with figures in reference unto the Leaguers page to show where each party or parties, parcel or parcels stand charged or discharged. And where any party or parcel is discharged, draw a line between the terms in form of a fraction, placing Debtor ●bove the said line, and Credito● beneath it, with the figures or numbers of the Leaguers page; one showing they are discharged, and the other where. But as for this, the figures only in most Journals are conceived to b● sufficient directions, as in the following Diary our Journal shall be made conspicuous. 000 0 0 Cr 2 Dr 1 000 0 0 Cr 2 000 0 0 Here insert the year of our Lord. ●he journal page. The Title of the Leaguer, Anno 1658. in LONDON. The form of ruling the Leaguer. Cash Debtor. Where Creditor in the L●ager. L S D 1658 1 First Jan. Creditor to stock for several Coins of money. 1 800 00 00 Which signifies no more but the figure 1 in the second column on the left hand points to you that Cash in the first page of the Journal is made Debtor, and the figure 1 in the first column on the right hand tells you that Cash has his Creditor entered in folio 1, viz. Stock is Creditor as may appear by the said question in the said page of the Journal and page of the Leaguer the like, mutatis mutandis is to be observed in the Creditors side of the Leaguer, will be conspicuous in the Leaguer hereunto annexed. THE JOURNAL; Number A in LONDON. 1658. Anno 1658. first Jan. in London. L S D The form of inscribing Debtors and Creditors in the Inventory. Dr Gr ●/● CAsh Debtor to Stock 800 l for several coins of gold and silver remaining, as by conclude of my former books appear, which amount to in Sterling money to 800 Goods or Merchandizes in the warehouse, or otherwise in my possession, and, are unsold. ●/1 Fustians Debtor to Stock 226 l 13 s 4 d. for 200 Pieces resting unsold, which cost 22 s. 8 d. per piece 226 13 4 ●/1 Spanish Tobacco Dr. to Stock 293 l 6 s 4 d. for ten Potacoos remaining unsold, weighing neat 880 poundat 6 s 8 d per pound, is 293 06 4 ●/1 Colchester Says Debtor to stock 34 l 6 s 8 d. for 20 Pieces unsold, which cost 34 s per Piece 34 6 8 2/1 Couchaneal Debtor to Stock 230 l 11 s 3 d. for 14 C 3 qu. 14 po. neat at 15 l 10 s per C. 230 11 3 Ships, Parts, Houses, Land, etc. Inventored. ●/1 Ship the Samson of London Debtor to stock 200 l for my ●/8 part thereof, which cost me 200 ●/1 House the Nag's head at Rumford Dr. to stock 350 l for the principal, it cost me 350 Money due upon Bond, Bill, or other agreement, insert thus. 3/1 Abraham Bland debtor to stock 387 l 11 s. payable the last of this month, as appears by his Obligation dated the 5. of December last 387 11 ●/1 Thomas Goodman debtor to stock 82 l 3 s 4 d, as by his bill under his hand appears payable on demand 82 3 4 3/1 William Lane of Ipswich debito to stock 36 l 11 s 4 d being the rest of an old account due on demand 36 11 4 Money due to others from us by bond, bill, or agreement. ¼ Stock debtor to Barnaby Clemens 300 l being so much due by my Obligation dated the 3. of November last, and payable the 5. of March next 300 ¼ Stock debtor to Thomas Spilman 36 l 11 s 4 d. by my bill payable on demand, for certain householdstuff bought of him, the particulars appears by his bill 36 11 4 ¼ Stock debtor to John Maz'oon 305 l payable on demand 305 Here ends the Inventory. Anno 1658. 3 Jan. in London. L S D The form of inserting Debtors and Greditors in Traffics continuace, and first of goods sold for ready money, secondly for time. ½ Cash debtor to Fustians 24 l 14 s. for 13 pieces sold to John Deport at 38 ● per Piece 24 14 4/2 John Thurrowgood of Chester debtor to Fustians 39 l 10 s for 20 Pieces sold 39 s per Piece, payable within 30 days 39 10 Jan. 15. 1658. Goods sold part for time, part for ready money. 4/2 John Benning debtor to Couchaneal 51 l 15 s for 3 C neat, at 17 l 5 s per cent. ½ at 6 month's rest instant 51 15 ¼ Cash debtor to Dito John 25 l 17 s 6 d. received in part this day 25 17 6 Goods bought, paying present money. 5/1 Led debtor to Cash 204 l 11 s 3 d for 95 Pigs, weighing 13 ton 12 C 3 qu. at 15 s per C. 204 11 3 Wares bought not paying present money. ¾ Claret Wines debtor to Jane de Clare for 5 ton at 12 l per ton 60 Anno 1658. 15 Jan. in London. L S D Commodities bought part for time, part to be paid ready money. 5/5 Sugars Debtor to James Wilson 54 l 7 s 4 d ½ for 10 hh. wai. gro. 42 C 3 qu. 11 pound tore, of each hh. 22 po. neat, 30 C 0 qu. 23 po. at 30 s per cent. all duties cleared, ⅓ to pay money ⅔ at a month 54 7 4½ 5/1 Dito James Debtor to Cash 18 l 2 s 5 d ½ for ⅓ paid in part 18 2 5½ Jan. 19 1658. Wares sold and delivered in Barter. 5/5 Edward Price Debtor to Sugars 12 l 12 s for 2 hh. wad. net. 6 C at 42 s per hh. to receive for the same Province oils in barter. 12 12 6/5 Province Oils Debtor to Dito Edward 9 l 3 s. for 1 Cask taken, content in part in truck for Sugars 09 03 ⅕ Cash Debit. to Dito Edward 3 l 9 s. received in full satisfaction to clear the said Truck 3 9 Jan. 29. 1658. Moneys borrowed at interest. ⅙ Cash Debtor to John Malthorse 300 l received of him upon interest 300 6/6 Interest reckoning our Profit and Loss Debtor to Dito John 4 l 10 s, for 3 month's allowance for 300 l at 6 per cent. 4 10 Anno 1658. 30 Jan. in London. L S D Moneys borrowed to be paid upon demand, or otherwise. ⅙ Cash Debtor to Simeon Peter's 50 l borrowed of him and payable on demand 50 Money let out at Interest. 5/1 Edward Price Debtor to Cash 100 l. delivered him at Interest 100 ⅚ Dito Edward Debtor to Interest reckoning or profit and loss 1 l 10 s for 3 month's allowance for a 100 l at 6 per cent. per ann. 1 10 House-keeping charges entered. 6/1 Profit and Loss Debtor to Cash 50 l paid my servant for 3 month's provision for house-keeping to end 25 March next 50 Febr. 2. 1658. Moneys paid that are due, the time of payment being com●. 4/1 Jane de Clare Debtor to Cash 60 l. paid him in full of all accounts 60 5/1 Jam. Wilson Debtor to Cash 36 l 4 s 11 d. paid in full for his Sugars as by his Acquittance appears 36 4 11 Febr. 9 1658. Moneys received that are due, time of payment b●ing come. ¼ Cash Debtor to John T●urrowgood 39 l 10 s. received in full of all demands to this day 39 10 Moneys received (for goods sold at time before due) upon rebate. ¼ Cash Debtor to John Benning 24 l. Received in full 24 6/4 Profit and Loss debtor to Dito John 1 l 17 s 6 d for payment before due 1 17 6 Anno 1658. 12 Febr. in London. L S D Moneys gained by Exchange. ⅙ Cash debtor to Profit and Loss 58 s 4 d for advance of 100 Dollars exchanged for English money at 4 s 7 d per Piece, which cost me but 4 s. the difference at 7 d per Piece is 2 18 − 4 To discharge a debt by Assignment. 4/3 Thomas Spilman debtor to William Lane for my Assignment, poid h●m in full of his debt 36 11 − 4 Debt sold to another from whom it was not due. ⅓ Cash debtor to Thomas Goodman 50 l, Received of William Short for Dito Thomas his Bill of 82 l 3 s 4 d. which I have sold for 50 6/3 Profit and loss debtor to Dito Thomas 32 l 3 s 4 d lost by the sale of his bill 32 − 3 − 4 Febr. 19 1659. Part of a debt lost by a Bankrupt and the rest received. ⅓ Cash debtor to Abraham Bland 259 l 7 s 4 d being but ⅔ Received in full of a debt of 387 l 11 s ⅔ 258 − 7 − 8 6/3 Profit and Loss debtor to Dito Abraham 129 l 3 s 8 d. lost by him when he failed − 129 − 3 Merchandizes sent into another Country, consigned to a Factor for my Account. 7/2 Voyage to Amsterdam consigned to Hans Butter box debtor to Fustians 80 l for 40 pieces at 40 s per piece, shipped by James Hope to Dito Hans to be sold for my account 80 7/2 Dito Voyage consigned to Dito Hans debtor to Spanish Tobaco 440 l for so Pot●en's w●d net 880 l at 10 s per pound, shipped by Dito James to the said Hans to be sold for my account − 440 Anno 1658. 19 Feb. in London. L S D Charges for a voyage, or otherwise. 7/1 Dito voyage debtor to Cash 9 l 16 s for charges upon the Fustians and Tobaco, for Fraight, Custom and Excise, etc. 9 16 Goods insured. 7/1 Dito voyage debtor to Cash 7 l 16 s for insurance of Dito Fustians and Tobacco, paid John Mazoone at 30 s per cent. is 7 16 Feb. 22. 1658. Money received for freight of a ships part. ⅓ Cash debtor to Ship the Samson of London, for freight received of John Wright Master thereof, for my ½ share 35 3 8 Money or goods given away to any person. 7/2 Profit and Loss debtor to Fustians 1 l 2 s 8 d for one piece given to A.B. 6/1 Profit and loss debtor to Cash 12 l given towards the relief of a fire at Enfield in the County of Middlesex 1 2 8 Gratuities received. ⅙ Cash debtor to Profit and Loss for 200 l received for a Legacy given m● by A. M. 12 Febr. 28. 1658. Commodities formerly shipped to another Country, and advice of the sale thereof. 7/7 Hans Butterbox at Amsterdam, my account currant, debtor to voyage to Amsterdam 613 l 15 s 4 d. as appear● by his account sent me, and dated at Amsterdam, Novemb. 25. instant, being the neat proceed of my goods sol● there 613 15 4 Anno 1658. 28 Febr. in London. L S D Commodities received to sell for another man's account in Commission. 7/1 Hans Butterbox at Amsterdam his account of Wheat debtor to Cash 18 l 5 s for several charges paid at the receipt of 100 quarters (received out of the Elephant of Amsterdam) as followeth: L S D For Fraight, at 1 s per qu. 05 00 00 For Custom, at 1 s 6 d per q. 07 10 00 For Excise, at 6 d per qu. 02 10 00 For Porterage, Literage and Cartage, at 6 d per qu. 02 10 00 For Meaters allowance 00 15 00 18 05 00 18 5 Commission Goods sold. 1/7 Cash debtor to Hans Butterbox his account of Wheat for 100 quarters so●d to John Sutton Junior at 58 s per qu. is 290 Provision for Commodities sold. 7/6 Hans Butterbox his account of Wheat debtor to profit and loss for my provision for the said Wheat at 12 d per l. 14 10 March 3. 1658. The proceed of commodities paid by bill of Exchange. 8/● Hans Butterbox his account currant debtor to Cash 257 l 5 s. remitted him for his account in Bills of Stephen Swabbers payable at sight to dito Hans by Simon Newman of Amsterdam, being the neat proceed of his wheat, all charges deducted 257 5 Anno 1658. 24. March in London. L S D Commodities bought for Companies Accounts. 8/8 Tobacco in Company between John Mazoon and myself debtor 210 l foe 30 hh of Vi●ginia each ½, the whole a● 7 l per hh is 210 ⅛ Cash debtor to John Mazoon his account by me in Company for his ½ part received 105 Commodities sold for Company Account. ⅛ Cash debtor to Tobacco in Company between John Mazoon and my sel● each ½ for 30 hh sold to Henry Beak at ●0 l per hh. 300 8/● Tobacco in Company debtor to John Mazoon his account by me in company 45 l for ½ advance 45 8/● Tobacco in Company debtor to profit and loss 45 l for my ½ of advance gain by the sale of the Tobaccoes 45 8/4 John Mazoon his account by me in Company debtor to dito John his account proper 150 l, v●z. for his principal brought in, and gains thereof made good upon his particular account 150 Anno 1659. 25 March in London L S D Order of Balancing. 2/9 Balance debtor to Hans Butterbox my account currant 613 l 15 s 4 d. due to me in ready money 613 15 4 4/9 John Mazoone debtor to Balance 455 l. due to him by conclude 455 4/9 Barnaby Clemens debtor to balance 300 due to him 5 March last, by my obligation 300 9/5 Balance debtor to Edw: Price 101 l 10 s due to me the 30 of April next 101 10 6/9 John Malthorse debtor to balance 304 l 10 s due the 29 of April next 430 10 7/6 Voyage to Amsterdam debtor to profit and loss 76 l 3 s 4 d. gained by sale of goods there 76 3 4 ⅞ Hans Butterbox his account of Wheat debtor to his account currant 256 l 5 s. being the neat proceed of his wheat 256 5 9/2 Balance debtor to Fustians 142 l 16 s for 126 pieces remaining unsold, at 22 s 8 d per piece 142 16 2/6 Fustians debtor to profit and loss 61 l 9 s 4 d gained by the sale of 73 pieces 61 9 4 2/6 Spanish Tobacco debtor to profit and loss 146 l 13 s 8 d gained by the sale of 10 Potacoes 146 13 4 9/2 Balance debtor to Colchester Says 34 l 6 s 8 d for 20 pieces unsold, at 34 s per piece 3 6 8 9/2 Balance debtor to Couchaneal 184 l 1 s 3 d for 11 C 3 qu. 74 pound, at 15 l 10 s per C. unsold − 184 1 3 ●/6 Couchaneal debtor to profit and loss 5 l 5 s gained by the sale of 3 C 5 5 9/5 Balance debtor to Led 204 l 11 s 3 d for 95 Pigs wa. 13 tun 12 C 3 q. at 15 s per cent. resting unsoed 204 11 3 9/5 Balance debtor to Claret wine 60 l for 5 tuns, at 12 l per tun resting unsold 60 9/5 Balance debtor to Sugars 43 l 11 s 4 d ½ for 24 C oh qu. 23 pound, at 36 shillings per cent. being unsold 43 11 4 ½ ⅙ Sugars debtor to profit and loss 1 l 16 s gained by the sale of 6 C. 1 16 9/6 Balance debtor to Province Oils 9 l 3 s for 1 Cask taken content unsold 9 3 6/6 Profit and loss debtor to Interest reckoning 3 l lost by giving of interest for money 3 9/3 Balance debtor to Ship the Samson of London 200 l for my ⅛ part thereof 200 ●/6 Ship Samson debtor to profit and loss 35 l 3 s 8 d. gained by freight 35 3 8 9/3 Balance debtor to the Nag's head at Rumford 350 l for the principal worth of it 350 9/1 Balance debtor to Cash 1561 l 4 s 1 d resting in hand 1561 4 ½ THE LEAGUER: NUMBER A. Anno Dom. 1659. in LONDON. The form of the Calendar belonging to the Leaguer. A Abrah: Bland. fol. 1 B John Benning. 4 Hans Butterbox my account currant. 7 Dito Butterbox his account of Wheat. 7 Dito Butterbox his account currant. 8 Balance. 9 C Cash. fol. 1 Couchaneale. 2 Barnaby Clemens. 4 Jane de Clare. 4 D E F Fustians. fol. 2 G Tho: Goodman. fol. 3 H House Nag's head. f. 3 I Interest reckoning. f. 6 K L William Lane. fol. 3 Led 5 M John Mazoon. fol. 4 John Malthorse. 6 John Mazoon his account by me in company. 7 N O Oils Province. fol. 6 P Edward Price. fol. 5 Simon Peter. 6 Profit and Loss. 6 Q R S Stock. fol. 1 Says. 2 Ship Samson. 3 Thomas Spilman. 4 Sugars. 5 T Tobacco. fol. 2 John Thurrowgood. 4 Tobacco in Company for John Mazoone and myself each ½. 8 V Voyage to Amsterdam. fol. 7 W Wines Claret. fol. 5 James Wilson. 5 X Y Z L S D Cash Debtor. 1658 1 First Jan. to stock for several coins amounting to Sterling money 1 − 800 3 3 Dito to Fustians for 13 pieces sold to John Deport at 30 s per piece 2 24 14 15 Dito to John Benning, received in part 4 25 17 6 4 19 Dito to Edward Price received to clear a Truck 5 3 09 29 Dito to John Malthorse received at interest at 6 per cent. per an. 6 − 300 5 30 Dito to Simon Peter's being so much borrowed of him 50 9 Feb. to John Thurrowgood received in full demand 4 39 10 Dito to John Benning received in full upon rebate 24 6 12 Dito to profit and loss fo● gain in exchange of 100 Dollars 6 2 18 4 2 Dito to Thomas Goodman, received of William Short in full 3 50 19 Dito to Abraham Bland being 13 s 4 d per l for 387 l 11 s 3 − 258 07 4 7 22 Dito to Ship Samson for my ⅛ part for freight 35 03 8 Dito ●o profit and loss for a Legacy ●●●eiv'd, given me by A. M. 6 − 200 8 29 Dito to H. Butterb. his account of Wheat for 100 qu. sold at 58 s per qua●●e● ●7 − 290 9 24 Mar●●, to John Mazoon his account by me in company for his part received 8 − 105 - Dito to Tobacco in company for 30 ●●●●at 10 l per hh − 300 2508 19 10 Contra Creditor. L S D 1658 3 15 Jan. by Lead for 95 Pigs wa. 13 T. 12 C. 3 qui at 15 s per C. 5 204 ●1 3 5 - Dito by James Wilson for ⅓ of his debt paid him in part 5 18 2 5 ½ 30 Dito by Edw. Price delivered him upon interest at 6 per c. 3 more. 5 100 - Dito by profit and loss for house-keeping for 3 months 6 50 5 2 Feb. by Jane de Clare, paid him in full of all demands 4 60 5 - Dito by James Wilson paid him in full of all demands 5 36 4 11 7 19 Dito by Voyage to Amsterd. paid charges of Fustians & Tobac. 7 9 16 7 - Dito by dito voyage for Insurance at 30 s per cent. 7 7 16 7 22 Dito by profit and loss given in relief for a fire 6 12 8 28 Dito by H. Butterb. his account of Wheat at the receipt for charges 7 18 5 8 3 March, by H. Butterb. his account currant remitted him by bills 8 257 5 9 24 Dito by Tobac. in Company, paid in full for 30 hh at 7 l per hh 8 210 11 1659. 9 25 Dito by balance resting in Cash 9 1524 19 2 ½ 2508 19 10 Stock Debtor. 1658 2 1 Jan. to Barnaby Clemens by my Obligation due 15 March 4 300 6 2 Dito to Thomas Spilman due to him on demand 4 36 11 4 2 Dito to John Mazoon payable on demand 4 305 1659. 25 March, to balance for conclude carried thither 9 2359 4 1 3000 15 5 Contra Creditor. 1658 1 1 Jan. by Cash for several Coins of money amounting to Sterling money 1 800 - Dito by Fustians for 200 pieces cost 22 s 8 d per piece 2 226 13 4 - Dito by Spanish Tobaco for 10 Potatoes wa. net. 880 pound, at 6 s 8 d per pound 2 293 6 ● - Dito by Colchester Says, for 20 pieces unsold at 34 s per piece 2 34 6 8 - Dito by Couchancal for 14 C 3 qu. 14 po. at 15 l 10 s per C. 2 230 11 3 - Dito by Ship the Samson of London, for my ⅛ part 3 200 - Dito by house the Nagshead at Rumford, for what it cost 3 350 - Dito by Abraham Bland, for his Obligation due the last instant 3 387 11 - Dito by Thomas Goodman for his Bill 3 82 03 4 - Dito by William Lane being the rest of an old account 3 36 11 4 1659. 25 March, by profit and loss gained by this 3 months trading 6 359 12 2 3000 15 5 L S D Fustians Debtor. 1658 1 1 Jan. to stock for 200 pieces which cost 22 s 8 d per Piece 1 226 13 4 1659. 25 March to profit and loss gained by the sale of 73 Pieces 6 61 09 4 288 2 8 Spanish Tobaco Debtor. 1658 1 1 Jan. to stock for 10 Potacoes wa. net. 880 pound, at 6 s 8 d per pound 1 293 6 4 1659. 25 March to profit and loss gained by the sale of dito Tobaco 6 146 13 8 440 Colchester Says Debtor. 1658 1 1 Jan. to stock for 20 Pieces cost 34 s per piece 1 34 06 8 Couchaneale Debtor. 1658 1 1 Jan. to stock for 14 C. 3 qu. 14 pound, at 15 l 10 s per C. 1 230 11 3 1659. 25 March to profit and loss gained by sale of 3 C. 6 5 05 235 16 3 L S D Contra Creditor. 1658 3 3 Jan. by Cash for 13 pieces sold to John Deport at 38 s. pe● piece 1 241 14 3 - Dito by John Thoroughgood of Chester for 20 pieces at 39 s per pi ce t 30 days 4 39 10 6 19 Feb. by voyage to Amsterdam to be sold for my account, 40 pieces at 40 s. per piece 7 80 7 22 Dito by profit and loss for one piece given away to A. B. 6 1 2 8 1659. 25 March by balance for 126 pieces unsold at 22 s. 8 d. per piece 9 148 16 288 02 8 Contra Creditor. 1658 6 19 Feb. by voyage to Amsterd. consigned to Hans Butterbox to be sold for my use 10 Potacoes of Spanish Tobaccoe wa. net. 880 pound, at 10 s. per po. 7 440 440 Contra Greditor. 1659. 25 March by balance for 20 pieces resting uns●●d, which cost 34 s. per piece 9 34 6 8 Contra Creditor. 1658 3 15 Jan. by John Benning for 3 C. at 17 l. 5 s. per C. ½ instant, rest at 6 months 4 51 15 1659. 25 March by balance for 11 C. 3. qu. 14 pound unsold, cost 15 l. 10 s. per C. 9 148 1 3 235 16 3 L S D Ship the Samson Debtor. 1658 1 1 Jan. to stock for my ⅛ part, which cost me 1 200 1659. 25 March, to Profit and Loss gained by freight 6 35 3 8 235 3 8 House the Nag's head at Rumford Debtor. 1658 1 1 Jan. to stock for the principle it cost me 1 350 Abraham Bland Debtor. 1658 2 1 Jan. to stock for his bond dated 5 Decemb. last, due the last of this month 1 387 11 Thomas Goodman Debtor. 1658 2 1 Jan. to stock for his bill to be paid on demand 1 82 3 4 William Lane of Ipswich Debtor. 1658 2 1 Jan. to stock, being the rest of an old Account 1 36 11 4 L S D Contra Creditor. 1658 7 22 Feb. by Cash received for my ⅛ part for freight 1 35 3 8 1659. 25 March, by Balance for my ⅛ part thereof 9 200 235 3 8 Contra Creditor. 1659. 25 March by Balance for the principal it cost 9 350 Contra Greditor. 1659. 6 19 Feb. by Cash received at 13 s 4 d per l for 20 s 1 264 7 4 16 - Dito by Profit and Loss lost by him when he failed 6 123 3 8 387 11 Contra Creditor. 1658 6 12 Febr. by Cash received of Will: Short in full of my said debt 1 50 6 Dito by Profit and Loss lost by the sale of my said debt to dito William 6 32 3 4 82 3 4 Contra Creditor. 1658 6 12 Feb. by Tho: Spilman ordered to receive of dito William 4 36 11 4 L S D Barnaby Clemens Debtor. 1659. 25 March to balance due to him by my obligation the 5 of March last 9 300 Thomas Spilman Debtor. 1658 6 12 Feb. to William Lane for my assignment 3 36 11 4 John Mazoon Debtor. 1659. 3 25 March to balance due to him by conclude 9 455 John Thoroughgood Debtor. 1658 3 3 Jan. to Fustians for 20 pieces ●t 39 s. per piece, to be paid with●n 30 days 2 39 10 John Benning Debtor. 1658 3 15 Jan. to Couchaneal for 3 C. neat, at 17 l. 5 s. per C. ½ instant rest at 6 months 2 51 15 Jane de Clare Debtor. 1658 5 2 Feb. to Cash paid in full of all demands 1 60 L S D Contra creditor. 1658 2 1 Januar. by stock payable 5 March next 1 300 Contra creditor. 1658 2 1 Jan. by stock payable on demand 1 36 11 4 Contra credito. 1658 2 1 Jan. by stock payable on demand 1 305 8 24 March by his account by me in company 8 150 455 Contra cred tor. 1658 5 9 Feb. by Cash received in full of all demands 1 39 10 Contra creditor. 1658 3 15 Jan. by Cash paid in part 1 25 17 6 5 9 Feb. by Cash received in full upon rebate 24 - Dito by profit and loss allowed for payment before due 6 1 17 6 51 15 Contra creditor. 1658 3 15 Jan. by Claret wines being 5 Tun, at 12 l. per Tun 5 60 L S D Lead Debtor. 1658 3 15 Jan. to Cash paid for 95 Pigs wa. 13 Tun 12 C. 3 qu. at 15 s per C. 1 204 1● 3 Claret Wines Debtor. 1658 3 15 Jan. to Jane de Clare for 5 ●uns at 12 l per tun 4 60 Sugars Debtor. 1658 4 15 Jan. to James Wilson for 30 C. 0 qu. 23 pound neat at 36 s per C. ⅓ instant, rest at 9 months 5 54 7 4 ½ 1659. 25 March to Profit and Loss gained by the sale of 6 C neat 6 1 16 56 3 4 ½ James Wilson Debtor. 1658 4 15 Jan. to Cash for ⅓ paid him in part. 1 18 2 5 ½ 5 2 Feb. to Cash paid in full of all demands 36 4 1 54 7 4 ½ Edward Price Debtor. 1658 4 19 Jan. to Sugars for 2 hh wa. neat 6 C. at 42 s per C. 5 12 12 5 30 Dito to Cash dd. him upon interest at 6 l per C. 1 100 - Dito to Interest-reckoning for allowance of dito C l. for 6 months 6 1 12 1●4 2 L S D Contra Creditor. 1659. 25 March by Balance for 95 Pigs resting unsold 9 204 11 3 Contra Creditor. 1659. 25 March by Balance resting unsold, 5 tuns at 12 l per tun 9 60 Contra Creditor. 1658 4 19 Jan. by Edward Price for 6 C. at 42 s per C. in barter 5 12 12 1659. 25 March by Balance for 24 C 0 qu. 23 l resting unsold 9 4● 11 4 ½ 56 3 4 ½ Contra Creditor. 1658 4 15 January by Sugars for 30 C. 0 qu. 23 l. neat at 36 s per C. ⅓ instant, rest at 1 month 5 54 7 4 ½ Contra Creditor. 1658 4 19 Januar. by Province Oils 1 Cask received content 6 9 3 Dito by Cash paid in full satisfaction of dito truck 1 3 9 1659. 25 March by Balance due to me 30 April next 9 10 10 114 2 L S D Province Oils debtor. 1658 4 To Edward Price for 1 Cask taken content 5 9 3 John Malthorse debtor. 1659. 25 March to Balance due to him 29 April next 9 304 10 Interest reckoning debtor. 1658 4 29 Jan. to John Malthorse for 3 month's allowance of 300 l. at 6 per cent. for 3 months 6 4 10 Simon Peter debtor. 1659. 25 March to Balance due upon demand 9 50 Profit and loss debtor. 1658 5 30 Jan. to Cash paid A B. my servant for 3 months house-keeping 1 50 9 Feb. to John Benning allowed for payment before due 4 1 17 6 6 12 Dito to Thomas Goodman lost by sale of his debt to Will: Short. 3 32 3 4 19 Dito to Abraham Bland lost by taking 13 s. 4 d. per l. when h● failed 3 129 3 8 7 22 Dito to Fustians for 1 piece given to A.B. 2 1 2 8 - Dito to Cash given towards the relief of a Fire at Enfield in ●he County of Middlesex 1 12 1659. 25 March to Interest reckoning lost by the same 6 3 - Dito to Stock gained by this 3 months trading 1 359 12 2 5●9 19 4 Contra Creditor. L S D 1659. 25 March by Balance for one Cask unsold 9 9 3 Contra Creditor. 1658 4 29 Jan. by Cash received upon interest at 6 per cent. for 3 months 1 300 - Dito by Interest reckoning for 3 month's allowance for 300 l. at 6 per cent. 6 4 10 Contra Creditor. 304 10 1658 5 29 Jan. by Edward Price for allowance of 100 l. for 3 months at 6 per cent 5 1 10 1659. 25 March by Profit and Loss lost per sum 6 3 Contra Creditor. 4 10 1658 5 30 Jan. by Cash borrowed and payable on demand 1 50 Contra Creditor. 1658 6 12 Feb. by Cash gained by exchange of 100 dollars for Eng. more. 1 2 18 4 7 22 Dito by dito given as a Legacy by A. M. 200 8 28 Dito by Hans Butterbox hi● account of Wheat for my provision at 12 d. per l. 7 14 10 24 March by Tobaccoe in company for ½ of my advance 8 45 1659. 25 Dito by voyage to Amsterd. gained by the sale of goods there 7 76 3 4 - Dito by Fustians gained by the sale of 73 pieces 2 61 9 4 - Dito by Spanish Tob. gained by the sale of 10 Potacoes 14● 13 8 - Dito by Couchaneale gained by the sale of 3 C. 5 5 - Dito by Sugars gained by the sale of 6 C. 1 16 - Dito by Ship the Samson gained by freight 3 35 3 8 588 19 4 L S D Voyage to Amsterdam configned to Hans Butterbox Deb. 1658 6 19 Feb. to Fustians for 40 Pieces at 40 s per Piece 2 80 - Dito to Spanish Tobacco for 10 Potacoes neat 880 pound at 10 s per pound, to be sold for my account 440 7 - Dito to Cash for charges upon the Fustians and Tobacco 1 9 16 - Dito to Cash for Insurance John Mazoon at 30 s per cent. 7 16 1659. 25 March to Profit and Loss gained by sale of dito goods 6 76 3 4 613 15 4 Hans Butterbox at Amsterdam my Account currant debtor. 1658 7 2 Feb. to Voyage to Amsterdam for the neat Proceed of my goods there 7 613 15 4 Hans Butterbox at Amsterdam his account of Wheat Deb. 1658 8 28 Feb. to Cash for charges at the receipt of 100 quarters 1 18 5 - Dito to Profit and Loss for my provision at 12 d per l. 6 14 10 1659. 25 March to dito Hans his account currant for the neat proceed thereof 8 257 5 290 L S Contra creditor. 1658 7 28 Feb. by Hans Butterbox my account currant 7 613 15 Contra creditor. 1659. 25 March by Balance due to me in ready money 9 613 15 4 Contra creditor. 1658 8 29 Feb. by Cash for 100 quarters sold to John Sutton at 58 s per qu. 1 290 L S D Hans Butterbox at Amsterdam his account currant debtor. 1658 8 3 March to Cash remitted him payable by Simon Newman of Amsterdam 1 257 5 Tobaccoe in company for John Mazoon and myself debtor. 1658 9 24 March to Cash for 30 hh. at 7 l. per hh. 8 210 - Dito to John Mazoon for his ½ advance 45 - Dito to profit and loss for my ½ of advance 6 45 300 John Mazoon his account by me in company debtor. 1658 9 24 March to dito John his account proper for principal and gains 4 150 L S D Contra creditor. 1659. 25 March by dito his account of wheat, being the neat proceed thereof 7 257 5 Contra creditor. 1658 9 24 March by Cash for 30 hh. sold to John Brown at 10 l per hh. 1 300 Contra creditor. 1658 8 24 March by Cash received for his ½ part 1 100LS 9 24 March by Tobaccoe in company his ½ advance 8 45 150 L S D Balance Debtor. 1659. 5 25 March, to Hans Butterb. my account currant due in ready mon. 7 613 5 4 - Dito to Edward Price due to me 30 April next 5 101 10 - Dito to Fustians for 126 pieces resting unsold, at 22 s 8 d per piece 2 142 16 - Dito to Colchester Says for 20 pieces unsold 34 6 8 - Dito to Couchaneal for 11 C. 3 q. 14 l at 15 l to s per C. unsold 2 184 1 3 - Dito to Led for 95 pigs remaining unsold 5 204 11 3 - Dito to Claret Wines for 5 tuns, at 12 s per tun unsold 60 - Dito to Sugars for 24 C. 0 qu. 23 l unsold, at 36 s per C. 43 11 4½ - Dito to Province Oils for one Cask unsold 6 9 3 - Dito to Ship Samson for my ⅛ part thereof 3 200 - Dito to House Nag's head at Rumford for the principal worth 350 - Dito to Cash resting in bank 1 1524 19 2 ½ 3468 14 1 Contra Creditor. 1659. 25 March, by John Mazoon due to him by conclude 4 455 - Dito by Barnaby Clemens for my obligation due 5 March last 300 - Dito by John Malthorse due to him 29 April next 6 304 10 - Dito by Simon Peter's due on demand 50 - Dito by Stock for difference, there being my present estate 1 2359 4 3468 14 1 FINIS.