Socius mercatoris: or The merchant's companion: in three parts. The first, being a plain and easie introduction to arithmetick, vulgur and decimal, the extraction of the square and cube roots, with a table of 200 square roots, and their use in the resolution of square equations. The second, a treatise of simple and compound interest and rebate, with two tables for the calculation of the value of leases or annuities, payable quarterly, the one for simple, the other compound interest, at 6 per cent. per annum, with rules for making the like for any other rate. The third, a new and exact way of measuring solids in the form of a prismoid and cylindroid, with the frustums of pyramids and of a cone: whereunto is added, some practical rules and examples for cask-gauging. By John Mayne, philo-accomptant. Mayne, John, fl. 1673-1675. 1674 Approx. 145 KB of XML-encoded text transcribed from 113 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2003-11 (EEBO-TCP Phase 1). A50425 Wing M1484 ESTC R214155 99826364 99826364 30766 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A50425) Transcribed from: (Early English Books Online ; image set 30766) Images scanned from microfilm: (Early English books, 1641-1700 ; 1764:5) Socius mercatoris: or The merchant's companion: in three parts. The first, being a plain and easie introduction to arithmetick, vulgur and decimal, the extraction of the square and cube roots, with a table of 200 square roots, and their use in the resolution of square equations. The second, a treatise of simple and compound interest and rebate, with two tables for the calculation of the value of leases or annuities, payable quarterly, the one for simple, the other compound interest, at 6 per cent. per annum, with rules for making the like for any other rate. The third, a new and exact way of measuring solids in the form of a prismoid and cylindroid, with the frustums of pyramids and of a cone: whereunto is added, some practical rules and examples for cask-gauging. By John Mayne, philo-accomptant. Mayne, John, fl. 1673-1675. [16], 206, [2] p. printed by W[illiam] G[odbid] for N. Crouch, in Exchange-Alley, over against the Royal-Exchange in Cornhill, London : 1674. 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Keying and markup guidelines are available at the Text Creation Partnership web site . eng Interest -- Tables -- Early works to 1800. Interest rates -- Early works to 1800. 2000-00 TCP Assigned for keying and markup 2001-11 SPi Global Keyed and coded from ProQuest page images 2002-02 TCP Staff (Oxford) Sampled and proofread 2003-04 SPi Global Rekeyed and resubmitted 2003-08 Emma (Leeson) Huber Sampled and proofread 2003-08 Emma (Leeson) Huber Text and markup reviewed and edited 2003-10 pfs Batch review (QC) and XML conversion SOCIUS MERCATORIS : OR THE Merchant's Companion : IN THREE PARTS . The First , Being a plain and easie Introduction to Arithmetick , Vulgar and Decimal , the Extraction of the Square and Cube Roots , with a Table of 200 Square Roots , and their Use in the Resolution of Square Equations . The Second , A Treatise of Simple and Compound Interest and Rebate , with Two Tables for the Calculation of the Value of Leases or Annuities , payable Quarterly , the one for Simple , the other Compound Interest , at 6 per Cent. per Annum , with Rules for making the like for any other Rate . The Third , A new and exact way of Measuring Solids in the Form of a Prismoid and Cylindroid , with the Frustums of Pyramids and of a Cone : Whereunto is added , some Practical Rules and Examples for Cask-Gauging . By JOHN MAYNE , Philo-Accomptant . Nunquam nimis quod nunquam satis dicitur . Sen. LONDON , Printed by W G. for N. Crouch , in Exchange-Alley , over against the Royal-Exchange in Cornhill . 1674. To his Honoured Friend THO. WILLIAMS , M. D. Physician in Ordinary to His S. Majesty . SIR , THough the happiness which I formerly enjoyed in your Converse , hath been , to my great loss , for some years discontinued ; yet I easily perswade my self , that the Favour of a Great Prince , and the Best Master in the World , has not wrought such a change upon your even Virtue , but that you will still descend to remember him whom you were once pleased to honour with the Name of Friend . This Confidence has embold'ned me to present you with this small trifle ; too mean indeed and trivial for your acceptance , but that I know you are wont to admit of any thing that proceeds from an honest undesigning Gratitude . And though I am not at all inclin'd to vanity from the merits of the Work it self , yet I am proud that it affords me an opportunity to discover the lasting impressions , which your many Favours have made upon my Breast . Geometry , with Arithmetick her Woman , are Beauties , that having Truth written in their Foreheads , dare appear in the Court of the greatest Monarch , and I doubt not but they will find very easie and courteous admittance into your Appartment ; where if they shall afford you any divertisement when you return wearied from your ingenious Elaboratory , I shall then accompt that I have written to very good purpose . However , they certainly assure you , that it is impossible the teeth of Time should obliterate the honourable esteem conserved for you , in the heart of , Sir , Your most humble Servant , IOHN MAYNE . THE PREFACE . I Shall not trouble thee , Reader , nor my self , with a long Apology for the publishing this Treatise . How demurely soever I should pretend to the contrary , I fear thou wilt still be apt to imagin , that I had a tolerable good opinion of it , before I ventur'd it to the Press ; and truly I my self cann't well conceive , how any man should be over ambitious of being publickly laught at . If it be in any measure suited to the General Good , ( for which I intended it , ) I may at least expect thy pardon ; but if upon the perusal thou shalt find it otherwise , I ingenuously acknowledge my self to have been mistaken . My Design in this Work is , to render the Rules of those excellent Arts , which the Title-page pretends to , so plain and obvious , as that they may be easily apprehended without the Assistance of a living Master . And if there were nothing new in the whole , but the perspicuity of the Principles , and easiness of the Method ( which out of civility to my self I must deny ) yet those alone are sufficient to vindicate me in this Publication ; and I hope thou wilt not be angry , that I am a Well-wisher to thy Vnderstanding . For when I consider'd , that among the many good Books of this Nature , that are abroad in the World ( though written by Persons of greater knowledge than I dare pretend to ) some were so learned and obscure , as not to be understood , unless by those who have already made a considerable proficiency in these things ; others , so voluminous and prolix that they fright the endeavours of such who cannot spend their whole time this way : I was willing , according to my abilities , to obviate both these inconveniences , and accordingly applied my self to the composure of something , which for its plainness and brevity might be accommodated to those of mean Capacities and small Leisure ; and this Book is the result of those Contemplations . Whether I have accomplish'd my purpose or no , I make thee the Iudge ; requiring only that thou censure impartially of the Author and his Endeavours , without being offended that he is desirous to do thee a courtesie . I shall not here expatiate in the praise of the Arguments I treat of , nor give thee one line of Encomium : though out of the great affection I bear to these Arts , I find a strange inclination in my self to be rhetorical , yet I am resolv'd not to affront thee ; for truly their usefulness and excellency is so universally known , that to tell thee of it as a new thing , were to suppose thee a Person of more than ordinary ignorance ; only ( as I said before ) I must be so civil to my self too , as to inform thee , that ( besides the Introduction to Arithmetick and the Treatise of Interest , of which I challenge no more than thou shalt find thy self very willing to give me ) that Part which concerns the measuring of Solids , viz. the Prismoid , Cylindroid , &c. is wholly new , and never before made publick . The bulk is bigger than at first by me intended ; but to gratifie the Book-seller , the Vulgar Arithmetick was an Appendix , though previous to the rest . But if one , or other , or all , prove either profitable or pleasant to thee , I am sufficiently oblig'd to subscribe , Thy Friend , John Mayne . From my House at the Golden-Ball in Shaws-Court , near St. Georges Church , Southwark ; Iuly 29. 1673. To the Ingenious Author , concerning his DECIMALS . 1 SIr , by your Art , and Pythagorean Pen , I 'd prove a Metempsychosis agen ; And were His Soul of Decimals but made , 2 As Plato's Soul o' th' world of Seven is said , I 'd swear 't was slunk to you ; but that you shew 3 More Skill than e're his rambling fancy knew . Let roving Rabbies praise their Seven and Four ; 4 We 'l shew them Mysteries enough and more : The Heav'nly Orbs are Ten , their Motions all 5 Conspire to make a perfect Decimal : This is their Musick , and they shall be thus , 6 In spight of Tycho or Copernicus . 'T is said the Muses are but Nine , but who 7 ( Rather than fail ) cann't add Apollo too ? Thus may we range the world , and quickly find , 8 We all to th' number of our Fingers bind . Thus Logick all the wandring Species brings , 9 And places under tenfold Heads of things . Thus I , to give the Author praise in all , 10 Reduce my Verses to a Decimal . On his GAUGING . YOur Circles , Sir , would make my folly ghess , You were a Conjurer , though you wo'n't confess . And Gauging is the rugged dev'lish Name Of some Hobgobling Imp , the very same That brought in Custome ; but what e're he be , He 's a rare Fellow at the Rule of Three : He doth just square the Circle ; nay so true , That the King 's Right is given to a Cue . There 's none else such Impossibles can do : You give the King's , I give this Right to you . I. W. On his worthy Friend Mr. J. Mayne , the Author of this BOOK . Ingenious Artist , whither do'st aspire ? Or why t' outvye the Ancients do'st desire ? Have they not left enough to following Ages ? No : Thou their Master art , they but thy Pages . My feeble Muse can never soar so high , As thy Deserts herein extend , nor nigh . Yet give me leave hereof to speak my mind : No Man could better teach us in this kind , Each Part so useful , and so plain I find . T. W. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . To his ingenious Friend the Author Mr. John Mayne . WHo reads thy Book with an impartial Eye , Will see how plain , and how ingeniously Thy Rules are fram'd ; here every Child may learn Arithmetick , which doth the Truth discern . The Iudges of our Realm could not dispence To all Men Iustice , were 't not fetch'd from hence : Those Sons of Mars that furrow Neptune's Brow , Unto this Science must their Labours bow : The wealthy Merchant , and all Traders , hence Must calculate their Gain , or their Expence : The greedy Miser , here may plainly see His Pelf's increase at Compound Vsurie : The Purchaser of Farms , may also here , Value his Lordships , whether cheap or deer . Thy Squares and Cubes , methinks , so plain do seem , That I old Euclid should thy Father deem . All Humane Arts , Mechanical and Free , For this Companion are oblig'd to Thee . By Lines and Numbers , we our Buildings bring In due proportion , framing every thing . By these our Wooden Walls and Towers are fram'd , Which guard our Island , and the Seas command : These fill our Stores with rich and costly things , Born from both Indies under Canvas Wings : These fortifie our Towns with Forts , by Line ; By these we learn our Foes to undermine : By these th' Excise and Customs we do scan , Without Injustice to the Trading Man. Thanks to our Author then , that hath set forth These Arts so plain , and of abundant worth : Which do to Sea and Land such Profit yield , In Court , in City , Garrison , and Field . Hugh Handy , Philomath . A TABLE OF THE CONTENTS . The First BOOK . NOtation or Numeration . pag. 1 Addition . 4 Subtraction . 8 Multiplication . 12 Division . 17 Reduction . 20 The Rule of Three . 22 The Rule of Practice . 28 Notation of Decimals . 34 Addition of Decimals . 37 Subduction of Decimals . 38 Multiplication of Decimals . ib. Division of Decimals . 41 Reduction of Decimals . 48 A Table of Reduction . 52 The Golden Rule . 54 The Double Golden Rule . 61 Of the Square Root . 66 A Table of Square Roots . 71 Of Quadratique Equations . 80 Of the Cube Root . 91 The Second BOOK . OF Simple Interest . pag. 99 Prop. 1. To find the Interest of any Sum , forborn any time , at any given rate . 100 Prop. 2. To find the present worth of any Sum , due at any time hereafter , at any given rate of Interest . 102 Prop. 3. Having the Principal , Amount , and Rate of Interest , to find the Time of forbearance . 103 Prop. 4. Having the Principal , the Time , and the Amount , to find the Rate . 105 A Table of the Amounts of 1 l. from one to twelve months . 107 To find the Interest or Discompt of any Sum of Money by that Table . 107 , 108 A Table for Equation of Time. 110 A more exact way of Equation . 111 , 112 , 113 A Decimal Table of the present worth of 1 l. per Quarter for 124 Quarters , at 6 per Cent. per Annum , Simple Interest . 114 The Vse of the Table . 117 Of Compound Interest . 118 Prop. 1. To find the Increase of any given Sum , forborn any known Time , at a known Rate per Cent. per Annum . 118 Prop. 2. The Amount of a Sum of Money , the Rate of Interest per Cent. per Annum , and the Time being known , to find what was the Principal . 120 Prop. 3. The Principal , the Time , and the Amount of a Sum of Money being known , to find the Rate of Interest per Cent. per Annum . 121 Prop. 4. The Principal , the Rate , and the Amount being known , to find the Time in which it hath so increased . 122 Of Compound Interest Infinite . 123 Prop. 1. To find the present worth of an Estate in 〈◊〉 Simple , at any Rate of Interest per C. per Ann. 123 Prop. 2. To find what Free-hold Estate any Sum of Money will buy , as any Rate of Interest per C. per Ann. 126 Prop. 3. An Estate being offered for a Sum of Money , the annual Rent being known , to find what Rate of Interest the Purchaser shall have for his Money . 126 To find how many years purchase any Free-hold Estate is worth at any given Rate of Interest . 128 The number of years purchase being propos'd , to find the Rate of Interest it is offered at . 129 A Decimal Table for the Valuation of Leases or Annuities● payable Quarterly , at 6 per Cent. per Annum , Interest upon Interest for 31 years . 130 The way of Making the Table before-mentioned , for this , or any other Rate of Interest . 133 The Vse of the Table . 134 Six Questions performed by aid of the Canon of Logarithms . 137 The Third BOOK . THe Definition of a Prismoid . pag. 151 To find its Solidity 153 Prop. 1. An Example in Numbers . ib. Prop. 2. The Inversion of the former Solid upon its opposite Base , the Rule and Example . 157 Prop. 3. Another Example . 161 Prop. 4. The Inversion . 163 The Definition of a Pyramid . 171 To find its Solidity . 172 A Table of Divisors for Reduction of the Polygons , and the Cone , to Cubick Inches or Gallons . 173 An Example of a Trigon . 174 Of a Tetragon . 179 Of a Pentagon . 18● Of a Cone . 184 To find the Fall of a Conical Tunn . 186 Of Ca●k-Gauging . 191 To find a Casks Length . 192 To find the Head-diameter . 193 To find the Diagonal . ib. To find the Content as Spheroidal . 194 As Parabolical . 196 As Conical . 197 By a Table of Area's . 199 , 200 , &c. To find the Vllage . 202 , 205 ARITHMETICK IN Whole Numbers . NOTATION . IT is necessary , that all Persons that would acquaint themselves with the Nature and Use of Numbers , do first learn to know the Characters by which any Quantity is expressed . These Characters are in number nine , who with a Cypher are the Foundation of the whole Art of Arithmetick . Their form and denomination as in this Example . 0. Cypher 1. One. 2. Two. 3. Three . 4. Four. 5. Five . 6. Six . 7. Seven . 8. Eight . 9. Nine . These Characters standing alone express no more than their simple value , as 1 is but one , 2 standing by it self signifies but two , and so of the rest ; but when you see more than one of those Figures stand together , they have then another signification , and are valued according to the place they stand in , being dignified above their simple quality , according to the Examples in this Table .                 Unites .               Tens . 1             Hundreds . 1 2           Thousands . 1 2 3         Ten Thousands . 1 2 3 4       Hundred Thousands . 1 2 3 4 5     Millions . 1 2 3 4 5 6   Ten Millions . 1 2 3 4 5 6 7 Hundred Millions . 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 The denomination of Places according to this Table , must be well known , and are thus exprest ; those standing in the place of Unites , signifie no more than their value before taught ; but standing in the second place toward the left hand , they are increased to ten times the value they had before , 1 or One in the Unite place signifies but One ; if it stand in the second place toward the left hand , and a Cypher before it thus 10 , it hath ten times its simple value , and is called Ten ; if 2 stand in the place of the Cypher thus 12 , it is then Twelve , being Ten and two Unites ; 1 , 2 , or 3 , standing in third place , with Figures or Cyphers toward the right hand of it , doth signifie Hundreds , as 100 is One hundred , 123 is One hundred twenty three , 321 is Three hundred twenty one , 213 is Two hundred thirteen ; and so any three of the other Figures have like value , according to their Stations , the first to the right hand in the Unite place signifies so many Unites , the second , or that in the place of Tens , is increased to ten times its simple value , and in the third place , or place of Hundreds , any Figure there standing hath a hundred times the value it would have had were it in the Unite place . The fourth place is the place of Thousands , any Figures standing there , with three Figures or Cyphers to the right hand of it , is so many Thousands as simply it contains Unites , so 3000 is Three thousand , 9825 is Nine thousand eight hundred twenty five , &c. The fifth place is Ten thousands , and any five Figures placed together , are to be read after this manner : Example . 45326 Forty five thousand three hundred twenty six . 12345 Twelve thousand three hundred forty five . The sixt place hath the denomination of Hundred thousands , and those six in the Table that stand in a rank are to be read , One hundred twenty three thousand four hundred fifty six . The seventh is the place of Millions , and the seven in the Table are , One million two hundred thirty four thousand five hundred sixty seven . And the eighth Rank of Figures are to be read , Twelve millions three hundred forty five thousand six hundred seventy eight . The ninth rank is , One hundred twenty three millions four hundred fifty six thousand seven hundred eighty nine . And so any greater number of places , every figure one place more toward the left hand , is increased ten times in value more than in the place it stood before . ADDITION . ADdition , is a gathering or collecting of several Numbers or Quantities into one Sum , by placing all Numbers of like Denomination under one another , carrying all above ten to the next place , as in these Examples . There is likewise another kind of Addition , that is not of whole Quantities , wherein is necessary to be known the number of Parts the Integer or whole Number is divided into , as Pounds and Shillings , every Pound is divided into 20 Shillings , and one Shilling is divided into twelve Pence , one Penny into four Farthings . Now being to add a Number of Pounds and Shillings together , they are thus set down with a small Line or Point between them . 3-5 6 : 16 If these be added together , observe in casting up your Shillings , so many times as you have 20 in the Shillings , you must carry Unites to the Pounds , and set down the Remainder , being under 20 , as in these Examples . l. s. 3 : 5 6 : 16 10 : 01 l. s. 4 : 17 3 : 15 5 : 9 6 : 12 19 : 13 In the first Example , I find in adding the Shillings together , they make 21 , so I set down 1 and carry 1 Pound to the Pounds : In the second Example , I find among the Shillings 53 , which is 2 Pounds 13 Shillings , so I set down 13 under the Shillings , and 2 to the Pounds . Any number of Shillings and Pence being to be added together , if your number of Pence amount to above 12 , carry 1 to the Shillings , and set down the remainder under the Pence ; if they make above 24 , carry 2 Shillings , and set down the remainder , as before . Examples . s. d. 1 : 6 2 : 7 4 : 1 s. d. 8 : 9 2 : 8 3 : 10 15 : 03 s. d. 1 : 7 2 : 6 3 : 9 4 : 8 5 : 11 18 : 05 In the first Example , you carry one Shilling ; in the second , two ; and in the third , three . In Addition of Pence and Farthings , carry so many times four as you find in the number of Farthings to the Pence , setting down the remainder under the Farthings , as in these Examples . When you would know the Sum of any number of Pounds , Shillings , Pence , and Farthing , they are to be placed thus : Addition of Weight and Measure is performed after the same manner . 16 Ounces Averdupois , make a Pound . 28 Pounds , make a Quarter . 112 Pound , or 4 Quarters , make an Hundred gross . 20 Hundred , make a Tun. Examples . Where observe , that so oft as I find 16 Ounces , I carry 1 to the Pounds ; so often as I find 28 Pounds , I carry 1 to the Quarters ; and as many times as I find 4 in the Quarters , so many times 1 do I carry to the Hundreds . SUBTRACTION . SVbtraction is the taking a lesser Number from a greater , and exhibits the Remainder . In Subtraction the Numbers are placed one under another , as in Addition , thus : The first of these Numbers is called the Minorand , the second the Subducend , and the third Number , or the Number sought , is the Residuum . 8 The Minorand 6 The Subducend 2 The Residuum or Remainder EXAMPLES of COINS . But when the number of Pence or Shillings , are greater than the number that stands over it in the Minorand , you must borrow the next Denomination , as in this Example . This Example I work after this manner , saying 9 d. out of 3 d. I cannot have , wherefore I borrow 1 s. from the Shillings , and subduct the 9 d. from that , and there will remain 3 d. which added to the other 3 d. maketh 6 d. I place therefore 6 d. in the Place of Pence , and proceed saying , 1 s. that I borrowed and 19 is 20 from 1 I cannot , wherefore I borrow 1 l. from the Pounds , and subduct from that the 20 s. and there remains nothing but the 1 s. which I place under the Shillings , and say , 1 that I borrowed and 6 is 7 from 7 and there remains nothing , then I place a Cypher under the 6 , and say , 1 from 2 and there remains 1 , which I set down , and 1 from 1 and there resteth nothing . After this manner is performed Subduction of Weight and Measure . Examples . By which Examples , the Learner may perceive , that where the number to be subducted is greater than the number standing over it , I then borrow one from the next greater denomination , adding the remainder , if any be , to the lesser number before-mentioned , and setting them underneath those of like denomination with them . The Proof of Subtraction is by adding the Subducend and Remainder together , and their Aggregate must always be equal to the Minorand , as you may see by the last Example . I could here add many more Examples of Weight and Measure , but to the ingenious Practitioner I hope it will be enough , all other being wrought af●er the same manner , respect being had to the number of lesser denominations contained in each greater . As In Troy Weight , 24 Grains make a Penny-weight . 20 Penny-weight one Ounce . 12 Ounces one Pound . Long Measure . 4 Nails make a Quarter of a Yard . 4 Quarters one Yard . 5 Nails one Quarter of an Ell. 4 Quarters one Ell. 12 Inches a Foot. 3 Feet a Yard . 16½ a Perch . 40 Perches a Furlong . 8 Furlongs make an English Mile . Liquid Measure . 8 Pints make a Gallon . 63 Gallons make a Graves Hogshead . 4 Hogsheads make a Tun. 36 Gallons make a Beer Barrel . 32 Gallons make an Ale Barrel . Dry Measure . 8 Gallons of Corn make a Bushel . 8 Bushels make a Quarter . MULTIPLICATION . MVltiplication is a kind of Addition , and resolveth Questions to be performed by Addition in a different manner : In order whereunto , it is necessary the Learner do well acquaint himself with this Table ; the having this Table perfectly by heart , will make both this Rule and Division also very facile , otherwise they will be both troublesome and unpleasant . 1 2 3 4 5 6 7 8 9 10 11 12 2 4 6 8 10 12 14 16 18 20 22 24 3 6 9 12 15 18 21 24 27 30 33 36 4 8 12 16 20 24 28 32 36 40 44 48 5 10 15 20 25 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 66 72 7 14 21 28 35 42 49 56 63 70 77 84 8 16 24 32 40 48 56 64 72 80 88 96 9 18 27 36 45 54 63 72 81 90 99 108 10 20 30 40 50 60 70 80 90 100 110 120 11 22 33 44 55 66 77 88 99 110 121 132 12 24 36 48 60 72 84 96 108 120 132 144 In the first Rank of this Table , you have an Arithmetical Progression from 1 to 12 , and also in the first Column toward the left hand downwards . This Table doth at first sight exhibit the Sum of any number , so often repeated as you shall require , provided the numbers do neither of them exceed 12. Multiplication hath three Members , thus called , a Multiplicand , a Multiplicator , and a Product : The Multiplicand , is the number to be repeated ; the Multiplicator , is the number of times the first is to be repeated ; and the Product , is the Sum of the Multiplicand so often repeated . As for Example . A Countrey-man sold 6 Bushels of Wheat for 5 s. how many Shillings ought he to receive ? By Addition 6 must be 5 times set down thus : Or 5 six times repeated thus : But by Multiplication it is done thus : 6 The Multiplicand . 5 The Multiplicator Now if you look in the Table precedent , in the first Column find 5 , then look in the first Rank for 6 , and cast your Eye down to their Angle of meeting , and you will find 30 standing under 6 and against 5 , I then conclude that 5 times 6 is 30 ; that is called the Product , and they will stand thus : 6 The Multiplicand . 5 The Multiplicator . 30 The Product . But when you have a number to multiply , greater than any in the Table , as for Example : A Gentleman having forborn his Rent of a Farm , at 157 l. per Quarter , for 3 Quarters , what ought he to receive ? The Multiplication will stand thus : 157 The Multiplicand . 3 The Multiplicator . 471 The Product . I then say , 3 times 7 is 21 , I set down 1 and carry 2 ; then , 3 times 5 is 15 and 2 is 17 , I set down 7 next the 1 , and carry 1 ; saying , 3 times 1 is 3 and 1 is 4 , as in the Example before-going ; and the Product is 471 l. There is yet more variety , of which take these Examples following . If 65 Ships do carry 536 Men in every Ship , how many Men will there be in all ? I say 5 times 6 is 30 , set down 0 and carry 3 ; then 5 times 3 is 15 and 3 is 18 , set down 8 and carry 1 ; then 5 times 5 is 25 and 1 is 26 , which I set down : Then for the next Figure , I say , 6 times 6 is 36 , I set down 6 one place short of the former rank , and carry 3 ; then 6 times 3 is 18 and 3 is 21 , set down 1 and carry 2 ; again , 6 times 5 is 30 and 2 is 32 , these I set down : Then draw a line , and cast them up as they are placed , and the Sum is the Product and Answer to the Question , viz. 34840 Men. In Multiplication , always make the lesser Number the Multiplicator , for it is all one whether I multiply 5 by 15 , or 15 by 5 , the Product is always the same . If 128 Men of War have each made 746 Shot , how many Shot were made in all ? Begin as before with the Unites place , and say , 8 times 6 is 48 , set down 8 and carry 4 ; 8 times 4 is 32 and 4 is 36 , set down 6 and carry 3 ; then 8 times 7 is 56 and 3 is 59 , which set down : Then go forward with the 2 , ( but remember to place your remainder one Figure short of the former ) saying , 2 times 6 is 12 , set down 2 under the 6 and carry 1 ; 3 times 4 is 8 and 1 is 9 , which set down ; twice 7 is 14 , which set down : Also then , once 6 is 6 , which place under the 9 ; once 4 is 4 , which set under the 4 ; and once 7 is 7 , which set under the 1 : Then cast th●● up , as in Addition , and the Sum is the Product , and answers the Question , viz. 95488 Shot . If any number be to be multiplied by 1 with Cyphers , it is but adding so many Cyphers to the Multiplicand as there is in the Multiplicator . As for Example . If 35678 be to be multiplied by 10 , add one Cypher to the Multiplicand , thus , 356780 ; if by 100 , add two Cyphers , thus , 3567800 ; &c. And when any number is to be multiplied by any other number , that hath Cyphers annexed , always place the Cyphers immediately under the Line , as in these Examples . DIVISION . DIvision is also a kind of Subduction , and informs the Querent , how many times one number is contained in another . There is in Division these three things to be observed , viz. the Dividend , the Divisor , and the Quotient . The Dividend is a number to be divided into parts , the Divisor is the quantity of one of those parts which the former is to be divided by , the Quotient is the number of such parts as the Dividend doth contain . There is also by accident a fourth number in this Rule necessary to be known , which is a Remainder , and that happens when the Dividend doth not contain an equal number of such quantities as it is divided by , as when 15 is to be divided by 4 , the Dividend is 15 , the Divisor is 4 , and there is a Remainder 3. In Division you may place your numbers thus . Multiplication is positive , but Division is performed by essays or tryals , after this manner : Here I first inquire how many times 3 I can have in 14 , I find 4 times , I place 4 in the Quotient , and then multiply the Divisor by that 4 , placing the Product underneath the Dividend , as in the Example ; say , 4 times 5 is 20 , set down a Cypher under the 6 and carry 2 , then 4 times 3 is 12 and 2 is 14 , which I set down also , as in the Example ; then subduct this Product from the Figures standing over them , and set down the Remainder . Then for a new Dividend , I bring down the next figure , and postpone that to the Remainder , and inquire how many times 3 in 6 , I cannot have twice , bacause I cannot have twice 5 from 5 , I say then once , and place 1 in the Quotient , proceeding as before saying , once 5 is 5 , which I place under the first 6 toward the right hand , and once 3 is 3 , which I set down under the other 6 ; subducting these as the former , I find the Remainder to be 31. A●ter which I bring down the next figure in the Dividend , and postpone it to the Remainder , as in this Example : Then I inquire how many times 3 in 31 , I suppose 9 times , placing 9 in the Quotient I multiply again , saying 9 times 5 is 45 , 5 and carry 4 ; then 9 times 3 is 27 , and 4 is 31 ; these being set down , as before directed , and subducted , there will remain nothing . I then conclude , that the Divisor is so often contained in the Dividend as i● expressed in the Quotient , viz. 419 times . For further Instructions , take these Examples . REDUCTION . REduction is twofold , viz. bringing greater denominations into smaller , and that by Multiplication , as Pounds into Shillings , Shillings into Pence , &c. Also lesser denominations are reduced into greater , by Division as Pence into Shillings , Shillings into Pounds Minutes into Hours , Hours into Days , and Days into Years , &c. Having any number of Pounds to reduc● into Pence , multiply them by 240. Example . In 869 Pounds how many Pence ? Answ. 208560 Pence . In 2486 Shillings how many Farthings ? Answ. 119328 Farthings . How many Minutes are there in 9476 Hours ? The Answer 568560 minutes . How many Pounds , Shillings , and Pence , are contained in 22929 Farthings ? In 544542 Cubique Inches , how many Beer Barrels , Firkins , and Gallons ? Inches in 1 B. Bar. 10152 ) 544542 ( 53 : 2 : 5 Bar. firk . g. Inches in 1 Firkin 2538 ) 6486 Inches in 1 Gallon 282 ) 1410 THE RULE OF THREE . THis Rule is so called , because herein are three numbers given to find a fourth ; of these three numbers , two are always to be multiplied together , and their Product is to be divided by the third , and the Quotient exhibits the fourth number , or the number sought . And here note , That of the three given numbers , if that number that asketh the Question be greater than that of like denomination with it self , and require more , or if i● be less , and require less , then the number of like denomination is the Divisor . Or , if the number that asketh the Question be less than that of like denomination , and require more ; or if it be more , and require less , then the number that asketh the Question is the Divisor . Example . If 3 Yards of Sarcenet cost 15 s. what shall 32 Yards cost ? Which 3 numbers if you please may stand thus : Here you may see the term that asketh the Question is greater than that of like denomination , being 3 , and the other 32 , and also requires more , viz. a greater number of Shillings ; therefore , according to the Rule , the first term , or the term of like denomination to that which asketh the Question , is the Divisor . And the Answer is 160 Shillings , which being divided by 20 will be found 8 l. Again , If 32 Ells of Holland cost 160 s. what shall 3 Ells cost ? In this Question ( being the Converse of the former ) you may see the term that asketh the Question , here 3 , is lesser than that of like denomination , being 32 Ells , and also requires less ; therefore the first term here also is the Divisor . And the Answer is 15 s. If 36 Men dig a Trench in 12 Hours , in how many Hours will 144 Men dig the same ? 144 ) 432 ( 3 Hours , the fourth number . In this Question , the term that asketh the Question is greater than that of like denomination , and requireth less ; wherefore the term that asketh the Question is the Divisor . If 144 Workmen build a Wall in 3 Days , in how many Days will 36 Workmen build the same ? This Question you may perceive to be the Converse of the former , here the term that asketh the Question is less than that of like denomination , and requires more , the term that asketh therefore is the Divisor . If 125 lb. of Bisket be sufficient for the Ships Company for 5 Days , how much will Victual the Ship for the whole Voyage , being 153 Days ? This Question is of the same kind with the first Example ; here the two terms of like denomination are 5 Days and 153 Days , the term that asketh the Question being more than the term of like denomination , and also requiring more ; so , according to the general Rule , the term of like denomination to that which asketh the Question is the Divisor . It matters not therefore in what order they ar● placed , so you find your true Divisor ; but if you will you may set them down thus : The Answer is 3825 lb. weight of Bisket . A Ship having Provision for 96 Men during the Voyage , being accompted for 90 Days , but the Master taking on boord 12 Passengers , how many Days Provision more ought he to have ? Which is no more than this : If 96 Men eat a certain quantity of Provision in 90 Days , in how many Days will 108 Men eat the same quantity ? The Answer is 80 , so that for 108 Men he ought to have 10 Days Provision more . If the Assize of Bread be 12 Ounces , Corn being at 8 s. the Bushel , what ought it to weigh when it is sold for 6 s. the Bushel ? In this Question , the term inquiring being less than the term of like denomination , and requiring more ; therefore is the term so inquiring the Divisor . The Answer is 16 Ounces . THE RULE OF PRACTICE . IT is necessary that the Learner get these two Tables perfectly by heart , which are only the aliquo● parts of a Pound and of a Shilling . The Parts of a Shilling . d. q.   0 1 Forty eighth . 0 2 Twenty fourth . 0 3 Sixteenth . 1 0 Twelfth . 1 2 Eighth . 2 0 Sixth . 3 ● Fourth . 4 0 Third . 6 0 Half. The Parts of a Pound . s. d. q.   0 00 1 The Nine hundred and sixtieth . 0 00 2 The Four hundred and eightieth . 0 00 3 The Three hundred & twentieth . 0 01 0 The Two hundred and Fortieth . 0 01 2 The Hundred and sixtieth . 0 02 0 The Hundred and twentieth . 0 03 0 The Eightieth . 0 04 0 The Sixtieth . 0 05 0 The Forty eighth . 0 06 0 The Fortieth . 0 08 0 The Thirtieth . 0 10 0 The Four and twentieth . 1 00 0 The Twentieth . 1 03 0 The Sixteenth . 1 04 0 The Fifteenth . 1 08 0 The Twelfth . 2 00 0 The Tenth . 2 06 0 The Eighth . 3 04 0 The Sixth . 4 00 0 The Fifth . 5 00 0 The Fourth . 6 08 0 The Third . 10 00 0 The Half. Having these Tables perfectly in memory , any Question propounded will be readily resolved , only by dividing the given number of Yards , Ells , Feet , Inches , Gallons , Quarts , Pounds , or Ounces . Of which take some Examples . 145 Ells of Cloth at 3d. per Ell. 36 s. 3 d. Three Pence being the fourth part of a Shilling , I divide the number by 4 , and the quote is the number of Shillings it is worth . 728 at 4.d . 242s . 8d . Four Pence being the third part of a Shilling , I divide by 3. 654 at 6d . 327 s. Here take the half . 321 at 1d . 2q . 40s . 1d . 2q . Here the eighth part . Here take the sixth and the eighth part of the quote . Having any number of Shillings to reduce into Pounds , cut off the last figure toward the right hand by a line , and the figures on the left hand of the line are so many Angels as they express Unites ; draw a line under them , and take the half of them , and you have the number of Pounds . Examples . Any Commodity , the value of 1 Yard being the aliquot part of a Pound , is thus cast up : 836 Yards of Broad Cloth at 6s . 8d . per Yard . 278l . 13● . 4d . Take the one third part , and that is the Answer in Pounds : 3 in 8 twice , and carry 2 ; 3 in 23 seven times , and carry 2 ; 3 in 26 eight times , and carry 2 ; the third part of 2 l. is 13 s. 4 d. where always observe , that the Remainder is always of the same denomination with the Dividend . 654 lb. of Cloves at 5s . per lb. 163l . 10s . Take the fourth part . 9464 Gall. of Brandy at 3s . 4d . 1577l . 6s . 8d . The Sixth . Where the Price is not aliquot . 625 at 3s . per Ounce . Here I take the tenth and the half of that tenth . 348 Dollers at 4s . 6d . The fifth and the eighth of that fifth . 245 lb. at 2s . 3d. The tenth and the eighth of that quote . To cast up the amount of any Commodity , sold for any number of Farthings by the Pound , I borrow from the Dutch a Coin called a Guilder , whose value is 2 s. English. Then if a Question be proposed of the Amount of an Hundred weight of any Commodity , by the Hundred Gross , viz. 112 lb. so many Hundred as there be , the Amount is so many Guilders so many Groats , as there are Farthings in the price of 1 lb. As for Example . A Hundred weight of Iron is sold for 5 Farthings the Pound , comes to 5 Guilders , that is 10 s. and 5 Groats , which together is 11 s. 8 d. Again . A Hundred weight of Lead is sold for 2 d. Farthing the Pound , that is 9 Guilders and 9 Groats , which is 21 Shillings . But if it be the subtil Hundred , it is then but so many Guilders so many Pence : As if a Hundred weight of Tobacco be sold for 5 d. Farthing the Pound , the Hundred comes to twenty one Guilders and twenty one Pence , that is forty three Shillings and nine Pence . ARITHMETICK IN DECIMALS . NOTATION . Integers . Decimals . 3 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 Thousand Millions . Hundred Millions . Ten Millions . Millions . Hundred Thousands . Ten Thousands . Thousands . Hundreds . Tens . Unites . Tenths . Hundredths . Thousandths . Ten Thousandths . Hundred Thousandths . Millioneths . Ten Millioneths . Hundred Millioneths . Thousand Millioneths . AS in Whole Numbers , the value or denomination of Places do increase by Tens , from the Unite place toward the left hand ; so in Decimals , the value or denomination of Places do decrease by Tens , from the Unite-place toward the right hand , according to the precedent Table . A Fraction or broken Number is always less than a Unite , as Pence are parts of a Shilling , and Shillings of a Pound , Inches of a Foot , and Minutes of an Hour , &c. Fractions are of two kinds , And are thus called Vulgar , & Decimal . A vulgar Fraction is commonly expressed by two Numbers set over one another , with a small line between them , after this manner ⅔ , the uppermost being called the Numerator , and the lower the Denominator . The Denominator expresseth into how many parts the Integer or whole Number is divided , and the Numerator sheweth how many of those parts is contained in the Fraction . Example . If the Integer be a Shilling ⅔ is 8 d. If it be 1 l. or 20 Shillings , it is 13 s. 4 d. If a Foot , it is then 8 Inches . Or if an Hour , it will be 40 Minutes . A decimal Fraction hath always a common Number for a Numerator , and a decimal Number for its Denominator . A decimal Number is known by Unity , wi●h one or more Cyphers standing before it , as 10 , 100 , 1000 , &c. A decimal Fraction is known from a whole Number by a point , or some other small mark of distinction , whether it stand alone , or be joyn'd with whole Numbers ; as in these following Examples . Or else with a point over the head of Unity , or the Unite place ; as in these Examples . In decimal Fractions , the Numerators only are set down , the Denominator being known by the last Figure in the Numerator . Example . .2 is Two tenths . .25 is Twenty five Hundredths . .257 is Thousandths . .2575 is Ten Thousandths , &c. As Cyphers before a whole Number have no value , so Cyphers after a decimal Fraction are of no signification : But Cyphers before a decimal Fraction , are of special regard ; for as Cyphers after a whole Number do increase that Number , so before a decimal Fraction they diminish the value of that Fraction . Example . .25 Twenty five hundredths . .025 Twenty five thousandths . .0025 Twenty five ten thousandths . Each Cypher so added removing the Fraction further from Unity , making it ten times less than before . ADDITION . ADdition in Decimals , whether in pure Decimals , or whole Numbers mixt with Decimals , differs not from Addition in whole Numbers , only care must be had to the seperating lines or points , that all places of like denomination stand one under another , both in the Addends and in the Sum ; as in these Examples . SUBTRACTION . AS in Addition , so in Subtraction care must be had to the placing each Figure under that of like denomination with it self , then it is the same with Subtraction in whole Numbers . Examples . MULTIPLICATION . MVltiplication in whole Numbers serveth instead of many Additions , and teacheth of two Numbers given to increase the greater as often as there are Unites in the lesser . It likewise consists of three Requisites , viz. a Multiplicand , a Multiplicator , and a Product . The Multiplicand is the Number to be increased . The Multiplicator is the Number by which it is to be increased . And the Product is the Sum of the first Number so often repeated as there are Unites in the second . In decimal Fractions , or whole Numbers mixt with Fractions , the two first Numbers are called Factors , and the last is called the Fact. Multiplication , whether in decimal Fractions , or whole Numbers mixt with Fractions , differeth not ( in the Operation ) from Multiplication in whole Numbers . The last Figures in both the Factors may be placed under one another , without respect to the distinction of places , or places of like denomination standing under one another , as in Addition and Subduction ; yet from the Product must be cut off by a line or point so many places as there are Figures in decimal Fractions in both Factors of the last Figures standing toward the right hand . Examples . If it happen when the Multiplication is ended , that there be fewer Figures in the Product● than there are places in Decimals in both the Factors , then put Cyphers before the Product till the number of places be equal to those in both the Factors : As in these Examples . Whereby may be observed , That the Multiplication of two Fractions doth not increase them as in whole Numbers , but they are hereby made less , and the Fact is removed further from Unity than either of the Factors . If a whole Number be to be multiplied by a decimal Number , put so many Cyphers after the whole Number as there are in the decimal Number , and that Number will be the Product . If 48 be multiplied by 10 , it will be 480 ; by 100 , 4800 ; &c. In multiplying decimal Fractions , or mixt Numbers , by a decimal Number , you need only remove the point or seperating line so many places toward the right hand as there be Cyphers in the decimal Number . If you multiply .2845 by 10 , the Fact will be 2.845 ; by 100 , it will be 28.45 ; by 1000 , 284.5 ; &c. DIVISION . DIvision , both in whole Numbers and Fractions , is by young Practitioners found to be more difficult than any of the four Species ; it will therefore require a little more industry in the Learner : But when once had , there will appear small difference between the Operation herein , as in any the precedent . Division is also constituted by three Requisites , and a fourth by accident , viz. a Dividend , a Divisor , and a Quotient : The fourth is a Remainder , which doth not always happen to be . The Dividend is the Number to be divided . The Divisor is the Number by which the other is to be divided . The Quotient is the Number found out by the Division . And the Remainder is that which is left of the Dividend after the Division is ended , and is always less than the Divisor . Example . If 12 be to be divided by 4 , then is 12 the Dividend , 4 the Divisor , and the Quotient will be 3. If 13 be divided by 3 , then 13 is the Dividend , 3 the Divisor , 4 the Quotient , and there will be a Remain , which is here 1. Decimal Fractions , or mixt Numbers , are divided after the same manner as whole Numbers are divided , only care must be had in giving a true value to the Quotient . To perform which , observe well this General Rule . The first Figure in the Quotient is always of the same denomination with that Figure which stands ( or is supposed to stand ) over the Unity place in the Divisor . As to the manner of placing your Figures , and the way of dividing , there are many published by divers Writers of Arithmetick : The way of placing the Divisor under the Dividend , is the most apt for giving a value to the Quotient ; but the rasing of Figures , and repeating the Divisor so often , is found an inconvenience ; which to avoid , observe the following Examples . Being to divide 2487.04 by 53.6 , I place them in this order : 53.6 ) 2487.048 ( Then I consider if the Divisor were placed under the Dividend , the Unity place in the Divisor , here 3 would stand under the 8 in the Dividend , I then set a mark over the head of the 8 , and conclude the first Figure in the Quotient to be of the same denomination with it , which is Tens , in whole Numbers . Having thus found the value of the first Figure in the Quotient , I proceed to the division , and inquire , how many times 5 in 24 ? I find 4 ; I then set 4 in the Quotient , and go back , multiplying the whole Divisor by that Figure , and subduct the Product out of the Dividend , placing the Remainder underneath as part of a new Dividend : Thus 4 times 6 is 24 , from 27 , and there remains 3 , which I place under the 7 ; again , 4 times 3 is 12 , and 2 that I borrowed is 14 , from 18 , and there remains 4 , which I place under the 8 , as in the Example ; then 4 times 5 is 20 , and 1 I borrowed is 21 , from 24 , and there remains 3 , which I place under the 4. For my new Dividend , I bring down the next Figure , here a Cypher , and postpone it to the Remainder , and the Example will stand thus : Then proceeding in my Division , I ask , how many times 5 in 34 ? finding 6 times , I then place 6 in the Quotient , and as before say , 6 times 6 is 36 , from 40 , and there remains 4 , which I set down under the Cypher ; then 6 times 3 is 18 , and 4 I borrowed is 22 , from 23 , and there remains 1 , which I place under the 3 ; then 6 times 5 is 30 , and 2 I borrowed is 32 , from 34 , and there will remain 2 , which I place under the 4 ; then to this Remainder I bring down the next Figure in the Dividend , postponing it as I did the Cypher , and they will stand thus : I now inqui●e , how many times 5 in 21 ? and find 4 times , I then place 4 in the Quotient , and go on as before ; there being yet a Remainder , I add a Cypher , and proceed as before ; and find , upon the adding one Cypher , my Divisor greater than the Dividend , I place a Cypher in the Quotient : Example . Having placed a Cypher in the Quotient , I add another to the Dividend , and make it 800 ; and then inquire , how many times 5 in 8 ? finding once , I put 1 in the Quotient , working as before : Where note , So long as there is a Remainder , if you add Cyphers and work after this manner , you may have as many Decimals as you please . It doth often happen in Division , in decimal Fractions , or mixt Numbers , that the Unite place in the Divisor will stand beyond all the significant Figures in the Dividend , either toward the right hand or toward the left ; in which case , that you may the better find out the value of the first Figure in your Quotient ( according to the precedent General Rule ) add Cyphers to the right or to the left hand of the Dividend , till you come over the Unity place in the Divisor , and what value or denomination that place is of , that is the denomination of the first Figure in the quote ; as in these Examples . If in Division in whole Numbers , there happen to be a Remainder , it is the Numerator of a Common Fraction , and the Divisor is the Denominator , and this Fraction is part of the quotient . Example . If you divide 66 by 8 , the quotient will be 8 and ● / 8 , according to the way of Vulgar Fractions , but in Decimal Fractions it will be 8.25 . Common way . Decimal way . If you be to divide a whole by a decimal Number , cut off so many places by a mark , as there are Cyphers in the decimal Number : If 468 be divided by 10 , the quote is 46.8 ; by 100 , 4.68 ; and by 1000 , quotes .468 . If a decimal Fraction , or a mixt Number , be to be divided by a decimal Number , remove your line or point so many places toward the left hand , as there are Cyphers in your decimal Number , supplying the vacant places with Cyphers , if there be occasion : 69.5 divided by 10 , is 6.95 ; by 100 , it will be .695 ; by 1000 , .0695 ; and by 10000 , quotes .00695 ; &c. Division being the Converse of Multiplication , as multiplying a mixt Number or decimal Fraction by a decimal Number , you remove your mark of distinction toward the right hand ; so in dividing a decimal Fraction or mixt Number by a decimal Number , the mark is removed toward the left hand , as in the foregoing Examples . REDUCTION . TO reduce a vulgar Fraction into a decimal Fraction , your Rule is : Divide your Numerator by your Denominator , and the Quotient will be a decimal Fraction of the same value with the vulgar Fraction . So ¼ , if reduced into a decimal Fraction , will be .25 . Example . Here note , That only the even parts of an Integer will be exactly reduced into a decimal Fraction , as ½ , 2 / 8 , 2 / 16 , &c. In all Surds , there will be some Remainder , but if you carry your decimal Fraction to four or five places , making the last one more than it is , if the sixth Figure be above 5 , or else leave them out , and your Calculation will come near the truth ; but if any desire to be more exact , he may take as many as he please . Examples . To reduce any decimal Fraction out of a greater denomination into a lesser , multiply the Fraction by those parts of the Integer into which you would have it reduced ; as .65 being the parts of a Pound , you would know how many Shillings are contained in the Fraction , multiply it by 20 : If you desire the Pence therein contained , multiply it by 240 ; or if Farthings , multiply by 960 , the number of Farthings in a Pound or 20 Shillings . The decimal parts of a Foot are reduced , by multiplying them by 12 ; if parts of a Foot Square , by 144 ; and the decimal parts of a Foot Solid , by 1728 , the Cubick Inches in a Foot of Solid . The decimal parts of a Pound , are reduced by 16 , the Ounces in a Pound Averdupois ; and 12 , the Ounces in a Pound Troy. The decimal parts of a Beer Barrel by 36 , and by 32 reduceth the parts of an Ale Barrel , into Gallons ; and Gallons into Pints , by 8 ; Gallon● into Cubick Inches , by 282 ; and for Win● Gallons , by 231 , the number of Cubick Inches in such a Gallon , &c. As greater denominations are reduced to lesser , by a multiplication of the several part● of the Integer ; so lesser denominations ar● reduced to greater , by division . Any number of Shillings are reduced into Pounds , and the decimal parts of a Pound , if you divide them by 20 ; and Pence , if divided by 240. Example . Hours are reduced into the decimal parts of a Day , if you divide them by 24 , the Hours in a Day Natural ; and Minutes into the parts of an Hour , if divided by 60. Perches are reduced into the decimal parts of an Acre , if you divide them by 160 , the number of Square Poles or Perches in an Acre ; and any number of Feet into Poles , and the decimal parts of a Pole , if you divide them by 16.5 the Feet in a Pole , or by 158.25 the number of Square Feet in a Square Pole ; but if Wood-land Measure by 18 , or if a Square Pole by 324 , the Square Feet in a Pole or Perch of such Measure . Any number of Inches are reduced into the parts of a Beer Barrel , if divided by 10152 ; and into Ale Barrels and parts , by 9024 ; &c. For the ease of the Reader here is made a Table of English Coin reduced into the decimal parts of a Pound sterling . A Table of Reduction of English Coin , the Integer being one Pound . Shillings . Decimals .   Pence . Decimals of a Pound . 19 .95   11 .0458333 18 .9   10 .0416667 > 17 .85   9 .0375 16 .8   8 .0333333 < 15 .75   7 .0291667 > 14 .7   6 .025 13 .65   5 .0208333 12 .6   4 .0166667 > 11 .55   3 .0125 10 .5   2 .0083333 < 9 .45   1 .0441667 > 8 .4       7 .35       6 .3       5 .25   Farthings . Decimals of a Pound . 4 .2   3 .15   3 .003125 2 .1   2 .0020833 1 .05   1 .0010417 > The Vse of the Table . Having any Quest. wherein Pounds , Shillings & Pence , are required to be under one denomination , viz. Pounds , and the parts of a Pound : First seek in the Column of Shillings for your Shillings , and set down the Fraction that stands against it ; then in the Column of Pence , seek your Pence ; in the Farthings , your Farthings ; add all these together , and the Sum is the decimal Fraction desired . Example . What is the decimal Fraction for 17 s. 9 d. ¾ ? First as the decimal parts of a Pound seek for 17 s. and the Fraction against it in the other Column is .85 ; Which set down thus .85 Then against 9 d. I find .0375 And against 3 Farthings .003125 Their Sum is .890625 Which is the Number required , and is the decimal Fraction for 17 s. 9 d. ¾ , as parts of a Pound . Again , having a decimal Fraction in the parts of a Pound , and its desired to know the value thereof in lesser denominations : Let it be the Fraction before found , viz. .890625 : I seek in the Table of Fractions for the neerest to it , and find .85 , and against it 17 s. I then set .85 down , and subduct it from the other , and there remains .040625 ; I look over the Table again , and find the next neerest is .0375 , against it 9 d. I subduct that ; and find the Remainder .003125 , stand against 3 Farthings . So finding the value of any other decimal Fraction : If any thing remain after the last subduction , being less than a Farthing , I cast it away as of small regard . THE GOLDEN RULE . THis Rule is called the Rule of Three , because herein are three Numbers given , to find a fourth . It is also called the Rule of Proportion , for as the first is in proportion to the second , so is the third to the fourth : And the Converse . This Rule is called the Golden Rule for its excellent use in the Solution of Questions of various kinds , and great advantage is made of it in almost all kind of Calculations Arithmetical . Two of the three Numbers given in every Rule of Proportion are of one denomination , and the third is of the same kind with the fourth sought ; and one of the two Numbers that are of like species doth always ask the Question . Arithmeticians distinguish this Rule by two denominations , one they call the Direct , and the other the Inverse or Backer Rule of Three . One of the three given Numbers of like denomination in any Rule of Proportion is a Divisor , the other remaining two are Multipliers . To find which of the forementioned Numbers is the Divisor , take these following Rules . 1. If that Term to which the Question is annexed be more than that of like denomination , and also requires more ; or if it be less , and require less than the Term of like denomination ; then that Term of like denomination to that which asketh the Question is the Divisor , and the Question is in the Direct Rule of Three . 2. If the Term which asketh the Question be more than that of like species , and requires less ; or less , and requires more ; then that Term which asketh the Question is the Divisor , and the Question is in the Backer or Inverse Rule of Three . Having by the precedent Rules discovered the Divisor , multiply the other two Numbers , and divide by the Divisor , your quote will be the Answer to the Question . Note , If any of the Numbers given be in several denominations , they must be reduced into one , either greater or lesser , as before directed . Example . Quest. 1. If 12 ½ Yards of Taffaty cost 5 l. 7 s. 9 d. 3 q. what shall 5 ½ Yards cost ? In this Example , of the three Numbers given there are two of like denomination , and they are 12 ½ and 5 ½ , the latter of which is the Term which asketh the Question , known always by the words what or how much . And this Term is less than that of like kind with it self , and also requires less , therefore according to the precedent Rule , this Question is in the Golden Rule Direct . These three Numbers may be placed in what order you please , provided you mistake not your Divisor , but according to the general way , being reduced into Decimals , and of one species , they will stand thus : Yards . l. Yards . 12.5 5.390625 ∷ 5.5 Then , as before directed , multiply the second and third Numbers , and divide by the first , and the quotient exhibits the fourth Proportional or the Number sought . The Answer is 2 l. 7 s. 5 d. 1 q. Quest. 2. If 6 Yards of Broad Cloth cost 4 l. what shall 32 Yards cost ? Here the Term which asketh the Question is greater than the Term of like denomination , and requires more ; therefore the Term of like denomination to the Term that asketh the Question is the Divisor . The Answer is 21 l. 6 s. 8 d. Quest. 3. If 320 Men raise a Breast-work in 6 Hours , in what time will 750 Men do the same ? Here the Term that asketh the Question is more than the Term of like denomination , and requires less ; therefore the Term that asketh the Question is the Divisor , and this is the Backer Rule of Three . The Answer is 2 Hours , 33 Minutes , and 36 Seconds . Quest. 4. If 756 Men dig a Trench in 12 Hours , in how many Hours will 126 dig the same ? Here the Term that asketh the question is less than the Term of like denomination , and requires more ; then according to the Rule the Term demanding is the Divisor , and this question is also in the Inverse Rule of Three . The Answer is 72 Hours . There is sometimes four Numbers given in a question , yet is it but a Single Rule of Three , for one of the four Numbers is of no signification , and might as well have been left out . Example . Quest. 5. If 10 Workmen build a Wall 40 Foot long in 3 Days , in what time might 50 Men have done the same ? Here note , there is four numbers given , and yet there is but three to be used in working the question , you must therefore find which those 3 are that are necessarily to be used : Thus , First , you must take the Term that asketh the question , here 50 Workmen ; secondly , you must have the Term of like denomination with it , which is 10 Workmen ; thirdly , the Term sought , being Days ; you must take the Term of like denomination with that also , which is here 3 Days : The superfluous Term then in the question is 40 , which might have been left out , and they will then stand thus : The Answer is Half a Day or 12 Hours . This question is in the Rule of Three Inverse . Quest. 6. If 100 l. gain 6 l. in 12 Months , what shall 32 l. gain in the same time ? In this question the 12 Months is the superfluous Term , being of no use in the Calculation , the Terms required being 100 l. 6 l. and 32 l. Note , Though the Terms in this question be all Money , and so may seem to be of one species , yet they are not ; 100 l. and 32 l. are of one kind , being both Principal , and the other Term is of the same denomination with the Term sought , viz. Gain or Interest . The Answer is 1 l. 18 s. 4 d. 3 q. ferè . And this question is in the Direct Rule of Three , the Term that asked the question being less than the Term of like denomination , and also requiring less , &c. THE DOUBLE GOLDEN RULE . THis Rule is called the Double Golden Rule , or Double Rule of Three , because it requires two distinct Calculations , before you can answer the question . And in this Rule there are five Numbers given to find a sixth sought . This differs not in the operation from the Single Rule , only the Calculation is twice repeated . Of the five Numbers given , the question is sometimes annexed to two , and sometimes but to one . If the question be annexed to two of the five given Numbers , then are there two of the other three of the same species with those that ask the question , and the third is proportional to the Number sought . For the due regulation of these two Calculations , wh●n the question is annexed to two of the five Numbers , take these Directions . First , take one of the Numbers demanding , and let that ask the question in the first operation ; secondly , take that of the same species , and also that of the like quality with the respondent , of these three constitute your first Rule of Proportion ; then find which is your Divisor , according to your Rule pag. 55. and proceed to find the fourth in proportion . Then for your second Rule of Three , take the other of the two Numbers to which the question is annexed , and let that ask the question ; take also the Number of like kind , and the fourth Number found in the first Calculation ; judge which is your Divisor , and work accordingly ; the last Quotient will be the sixth Number , or the Number sought . Example . If a Trench be 20 Perches in length , and made by 12 Men in 18 Days ; how long may that Trench be , that shall be wrought be 48 Men in 72 Days ? Here the question is annexed to two of the five Numbers , viz. 48 Men and 72 Days ; now according to the foregoing direction , take one of the two Numbers inquiring , 48 , and say , Then take the other of the two Numbers inquiring , and say , If 6 Lighters bring 60 Tuns of Ballast in 5 Tides , how many Tun will 15 bring in 12 ? If a Man travel 160 Miles in 4 Days , when the Days are 10 Hours long ; in how many Days will he travel 195 Miles , when the Days are 14 Hours long ? When a Question is stated in the Double Rule of Three , so that there is but one Number inquiring , First , take that Number , and let it ask the question in the first Rule ; take also the Number of like denomination , together with the Number joyn'd to that of like denomination ; and of these three Numbers constitute your first Rule of Proportion . Secondly , let that Number which was found in the first Operation , ask the question in the second ; then take the Number of like denomination to it , and also the Number joyn'd with that like Number ; of these three is your second compounded ; find your Divisor , and proceed ; the last quote exhibits the Answer . Example . If 4 Crowns at London make 2 Ducates at Venice , and 8 Ducates at Venice make 20 Patacoons at Genoa ; how many Patacoons at Genoa will make 120 Crowns at London ? Of the Square Root . A Square is a plain Superficies bounded with four right Lines of equal length , the Angles also are equal , being all right Angles , as ( a b c d ) The measure of a Square is by a Square , that is , when it is known how many Square Inches , Feet or Perches , is contain'd in any Superficies , the Content or Area of the said Superficies is then said to be known . And in a Square , it is found by multiplying the length by the breadth , which being equal , it is called Squaring of a Number , and by the Learned Dr. Pell , Involution , and the Product or Area is the second Power ; now the Side of such a Square is by Geometricians called a Root or the first Power . Let the Side a b be 222 Inches , Feet , or Perches , &c. 49284 the Area or q square . Number . Now having the Area of a Square or Square Number given , and the Side or Root be required . This is called the Extraction of a Square Root , and also Evolution of the second Power . Let the Number be as before 349284. The first thing to be done in the Extraction of a Root is punctation , or pointing the Number given ; which is thus done , first set a point over the Unite-place , and omitting one point every other Figure thus , 349284 ; there being three points in the Number , intimates three figures in the Root To proceed then , enquire the greatest Square Number contained in those figures , under the first point on the left hand ; the greatest Square Number in 34 is 25 , whose Root is 5 , which place in the quotient for the first figure in the Root , subduct its Square out of 34 , and set the Remainder 9 underneath as in the Example . Example . The first figure in the Root thus found , the rest are found by Division ; for a Dividend bring down the figures under the nex● point , and postpone them to the last Remainder , and the Example will stand thus , your Divisor being double the Root found . Then I proceed to Division , always supposing the last Figure in my Divisor standing unde● the last save one in the Dividend ; the Number to be subducted from the Dividend must always be the Square of the last Figure in the Root , and the Divisor multiplied by the last Figur● in the Root , so added together as in this Example , viz. so that the Unite-place in the last Number stand one place further to the righ● hand . Nine times the Divisor 90 And 9 multiplied by it self 81 981 Which being subducted from the Dividend will remain 11 , as part of a new Dividend , to them bring down the two next figures , and the Example will stand thus : The Divisor as before is double the whole Root found , and for the Number to be subducted , after you have made enquiry how many times the Divisor will be found in the Dividend , if so placed as aforesaid , it will here be found once , then place ●r in the quotient for the third figure in the Root , the Number to be subducted will be as before , and the Example will stand thus : The Divisor multiplied by the last Figure found , and the Square of that Figure placed as before directed . Rectangle — 118 Square — 1 1181 1181 subducted from the Dividend 1184 There will remain but — 3 Which sheweth the Number was not a Square Number ; but if you desire to have it further , add two Cyphers to the Remainder for a new Dividend , double your whole quotient for a new Divisor , and you may have as many Decimals as you please . Tabula Laterum Quadratorum ab Unitate ad 200. Quadrata . Latera . 1 1 , 00000 , 000000 2 1 , 41421 , 356237 3 1 , 73205 , 080757 4 2 , 00000 , 000000 5 2 , 23606 , 797750 6 2 , 44948 , 974278 7 2 , 64575 , 131106 8 2 , 82842 , 712474 9 3 , 00000 , 000000 10 3 , 16227 , 766017 11 3 , 31662 , 479036 12 3 , 46410 , 161514 13 3 , 60555 , 127546 14 3 , 74165 , 738677 15 3 , 87298 , 334621 16 4 , 00000 , 000000 17 4 , 12310 , 562562 18 4 , 24264 , 068712 19 4 , 35889 , 894354 20 4 , 47213 , 595500 21 4 , 58257 , 569496 22 4 , 69041 , 575982 23 4 , 79583 , 152331 24 4 , 89897 , 948556 25 5 , 00000 , 000000 26 5 , 09901 , 951359 27 5 , 19615 , 242271 28 5 , 29150 , 262213 29 5 , 38516 , 480713 30 5 , 47722 , 557505 31 5 , 56776 , 436283 32 5 , 65685 , 424948 33 5 , 74456 , 264654 34 5 , 83095 , 189485 35 5 , 91607 , 978310 36 6 , 00000 , 000000 37 6 , 08276 , 253030 38 6 , 16441 , 400297 39 6 , 24499 , 799840 40 6 , 32455 , 532034 41 6 , 40312 , 423743 42 6 , 48074 , 069841 43 6 , 55743 , 852430 44 6 , 63324 , 958071 45 6 , 70820 , 393250 46 6 , 78232 , 998313 47 6 , 85565 , 460040 48 6 , 92820 , 323028 49 7 , 00000 , 000000 50 7 , 07106 , 781185 51 7 , 14142 , 842854 52 7 , 21110 , 255093 53 7 , 28010 , 988928 54 7 , 34846 , 922835 55 7 , 41619 , 848710 56 7 , 48331 , 477355 57 7 , 54983 , 443527 58 7 , 61577 , 310586 59 7 , 68114 , 574787 60 7 , 74596 , 669241 61 7 , 81024 , 967591 62 7 , 87400 , 787401 63 7 , 93725 , 393319 64 8 , 00000 , 000000 65 8 , 06225 , 774830 66 8 , 12403 , 840464 67 8 , 18535 , 277187 68 8 , 24621 , 125124 69 8 , 30662 , 386292 70 8 , 36660 , 026534 71 8 , 42614 , 977318 72 8 , 48528 , 137424 73 8 , 54400 , 374532 74 8 , 60232 , 526704 75 8 , 66025 , 403784 76 8 , 71779 , 788708 77 8 , 77496 , 438739 78 8 , 83176 , 086633 79 8 , 88819 , 441732 80 8 , 94427 , 191000 81 9 , 00000 , 000000 82 9 , 05538 , 513814 83 9 , 11043 , 357914 84 9 , 16515 , 138991 85 9 , 21954 , 445729 86 9 , 27361 , 849550 87 9 , 32737 , 905309 88 9 , 38083 , 151965 89 9 , 43398 , 113206 90 9 , 48683 , 298050 91 9 , 53939 , 201417 92 9 , 59166 , 304663 93 9 , 64365 , 076099 94 9 , 69535 , 971483 95 9 , 74679 , 434481 96 9 , 79795 , 897113 97 9 , 84885 , 780180 98 9 , 89949 , 493661 99 9 , 94987 , 437107 100 10 , 00000 , 000000 101 10 , 04987 , 562112 102 10 , 09950 , 493836 103 10 , 14889 , 156509 104 10 , 19803 , 902719 105 10 , 24695 , 076596 106 10 , 29563 , 014099 107 10 , 34408 , 043279 108 10 , 39230 , 484541 109 10 , 44030 , 650891 110 10 , 48808 , 848170 111 10 , 53565 , 375285 112 10 , 58300 , 524426 113 10 , 63014 , 581273 114 10 , 67707 , 825203 115 10 , 72380 , 529476 116 10 , 77032 , 961427 117 10 , 81665 , 382639 118 10 , 86278 , 049120 119 10 , 90871 , 211464 120 10 , 95445 , 115010 121 11 , 00000 , 000000 122 11 , 04536 , 101719 123 11 , 09053 , 650641 124 11 , 13552 , 872566 125 11 , 18033 , 988750 126 11 , 22497 , 216032 127 11 , 26942 , 766958 128 11 , 31370 , 849898 129 11 , 35781 , 669160 130 11 , 40175 , 425099 131 11 , 44552 , 314226 132 11 , 48912 , 529308 133 11 , 53256 , 259467 134 11 , 57583 , 690279 135 11 , 61895 , 003862 136 11 , 66190 , 378969 137 11 , 70469 , 991072 138 11 , 74734 , 012447 139 11 , 78982 , 612255 140 11 , 83215 , 956620 141 11 , 87434 , 208704 142 11 , 91637 , 528781 143 11 , 95826 , 074310 144 12 , 00000 , 000000 145 12 , 04159 , 457879 146 12 , 08304 , 597359 147 12 , 12435 , 565298 148 12 , 16552 , 506060 149 12 , 20655 , 561573 150 12 , 24744 , 871392 151 12 , 28820 , 572744 152 12 , 32882 , 800594 153 12 , 36931 , 687685 154 12 , 40967 , 364599 155 12 , 44989 , 959799 156 12 , 48999 , 599680 157 12 , 52996 , 408614 158 12 , 56980 , 508998 159 12 , 60952 , 021292 160 12 , 64911 , 064067 161 12 , 68857 , 754045 162 12 , 72792 , 206136 163 12 , 76714 , 533480 164 12 , 80624 , 847487 165 12 , 84523 , 257867 166 12 , 88409 , 872673 167 12 , 92284 , 798332 168 12 , 96148 , 139682 169 13 , 00000 , 000000 170 13 , 03840 , 481041 171 13 , 07669 , 683062 172 13 , 11487 , 704860 173 13 , 15294 , 643797 174 13 , 19090 , 595827 175 13 , 22875 , 655532 176 13 , 26649 , 916142 177 13 , 30413 , 469565 178 13 , 34166 , 406413 179 13 , 37908 , 816026 180 13 , 41640 , 786500 181 13 , 45362 , 404707 182 13 , 49073 , 756323 183 13 , 52774 , 925847 184 13 , 56465 , 996625 185 13 , 60147 , 050874 186 13 , 63818 , 169699 187 13 , 67479 , 433118 188 13 , 71130 , 920080 189 13 , 74772 , 708488 190 13 , 78404 , 875209 191 13 , 82027 , 496109 192 13 , 85640 , 646056 193 13 , 89244 , 398945 194 13 , 92838 , 827718 195 13 , 96424 , 004377 196 14 , 00000 , 000000 197 14 , 03566 , 884762 198 14 , 07124 , 727947 199 14 , 10673 , 597967 200 14 , 14213 , 562373 The Use of the precedent Table is principally for the ease of the industrious Artist ; when he hath the Extraction of a Square Root in the Solution of any Question , it is but seeking the given Number in the Table , and just against it he shall find the Root . By the subsequent Examples will it plainly appear , how useful such a Table to 1000 Roots would be in quadratique Equations , and in the Cubes also , which ( were there incouragement given to the Sons of Art ) I doubt not but some ingenious Person would enrich the World therewith ; these being long since Calculated by Mr. Henry Briggs of Oxford , and given me by my honoured Friend , Mr. Iohn Collins , his desire being to have them made more publick , and the conveniency of such a Table ( before mentioned ) shewn , by some Examples upon this . Of Quadratique Equations . Mr. Dary , in his Miscellanies , chap. 8. saith to this , or the like purpose : 1. When any Equation propos'd is incumbred with Vulgar Fractions , let it be reduced to its least Terms in whole Numbers , if possible ; if not , let it be brought to its least Terms in Decimals . 2. It is evident from divers Authors , That if any Quant●ty shall be signed − , then the Square Root , or the Root of any even Power of such Quantity so sign'd , is inexplicable , for they cannot be generated from any Binomials that shall be equal . As for Example . − 9 being a Negative can be made of nothing ( if taken as a Square Number ) but + 3 and − 3 , which Roots are not equal , they being neither both Affirmatives nor both Negatives . 3. When you have cleared the Equation by the Second hereof , and that the Co-efficient in the highest Power is taken away , or be Unity , then will quadratique Equations resolve themselves into the four following Compendiums . 4. Let your Equation be so reduced , that the highest Power stand on the left side alone , the sign + being always annexed , or supposed to be annexed . Example , Quesita a. First Equation . Second Equation . Third Equation . Fourth Equation . Illustration by Numbers , Quesita a. First Equation . Proof of the Affirmative . The Square of 9 = 81 Six times the Root = 54 To which add 27 81 Which was to be proved . Proof of the Negative . Example 2. Second Equation . Example 1. Example 2. Third Equation . Example 1. Example 2. Fourth Equation . Example 1. Example 2. But if in a Square Equation there happen to be a Coefficient annexed to the highest Power , it is resolved by transferring the Coefficient with the Sign of Multiplication to the other side . Admitting the Equation be Then the Coefficient 2 being transferred ( as before directed ) they will stand as in this Example . First Equation . The Root of +25 being +5 , then is +5+3 = 8 , and a = +4 , the Affirmative Answer . And +3 − 5 is = − 2 , and a = − 1 , the Negative Answer . The Proof is easie : First , if a be = 4 , 2 aa is = + 32 , and 6 a is = +24 , to which +8 being added , the Sum is +32 which was to be proved . Again , a = − 1 , then 1 aa is = − 2 , whereto +8 being added , the Sum is = +6 , which also was to be done . Second Equation . Now + 13 + the √ 121 , viz. + 11 is = +24 , the ¼ whereof is = +6 = a , and aa = 36 , and 4 aa = +144 , +26 a = +156 , to which if − 12 be added , the Sum will be +144 also . Again , If to +13 you add − the √ 121 , viz. − 11 , 4 a will be = +2 , and consequently + a = +½ , 4 aa is then = +1 , and +26 = +13 , to which add − 12 , and the Sum is = +1 , which was to be proved . Third Equation . − 3+13 = +5 a , here a = +2 , 5 aa = +20 , − 6 a = − 12 , to which add +32 , the Sum is also +20. Again , − 3 − 13 = − 16 = − 5 a , and a = − 3.2 , 5 aa = +51.2 : Also − 6 a being = +19.2 , to which add +32 , the Sum is = +51.2 . Fourth Equation . Which was to be done . Note , Always where there is no Sign annexed to any Term in the Equation , the Sign + is supposed to be annexed . I have been the larger in these Examples , that the young Analist may with the more ease apprehend the several kinds by this variety ; in some o● the surd Roots I have on purpose omitted the large number of Places , four or five being sufficient for use in most cases ; but if any desire to be more exact , he may take them as far as he pleaseth , or the Table doth exhibit . Of the Cube Root . THe Cube is a Solid , and hath three dimensions , length , breadth , and depth , and is inclosed by six plain square Superficies . Example . Let the Side a , b , or c , d , &c. be 125 : To find the Content in Solid Feet or Inches , is the Involution of the Side or Root . Thus : The Solidity . And this is called the Third Power . The Evolution hereof , is also termed the Extraction of the Cube Root , wherein observe first your punctation , omitting two , point every third Figure . Example . The first Figure in the Root is found by taking the greatest Cube Number , contained in the Figure or Figures that stand under the first Point towards the left hand , here 71 , whose Root is 4 , therefore that 4 must be placed in the Quotient as the first Figure in the Root , and the Example will stand thus : Then the Cube of 4 is 64 , which subduct out of the first Figures , and set down the remainder if any be . The first Figure found in this peculiar manner , the rest are found by Division thus : The Dividend consists of the remainder , if any be , and the three Figures under the next Point postponed ; the Divisor is always three times the Square of the Root , and three times the Root it self : These two Numbers being so to be added together , as that the Unites of the first stand over the Tens of the second . Three times the q square . of the √ = 48 Three times the √ = 12 Divisor 492 Then will the Example stand thus : Then proceed to Division , always supposing the last Figure in the Divisor to stand under the last save one in the Dividend , and enquire , how many times 4 in 7 ? place 1 in the Quotient . Then for your Number to be subducted out of the Dividend , it always consists of three Numbers , viz. Three times the q square . of the first Figure = 48 Multiplied by the second 1 Product of the first by the second 48 3 times the q square . of the second by the first 12 And the Cube of the second 1 The Subducend 4921 The q square . of 41 = 1681 3 q square of 41 × 3 = 5043 Then for a new Dividend , bring down the three next Figures , postponing them as before . The Divisor thrice the q square . of the √ 41 = 5043 And three time the √ 41 = 123 50553 Which being set on the left hand the Dividend , stands thus : Then enquire , how many times 5 in 30 ? you will find ● times , which place in the Quotient . Your Subducend is as before , 3 times the Square of 41 × 5 = 25215 3 times the Square of 5 × 41 = 3075 And the Cube of 5 = 125 The Subducend 2552375 Which being subducted from 3032125 There will remain 479750 Which shews the Number was not a Cube Number ; if you add three Cyphers , and work as before , you may have as many Decimals Fractions as you please . In this Extraction I have not taken the same Number the Cube first mentioned did produce , but by adding another Figure , made the Number greater , that it might take in all Cases ; but in the following Extraction it is explicated . Three times the Square of the Root = 3 Three times the Root is also = 3 The first Divisor = 33 Three times the Square of 1 = 3 2 Three times the Square of 1 × 2 = 6 Three times the Square of 2 × 1 = 12 The Cube of 2 = 8 The first Subducend = 728 Three times the Square of the Root = 432 Three times the Root = 36 The second Divisor = 4356 Three times the Square of the √ 12 = 432 5 Three time the Square of 12 × 5 = 2160 Three times the Square of 5 × 12 = 900 The Cube of 5 = 125 The second Subducend = 225125 A SHORT TREATISE OF SIMPLE & COMPOUND INTEREST : WITH TWO TABLES FOR THE CALCULATION OF The Value of Leases or Annuities by Quarterly Payments , at 6 per Cent. per Annum . By Iohn Mayne . London , Printed by William Godbid , for Nath. Crowch , in Exchange-Alley . M. DC . LXXIII . Of Simple Interest . QUestion 's in Simple Interest are wrought by the Double Rule of Proportion , wherein five Numbers are given to find the sixt . And if you put P = 100 Principal , and T for Twelve Months , G = 6 l. the Rate of Interest , and p = any other Sum greater or lesser , t = any other Time ( above or under Twelve Months ) and also g = to the Gain thereof at that Rate . Then if any one Term of these six be unknown , it is explicated by the other five ( like Symbols having the same denomination ) as in this Equation . PTg = ptG That is the Fact of 100 l. multiplied by one year , and that Product by 6 the Interest of 25 l. for 4 years , is equal to the Fact of 25 multiplied by 4 years , and that Product by 6 the Interest of 100 l. for one year . Example . Which was to be proved . Now forasmuch as the usual Questions of Simple Interest , are proposed from a Sum presently due to the Gain thereof , & contra ; it will be requisite you put A = the Amount of a Sum , forborn or due hereafter , and then you will have A = p+g , as in the former Equation . Example . P T P+G As 100 in 1 to 106 So 25 in 4 to 31 = p+g From the precedent Analogism will arise these four Propositions . Prop. I. A Sum presently due = p , being forborn a certain time = t , at a certain rate = G , per Cent. per Annum : Q. the Amount = A ? Equation A = That is , the given Sum multiplied by the given Time , and that Product again multiplied by the given Rate of Interest , the last Product divided by the Principal , viz. 100 , in the Time , viz. 1 , exhibits the Gain of that Sum in that Time. Illustration . Quest. 1. 25 l. being forborn 18 Months , at 6 per Cent. per Annum ; what doth it amount to ? The Answer being 27 l. 5 s. the Amount in that time . Quest. 2. If 175 l. be forborn for 7. Years , at 6 per Cent. per Annum , Simple Interest ; what will it amonut to at the end of the said time ? The Answer = 248.5 That is 248 l. 10 s. Prop. II. A Sum of Money = A , due at a certain time hereafter = t , at a certain Rate of Interest = G , per Cent. per Annum . Q. The present worth = p ? Equation , p = That is , the Fact of the Amount multiplied by the Principal , 100 , in the Time , viz. 1 Year , divided by the said Principal multiplied into the said Time , more the Rate of Interest multiplied into the Time of Forbearance , the Quotient is equal to the present worth . Example . Quest. 1. If 248 l. 10 s. be due at the end of 7 years , what is it worth in ready money , discompting Interest , at 6 per Cent. per Annum . The Answer is 175 l. Quest. 2. If 950 l. be due at the end of 12 years , what is it worth in ready money , at 9 per Cent. per Annum Simple Interest ? The Answer is 456 l. 14 s. 7 d. ¼ ferè . Prop. III. A Sum presently due = p , having been forborn ● time unknown = t , did amount to a certain Sum = A , at a Rate of Interest = G , per Cent. per Annum . Q. the Time of forbearance = t ? Equation , That is to say , the Amount less the Principal , so increased , multiplied by 100 , and that Product divided by the Fact of the before-mentioned Principal , and Rate of Interest , quotes the Time of forbearance . Example . Quest. 1. If 175 l. hath been forborn till with the Interest at 6 per Cent. per Annum it is increased to be 248 l. 10 s. Q. How long hath it been forborn ? The Answer is 7 years . Quest. 2. If 25 l. hath been forborn till it is amounted to 27 l. 5 s. at 6 per Cent. per Annum , Simple Interest . Q. In what time is it so increased ? The Answer 1 year and an half . Prop. IV. A Sum of Money = p , being forborn a certain time = t , and at the end of that Term did amount to a Sum = A. Q. At what Rate of Interest ? Equation , Or from the Amount subduct the Principal , and the Remainder multiply by 100 , that Product divided by the Principal multiplied by the Time , the Quotient will be = G the Rate of Interest , per Cent. per Annum . Illustration . Quest. 1. If 250 l. forborn 3 year● and 6 months , did amount to 324 l. 7 s. 6 d. at what Rate of Interest did it so increase ? The Answer is 8 l. 10 s. Quest. 2. If 175 l. being forborn 7 years , did amount to 248 l. 10 s. what Rate of Simple Interest per Cent. per Annum was it accompted at ? The Answer is 6 l. If one month be taken for the 1 / 12 of a year , the business of Interest and Rebate is very easily performed by a small Table of the Amounts of 1 l. for any number of months , not exceeding 12 ; which Table is made by this Analogy , 100. 106 : : 1.1.06 . A Table of the Increase of 1 l. at 6 per Cent per Ann. Months . Value . 12 1.06 11 1.055 10 1.05 9 1.045 8 1.04 7 1.035 6 1.03 5 1.025 4 1 . 0● 3 1.015 2 1.01 1 1.005 If the Question be of the Amount of any Sum forborn any number of months , at 6 per Cent. per Annum , multiply the given Sum by the Tabular Number for that time , and the Product answers the Question . Example . If 125 l. be forborn 10 months , what will it amount to ? The Principal = 125 The Tabular Number for 10 months = The Answer is 131 l. 5 s. If the Question be only what is the Interest of any Sum for any time , then multiply the Sum for that time by the Tabular Number less an Unite . Example . What is the Interest of 125 l. for 10 months ? The Principal = 125 The Tabular Number less an Unite = .05 6.25 The Answer is 6 l. 5 s. prout suprà . For Discompt or Rebate of any Sum to be forborn , the present worth is found by dividing the given Sum by the Tabular Number . Example . What is the present worth of 131 l. 5 s. due at the end of 10 months ? The Answer is 125 l. But if any desire to be more exact , let him multiply the Interest of 1 l. for 1 day ( which is .000164384 ) by the number of days , and that Product by the given Sum , and the last Product will be the Interest for that Sum forborn the time given . Example . What is the Interest of 125 l. forborn from the Tenth of March to the Tenth of Ianuary following , viz. 305 days ? The Answer is 6 l. 5 s. 4 d ferè . Discompt is performed by Division , viz. get the Amount of 1 l. for the time required , by which divide the given Sum , and the Quote is the present worth . Of Mean Time. It hath been a custome amongst Merchants , in their Contracts upon Sale of Commodities , to agree upon divers times of payment , as two three-months , three six-months , &c. Now to find a time between these , wherein the whole Sum may be paid at one entire Payment without detriment to either Party , the subsequent Table doth shew upon the first inspection . A Table for Equation of Time.   1 is 1.5   2 is 3   3 is 4.5   4 is 6 2 5 is 7.5   6 is 9   7 is 10.5   8 is 12   9 is 13.5   1 is 2   2 is 4   3 is 6   4 is 8 3 5 is 10   6 is 12   7 is 14   8 is 16   9 is 18   1 is 2.5   2 is 5   3 is 7.5   4 is 10 4 5 is 12.5   6 is 15   7 is 17.5   8 is 20   9 is 22.5   1 is 3   2 is 6   3 is 9   4 is 12 5 5 is 15   6 is 18   7 is 21   8 is 24   9 is 27   1 is 3.5   2 is 7   3 is 10.5   4 is 14 6 5 is 17.5   6 is 21   7 is 24.5   8 is 28   9 is 31.5   1 is 4   2 is 8   3 is 12   4 is 16 7 5 is 20   6 is 24   7 is 28   8 is 32   9 is 36   1 is 4.5   2 is 9   3 is 13.5   4 is 18 8 5 is 22.5   6 is 27   7 is 31.5   8 is 36   9 is 40.5   1 is 5   2 is 10   3 is 15   4 is 20 9 5 is 25   6 is 30   7 is 35   8 is 40   9 is 45 The manner of making this Table , is no more than adding one Term to the given number of Terms , and take half the Sum. Example . Is three four-months given , add 4 to 12 , the Sum will be 16 , half that Sum , viz. 8 months , is the equated Time of Payment . This indeed is but an approximation , though near enough the truth for practice . That excellent Accomptant Mr. Collins , in a Sheet Printed Anno 1665. hath taught a more exact way of Equation : Simple Interest , prop. 4. Compute ( saith he ) all the present worths , and then by proportion , if all those present worths amounted to the total of all those Payments , what did 1 l. amount to in the said time ? From the result subtract an Vnite , the Remainder is the Interest of 1 l. for the time sought , which divide by the Interest of 1 l. for one Day , and the Quote is the Number of Days sought . Example . A Merchant sold Wines for 300 l. and hath given the Vintner three six-months for Payment , viz. to pay 100 l. at the end of 6 months , another at 12 , and the third 100 l. at 18 months end ▪ the Question is , At what time may this Vintner pay 300 l. together , without detriment to himself or the Merchant . 100l . at 6 months is worth — 97.087 100l . at 12 months is worth — 94.34 100l . at 18 months is worth — 91.826 The present worths = 283.253 The Interest of 1 l. for the time is .059123786 The Interest of 1 l. for 1 day is = .000164384 The Answer is 359 days and a half , ferè . By the Table , three six-months gives twelve months for the equated time , which you find above five days less than a year by this Calculation . A Decimal Table of the present worth of One Pound , Quarterly Payment , at 6 per Cent. per Annum , Simple Interest , for 124 Quarters . 1 .985222 2 1.956095 3 2.913033 4 3.856429 5 4.786662 6 5.704093 7 6.609071 8 7.501928 9 8.382985 10 9.252550 11 10.110919 12 10.958377 13 11.795197 14 12.621643 15 13.437970 16 14.244421 17 15.041234 18 15.828636 19 16.606846 20 17.376077 21 18.136533 22 18.888413 23 19.631907 24 20.367201 25 21.094474 26 21.813898 27 22.525642 28 23.229868 29 23.926732 30 24.616387 31 25.298981 32 25.974656 33 26.643553 34 27.305804 35 27.961542 36 28.610893 37 29.253979 38 29.890922 39 30.521837 40 31.146837 41 31.766032 42 32.379529 43 32.987432 44 33.589841 45 34.186856 46 34.778572 47 35.365082 48 35.946478 49 36.522847 50 37.094275 51 37.660848 52 38.222645 53 38.779748 54 39.332235 55 39.880180 56 40.423658 57 40.962742 58 41.497501 59 42.028005 60 42.554321 61 43.076514 62 43.594649 65 44.108787 64 44.618992 65 45.125321 66 45.627833 67 46.125686 68 46.621636 69 47.113036 70 47.600841 71 48.085103 72 48.565872 73 49.043199 74 49.517133 75 49.987721 76 50.455010 77 50.919048 78 51.379877 79 51.837543 80 52.292088 81 52.743556 82 53.191986 83 53.637421 84 54.079898 85 54.519459 86 54.956140 87 55.389980 88 55.821014 89 56.249280 90 56.674811 91 57.097644 92 57.517813 93 57.935349 94 58.350287 95 58.762658 96 59.172494 97 59.579826 98 59.984684 99 60.387099 100 60.787099 101 61.184713 102 61.579970 103 61.792897 104 62.363522 105 62.751872 106 63.137972 107 63.521849 108 63.903529 109 64.283035 110 64.660394 111 65.035628 112 65.408763 113 65.779820 114 66.148824 115 66.515796 116 66.880760 117 67.243736 118 67.604747 119 67.963814 120 68.320956 121 68.676196 122 69.029553 123 69.381047 124 69.730697 The Use of the precedent Table is principall● to shew the present worth of any Lease or A●nuity , payable Quarterly , for any term 〈◊〉 years under 31 , at 6 per Cent. per Annu● Simple Interest . Example . There is a Lease for 18 years to be sold , of the yearly value of 160 l. payable Quarterly , viz. 40 l. per Quarter , what is this Lease worth in ready money , allowing the Purchaser 6 per Cent. Simple Interest ? 72 Quarters per Table = 48.565872 The Answer is 1942 l. 12 s. 8 d. ¼ ferè . The Inversion of the Question , viz. What Quarterly Payment for 18 years will 1942 l. 12 s. 8 d. ¼ purchase ? As the former was done by Multiplication , where the Product exhibits the Answer ; so if the Sum proposed be divided by the Tabular Number , the Quote gives your Answer . Example . The Answer is 40 l. Of Compound Interest . AS Simple Interest is performed by a Serie● of Musical , so is Compound Interes● wrought by a Rank of Geometrical continua● Proportionals . The operation whereof by th● Canon of Logarithms , take under these four Considerations . Prop. I. If you shall put p = the Logarithm of a Principal or Sum forborn , and t = the time o● forbearance in years , quarters , months , or day , r = the Logarithm of the Rate of Interest , per cent . per annum , per mensem , or per diem , a = the Logarithm of the Amount of the said Principal for the said time , at the Rate also aforesaid : Then Q. The Amount = a ? Equation , a = rt + p. That is , Multiply the Logarithm of the Rate by the Number of Years , Quarters , &c. to which Product add the Logarithm of the Principal , and the Aggregate is equal to the Logarithm of the Amount . Example . Quest. 1. If 175 l. be forborn 7 years , what will it amount to at 6 per Cent. per Annum , Compound Interest ? Log. of the Rate = 0,02530586 = r Log. of the Sum = 2,24303805 = 175 = p The Answer 263 l. 2 s. 8 d. ¼ ferè . Quest. 2. If 1000 l. be forborn for 6 months , at 6 per Cent. per Annum , Compound Interest , what will it amount to ? Log. of the former Rate divided by 12 , the months in a year , is = 0,00210882 = r Add the Log. of 1000 viz. 3,00000000 1029.563 = Log. Amount 3,01265292 = a The Answer 1029 l. 11 s. 3 d. ferè . Prop. II. A Sum of Money unknown , being forborn a certain time = t , at a given Rate of Interest = r , is amounted to a given Sum = a ; Q What was p ? Equation , p = a − rt . From the Logarithm of the Amount , subduct the Logarithm of the Rate , multiplied by the time , and the Remainder is the Logarithm of the Principal . Example . Quest. 1. If 263 l. 2 s. 8 d. ¼ be the Amount of a Sum forborn 7 years , at 6 per Cent. per Annum , Compound Interest , what was the Principal ? Log. of the Rate = 0,02530586 = r Log. of the Amount 2,42017910 Log. of the Principal 2,24303805 = p = 175 The Answer 175 l. Quest. 2. If 102 l. 11 s. 3 d. be the Principal and Interest of a Sum of Money forborn 6 months , at 6 per Cent. per Annum , Compound Interest , what was the Principal ? Log. of Rate for 1 mo. 0,00210882 Log. of 1029.563 = 3,01265292 Log. of the Principal 3,00000000 = 1000 The Answer 1000 l. Prop. III. A Sum of Money = p , being forborn for a time = t , did amount to a given Sum = a , at a Rate of Interest unknown : Q. The Rate per Cent. per Annum = r ? Equation , Divide the Logarithm of the Amount , less the Logarithm of the Principal , by the Time , and the Quote is the Logarithm of the Rate . Example . If 25 l. forborn 4 years , did amount to 31 l. 11 s. 2 d. ¼ ; at what Rate of Compound Interest did it so increase ? Logarithm of the Amount = 1,49808345 Logarithm of the Principal = 1,39794001 a − p divide by 4 ) 0,10014344 The Log. of the Rate = r = 0,02503586 Prop. IV. A Sum of Money being forborn , at a given Rate , for a time unknown , but the Amount is known , how long was it so forborn ? Equation , Example . If 1000 l. be increased to 10●9 l. 11 s. 3 d. at 6 per Cent. per Annum , Compound Interest , in what time was it so increased ? The Answer 6 months . It may here be expected that I should lay down the Construction of the Logarithms , having made use of them in these Calculations , but this being design'd a small Enchiridion , and there being large Volumns of that Subject in the World already , by several more learned Pens , I think it unnecessary to say any thing further thereof , for as they are of excellent use , so are they easie to be had . COmpound Interest Infinite , may be so called as it relates to divers equal Payments at equal times , but the number of those equal times are infinite , ( i. e. ) when an Estate in Fee-Simple shall be sold for ever . Now there being usually an interval of time , between the Purchasers Payment and the reception of his first Rent , be it yearly , half yearly , or quarterly ; Any Question of this Nature may be wrought by the following Analogism : Putting V = the Rent ( yearly or quarterly ) and S = the Price paid for the Land , also R = the Common Factor of the Rate of Interest , per Cent. per Annum . Hence then may arise these three Propositions . Prop. I. There is a Fee Simple to be sold , what is it worth in ready money , so that the Purchaser may have 6 per Cent. per Annum , Compound Interest , allowed for his money . Quest. 1. There is a Manour to be sold of the clear yearly value of 969 l. 18 s. what Sum of ready money is this Estate worth , 6 per Cent. per Annum Compound Interest being allowed the Purchaser for his money ? Equation , The Annual ( or Quarterly ) Payment , divided by the Ratio , less Unity , exhibits the Sum in the Quotient . The Answer is 16165. Quest. 2. There is an Estate of 969 l. 18 s. per Annum ▪ payable Quarterly , what is it worth in ready money , allowing the Purchaser 6 per Cent. per Annum Compound Interest ? The Answer is 16524 l. 2 s. 6 d. ferè . The difference between Yearly and Quarterly Payments in this Purchase raiseth the value 359 l. 2 s. 6 d. ☞ Having the increase of 1 l for a Year , at any Rate of Interest , the Biquadrate Root of that Increase , is the Increase of 1 l. for a Quarter at Compound Interest . Prop. II. A Sum of money lying ready for a Purchase , and it be desired to know what Free-hold Estate such a Sum will purchase , if laid out at a given Rate per C. per Ann. Compound Interest . Theorem , V = S × R − 1. Or , in other terms , the Sum of Money multiplied by the Rate , less Unity , the Product shall be equal to the Annual half quarterly or quarterly Payment . Quest. A Gentleman upon Marriage of his Daughter promiseth to lay out 1600 l. for a Free-hold Estate , to be settled upon her and her Heirs , provided he meet with such a Pennyworth as shall bring 8 per Cent. per Annum , Compound Interest for his money : Q. What Annual Rent must it be ? The Answer 128 l. per Annum . Prop. III. An Estate being offered for a certain Sum of money , the annual Rent is also known : Q What Rate of Interest upon Interest shall the Purchaser have for his money ? Equation , V ÷ S = R − 1. The annual Rent being divided by the Sum demanded , quotes the Rate less Unity . Example . Quest. 1. There is a Free-hold Estate to be sold for 1600 l. the yearly Rent being 128 l. what Rate of Interest shall the Purchaser have for his money ? Quest. 2. Admit there be a small Farm to be sold of the Value of 35 l. per Annum for 500 l. what Rate of Compound Interest shall the Purchaser have for his money at that price ? Furthermore , if it be inquired how many years Purchase any Annuity is worth , putting R = the Ratio as before , and Y the number of Years , the Rule is : That is , Divide Unity by the Ratio less 1 , and the Quote informs the Number of Years . Example . There is a Free-hold Estate to be sold , Q. How many Years Purchase is it worth at 5 per Cent. per Annum ? The Answer is 20 Years Purchase . What is it worth at 6 per Cent. p●r Annum ? The Answer is 16 Years , and ⅔ of a Year . Again , if an Estate be offered at any number of Years Purchase , and it be demanded what Rate of Interest the Purchaser shall have for his Money , the Rule is : That is , Divide Unity by the number of Years propos'd , and the Quote exhibits the Ratio , less Unity . Example . An Estate is offered at 20 Years Purchase , what Rate of Interest shall the Purchaser then have ? The Answer is 5 per Cent. per Annum . There are many Tables of Compound Interest Printed in sundry Books for the valuation of Leases and Annuities , but they are generally made for yearly Payments , when indeed by the common and most usual Covenants in Leases the Tenant is obliged to pay quarterly ; and in Leases of great value , there will be found a considerable difference in the true worth , ( so great , that 25 l. per Quarter is as good as 102 l. 5 s. per Annum . ) I have therefore presented the Reader with a Table fitted to such Quarterly Payments , the Use of which Table I doubt not but will be very easily found by the Examples that follow . A Table of Interest , for the Valuation of Leases or Annuities for Quarterly Payments , at 6 per Cent. per Annum , Compound Interest , for 31 Years . 1 .985538 2 1.956824 3 2.914064 4 3.857459 5 4.787213 6 5.836372 7 6.609520 8 7.496573 9 8.373700 10 9.238139 11 10.090079 12 10.929724 13 11.757176 14 12.572685 15 13.376402 16 14.168496 17 14.949134 18 15.718484 19 16.476707 20 17.223982 21 17.960417 22 18.686219 23 19.401524 24 20.106484 25 20.801250 26 21.485968 27 22.160789 28 22.825841 29 23.481282 30 24.127242 31 24.763844 32 25.391256 33 26.009595 34 26.618988 35 27.219206 35 27.811474 37 28.394861 38 28.969764 39 29.536352 40 30.094714 41 30.645034 42 31.187396 43 31.721914 44 32.248000 45 32.767870 46 33.279531 47 33.783794 48 34.280753 49 34.843000 50 35.253244 51 35.728999 52 36.197819 53 36.659861 54 37.115237 55 37.564028 56 38.006330 57 38.442234 58 38.871893 59 39.295222 60 39.712487 61 40.123777 62 40.529043 65 40.928469 64 41.322133 65 41.710087 66 42.092531 67 42.469245 68 42.841611 69 43.206601 70 43.567307 71 43.922768 72 44.273138 73 44.618415 74 44.958701 75 45.194062 76 45.624577 77 45.950311 78 46.271331 79 46.587715 80 46.899521 81 47.206817 82 47.509668 83 47.808141 84 48.102298 85 48.392200 86 48.677877 87 48.959492 88 49.236993 89 49.510486 90 49.780023 91 50.045601 92 50.307460 93 50.665471 94 50.819753 95 51.070356 96 51.317335 97 51.560742 98 51.841984 99 52.037044 100 52.271047 101 52.499677 102 52.725986 103 52.949021 104 53.134301 105 53.385474 106 53.598963 107 53.809375 108 54.016743 109 54.221113 110 54.422527 111 54.620966 112 54.816601 113 55.009461 114 55.199474 115 55.386751 116 55.571297 117 55.753185 118 55.932443 119 56.109107 120 56.283219 121 56.454811 122 56.623921 123 56.790588 124 56.954843 The Calculation of a Number in the precedent Table , by aid of the Canon . The Question being , What is the present worth of 1 l. per Quarter for 21 Years ? The Logarithm of the Increase of 1 l. at 6 per Cent. for three months = 0,0063264664 The Number of Quarters = 84 Discompt of 1 l. for 84 Qrs. 0,5314231776 Then by the Rule of Proportion : If 1,467384617 com . arithm . 9,8334569382 have 100 l. for its Principal , 2,0000000000 what shall 1 l. have for its Pr. 0,0000000000 68.148601 = 1,8334569382 The Log. of the Discompt = 0,5314231776 The Answer is = 48 l. 2 s. 0 d. ½ ferè . And after this manner may a Table be Calculated , or the Value of a Lease for any Number of Years , may be found at any Rate of Interest required . The VSE of the TABLE . This Table sheweth the Discompt of 1 l. per Quarter at 6 per Cent. per Annum , Compound Interest , and if the Tabular Number for so many Quarters as the Lease is to continue be multiplied by the Quarterly Payment , that Product is the present Value of that Lease in ready money . Example . A Lease of 40 l. per Annum ( viz. 10 l. per quarter ) for 21 years , being to be sold , what is it worth in ready money ? 84 quarters per Table = 48.102221 The quarterly Rent = 10 The Answer is 481 l. 0 s. 5 d. ● / 4 ; ferè . But if the question be , What quarterly Rent for 21 years will a given Sum purchase ? Then divide the given Sum by the Tabular Number for so many quarters . Example . A Gentleman having a Lease of certain Church Lands , worth 200 l. per Annum more than the reserved Rent , for 14 years to come , surrenders the same , upon condition the Chapter shall make him a new Lease for 31 years without a present Fine , but advancing the old Rent 10 l. per quarter during the whole term of 31 years ; what doth he gain by the bargain , accompting Compound Interest on both sides ? 56 quarters per Table = 38.006 The quarterly Rent = 50 124 quarters per Table = 56.955 The Answer is 377 l. 18 s. the new Lease being so much more worth than the old one . 240 l. is demanded for the Lease of a House for 7 years , the Tenant offers 100 l. and an advance of Rent equivalent to the rest of the Fine required , what ought this Rent to be ? The Advance of Rent ought to be 6 l. 2 s. 8 d. per quarter . There is a Lease of 200 l. per Annum , viz. 50 l. per quarter , for 13 ¼ years , to be sold , what is it worth at 6 per Cent. Simple , and what at 6 per Cent. Compound Interest ? Simple . 53 quarters per Table = 38.779748 Compound . 53 quarters per Table = 36.659861 Which subtracted from 1938.987440 Leaves 105.99439 Whereby it appears , that it is cheaper to the Purchaser at Compound Interest than at Simple Interest by 106 l. Six Questions performed by the aid of the Canon of Logarithms . Quest. 1. A Gentleman pays 350 l. for a Lease in Reversion , to commence at the end of 13 years and a quarter , and to continue for 21 years and 3 quarters , what quarterly Rent may he lett the Premises for , after he comes to be in possession thereof , so as to gain 8 per Cent. Compound Interest for his money ? The Logarithm of 350 l. = 2,544008 Worth of 1 l. forborn 53 quarters = 0,442865 Log. of the Increase of 350 l. = 2,986873 Worth of 1 l. for 87 quarters = 1,621420 Log. of 23.198 = 1,365453 The Answer = 23 l. 3 s. 11 d. ½ ferè . Quest. 2. A Citizen having taken a Lease of a House and Shop for 21 years , at 370 l. Fine , and 100 l. per Annum , viz. 25 l. per quarter , Rent , at the end of two years is willing to leave it for 300 l. and the old Rent , or to have such an increase of Rent , during the whole term yet to come , as may reimburse him his Fine paid , with Compound Interest at 6 per Cent. per Annum : What ought he to receive in advance of Rent , and what doth he offer to lose of his Fine paid in taking 300 l. 2,568202 = 370l . 1,682165 = The worth of 1 l. per quarter for 84 quarters . 0,886037 The Advance of Rent ought to be 7 l. 13 s. 10 d. ¼ ferè . 1,652198 The worth of 1 l. per quarter for 76 quarters . 2,545235 The present Fine ought to be 350 l. 18 s. 10 d. Whereby it appears , there is 50 l. 18 s. 10 d. offered to be lost in putting off the House and Shop aforementioned . Quest. 3. A sells a House to B for 800 l. to be paid with Interest upon Interest by 100 l. per Annum , viz. 25 l. per quarter , how many quarters Rent ought B to pay before A is satisfied for his 800 l. with Compound Interest at 6 per Cent. per Annum , and what ought the last Payment be ? The last Payment 13 l. 3 s. 4 d. ferè . Quest. 4. A lends unto B a certain Sum of ready money , and accepts a Rent Charge of 40 l. quarterly for 7 years in satisfaction , finding it paid him his Principal with Interest upon Interest at 8 per Cent. within 13 l. 4 s. 6 d. what was the Money lent ? 1,602060 The Logarithm of 40 l. 1,565196 The Logarithm of the worth of 1 l. quarterly for 48 quarters . 3,167256 The Logarithm of the worth of 40 l. per quarter for 48 quarters . 0,177140 The Logarithm of the increase of 1 l. forborn 28 quarters . 2,986188 The Logarithm of 968.7 . The Money lent was 968 l. 14 s. Quest. 5. A Testator leaving one Son and two Daughters , bequeaths out of his Estate ( being 600 l. per Annum for 11 years ) to his eldest Daughter 500 l. per Annum for 4 years next coming , at the end whereof , to his younger Daughter 300 l. per Annum for 7 years , and to his Son the Remainder of the Estate for the whole time : Q. Which had the greatest Portion , and by how much , calculating their several Annuities at 6 per Cent. Compound Interest ? 0,539716 Logarithm of the worth of 1 l. per annum for 4 years . 2,698970 The Logarithm of 500 l. 3,238686 = 1732.55 . 0,746820 The Logarithm of the worth of 1 l. per annum for 7 years . 2,477121 The Logarithm of 300 l. 0,101232 The Logarithm of the worth of 1 l. forborn 4 years . 3,122709 = 1326.47 . 0,896905 The Log. of the present worth of 1 l. per annum for11 years . 2,896905 = 788.68 0,746820 The Log. of the present worth of 1 l. per annum for 7 years . 2,301029 The Logarithm of 200 l. 0,101232 The Logarithm of the worth of 1 l. forborn 4 years . The Sons Portion — 1673 : 00 : 05 The eldest Daughters Portion 1732 : 11 : 00 The youngest Daughters — 1326 : 09 : 04¾ Proof . 0,896905 The Logarithm of the worth of 1 l. per annum for 11 years . 2,778151 The Logarithm of 600 l. 3,675056 The Logarithm of 4732 l. Quest. 6. A Merchant sold 16 Kintals of Cyprus Cottons for 320 l. to be paid at two six-months ; the Buyer having Money by him , offers to pay the Money presently , provided the Merchant allow him Discompt at 6 per cent . Compound Interest . Q. What ought the Merchant to receive ? 2,505149 The Logarithm of 320l . 0,018979 The Logarithm of 1 l. forborn 9 months , the equated time , acording to the Table of Mean Time , Pag. 110. 2,486170 The Logarithm of 306.316 . The Answer 306 l. 6 s. 4 d. ferè . STEREOMETRY : OR , A New and the most Practical Way OF Gauging Tunns In the form of a PRISMOID & CYLINDROID : ALSO The Frustums of Pyramids and of a Cone . Together with The Art of CASK-GAUGING . By Iohn Mayne . London , Printed by William Godbid , for Nath. Crowch , in Exchange-Alley . M. DC . LXXIII . TO THE Young Geometrician . I Hope by this time thou art so sufficiently acquainted with the Nature and Use of a Decimal Fraction , that any Operation in the six Species , viz. Addition , Subtraction , Multiplication , Division , Involution and Evolution of the Second and Third Powers , will not appear difficult to thee ; and these being familiar , any Calculation in Arithmetick , Geometry , Trigonometry , or other Mathematical Arts , will not seem strange : Amongst the many pleasant Walks in this Tempe , I have made it my present design to give thee some diversion in that part of Solid Geometry called Gauging , and herein passing by those Blossoms that kiss the hand of every Passenger , I have endeavoured ( and I hope not altogether without success ) to shew thee how to gather a Rose without danger of its Thorn : For the Invention , the World is obliged to the Ingenious Mr. Michael Dary , the Roots of these , and many other choice Mathematical Flowers , lying crowded together in a small Trea●ise called Dary's Miscellanies , Printed 1669. Here , as in the former Part , thou hast both Precept and Example in the plainest method I could possibly express them . That they may by no means seem obscure to any ingenious Student , is the hearty desire of Thy Friend , J. M. The Explanation of the Signs or Characters . + More . − Less . = Equal . > Greater . < Lesser . × Multiplied . √ Square Root . q square Square . ⊙ Circle . ∷ Proportional . ∽ Difference . STEREOMETRY : OR , A New and the most Practical Way of GAUGING TUNNS , &c. A plain and easie Method for finding the Solid Content of a Prismoid . DEFINITION . BY the word Prismoid is to be understood a Solid contained under six plain Surfaces , whereof the two Bases ought to have these three qualifications : 1. Rectangular Parallelograms . 2. Parallel . 3. Alike Situate . i. e. So situate , that the Rectangular Conjugates in both Bases may be inserted by two and the same Planes , and a Right Line extended from the Center of one Base to the other may be called the Axis , and the other remaining four Planes are the Peripatasma . Under this Definition is comprehended the Frustums of Pyramids and Prisms . Note also , If the Peripatasma be not made by the four flat Sides ( spoken of before ) but shall be constituted by Curveture from Circles or Elipse's , the Solid is then called a Cylindroid , and under this Definition is comprehended Frustums of Cones and Cylinders . PROBL. If in a Prismoid you put C = the whole Content thereof . A & B = the two Rectangular Conjugates above . G & H = the two Rectangular Conjugates below . A & G opposite = their two Correspondents one above the other below inserted by one Plane . B & H opposite = their two Correspondents above and below , and also inserted by one Plane . P = the perpendicular height of the Prism or Prismoid . K = the increment of any two Diameters , to be taken between A and G in the same Plane with them , at one Inch distance of the perpendicular . L = the increment of any two Diameters , to be taken between B and H in the same Plane with them , at one Inch distance of the perpendil●r . Then , Analogism . Or , The Rectangle of the two Diameters at the Base multiplied into the Perpendicular , more the Semi-sum of G L and H K , ( viz. those two Diameters multiplied into their altern increments ) multiplied into the Square of the Perpendicular , to which add one third of the Rectangle of K L ( i. e. ) the two increments multiplied into the Cube of the Perpendicular is equal to the Content in Cubick Inches . By which Theorem you find three fixed or stationary Numbers , which Mr. Dary calls reserved Coefficients , wherefore you shall find them hereafter called by that denomination : These three reserved Coefficients thus multiplied into the Perpendicular , the Product is equal to the whole Content , or by any part of the Perpendicular gives the Solidity of that part . Prop. I. Having a Tunn in the form of a Prismoid , the Dimensions being , Inches . above . A = 126 and B = 144 below . G = 102 and H = 108 P = 60 What is the Solidity of this Tunn in Cubick Inches ? First then to find the three reserved Coefficients . i. e. The difference between A and G ( the two opposite Diameters above and below ) divided by the Perpendicular quotes K , the increment of any two Diameters to be taken between them , at one Inch distance in the Perpendicular , and in the same Plane . That is , the difference between B and H ( the two Diameters opposite the one above the other below ) divided by the Perpendicular quotes L , the increment of any two Diameters to be taken between them , at one Inch distance in the Perpendicular , and in the same Plane with them . .6 = L .4 = K .24 = The Rectangle of LK .08 = ⅓ of LK , the first Coefficient . Now having found these reserved Coefficients , I proceed , and finding that ⅓ of K L must be multiplied by the Cube of the Perpendicular , I begin with it , and call that the first Coefficient ; then ½ G L H K being to be multiplied by the Square of the Perpendicular , I add that to the first Fact , and call it the second Coefficient ; lastly , G H being to be multiplied by the Perpendicular , I add that to the second Fact , and call it the third Coefficient ; then will the Work stand thus : Example . The Answer 866160 Cubick Inches . Now admitting this Tunn have but 33 wet Inches , what is the Content thereof ? The Answer = 423248.76 Cubick Inches . Prop. II. Having a Tunn in the form of a Prismoid , the Dimensions being , Inches . above . G = 102 and H = 108 below . A = 126 and B = 144 P = 60 What is the Solidity in Cubick Inches ? To find this Tunns Solidity , the Rule is : i. e. The Fact of A B ( the rectangular Congates at the Base ) multiplied by the Perpendicular , from whence subduct the Semi-sum of the two Facts ( A and its altern decrement , B and its altern decrement , multiplied into the Square of the Perpendicular ) more the one third of the Rectangle of K L , viz. the two decrements , multiplied into the Cube of the Perpendicular , and that Remainder is the Content in Cubick Inches . To find the Coefficients . This Rule being the Converse of the former , these Numbers K and L which before were Affirmatives are now become Negatives ( then increment , now decrement ; ) the greater Conjugates being subducted from the lesser makes the Dividends so much less than nothing , and consequently the Quotes , the Divisor being an Affirmative , yet these two Negatives being multiplied together , their Fact becomes Affirmative , according to the Rule of Algebra , the Signs of the Factors being homogeneal ( or alike ) makes the Fact more , as in this Example . The Factors in these Rectangles being heterogeneal ( or unlike ) the Fact is made less . These two Factors being both Affirmatives , the Fact is + . With these three reserved Coefficients I proceed to the Calculation , according to the precedent Theorem . The Answer = 866160 Cubick Inches . But if this Tunn have only 27 Inches of the Perpendicular wet , the Content then being required : Prop. III. There is a Tun in the form of a Prismoid , the Dimensions are , Inches . above . A = 132 and B = 144 below . G = 108 and H = 108 P = 60 What is the Solidity in Cubick Inches ? The Answer = 911520 Cubick Inches . Prop. IV. There is a Tunn in the form of a Prismoid , the Dimensions being , Inches . above . G = 108 and H = 108 below . A = 132 and B = 144 P = 60 What is its Solidity in Cubick Inches ? The reserved Coefficients are found to be : The Calculation . The Answer = 911520 Cubick Inches . Here note , if any of the precedent Tunns be cloathed by Curveture , ( i. e. the Bases being Circular or Elliptical ) the last Product ought to be divided by 1.27324 , then will the Quotient exhibit the Cubick Inches in that Solid . But if the Question be Ale Gallons , let your Coefficients be divided by 282 ; if Beer Barrels be required , divide the Coefficients by 10152 , the number of Inches in a Beer Barrel . In all flat sided Figures , and for those Solids , whose Peripatasma is constituted by Circles or Ellipsis , the Divisor for Beer Barrels is 12926 > , for Ale Barrels 11490 > , and for Ale Gallons 359 ; of which take these Examples . What number of Beer Barrels and Gallons doth the last mentioned Tunn contain ? The three Coefficients for Beer Barrels divided by 10152 are : The first = +.000007880220646 < The second = − .0067375888 > The third = +1.8723404 < The three Coefficients for Gallons being divided by 282 are : The first = .0002837 > The second = .2425532 > The third = 67.4042553 > The Coefficients being thus fitted , the Calculation is after this manner : For Barrels . The Answer is 89 Barrels , 3 Firkins , and 1 Gallon , or 89 Barrels and 28 Gallons . For Gallons . The Answer is 3232 Gallons , which divided by 36 quotes 89 Barrels 28 Gallons , as before . Example . A New Way of GAUGING THE Frustum of a PYRAMID OR CONICAL TVNN . A New Way of GAUGING THE Frustum of a Pyramid , &c. DEFINITION . A Pyramid is a Solid Figure , contained under many Superficies , whereof one is the Base , and the rest arise from the Base to the Vertex , and there meet in a Point . The Frustum of a Pyramid is a Solid , cut with a Plane parallel to the Base , and the part cut off is also a Pyramid . The Frustum of a Cone , may not improperly be termed the Frustum of a round Pyramid , ( the Base being circular ) nor do I think it an Heresie to call a Cylinder a round Prism . The Frustum of a Pyramid , whose Bases are in the form of any ordinate Polygon , being alike , and alike situate , and also if a Right Line may be every where applied in the Peripatasma from Base to Base , moreover a Right Line being extended from the Center of one Base to the other , may be called the Axis . Then if you put S = the whole Solidity . B = a Side above . A = a Side below . P = the Perpendicular . d = the common Addend at one Inch distance of the Perpendicular , and is thus made , that is , the difference between a Side above and a Side below , divided by the Perpendicular , quotes the increment , &c. G = the Divisor . The Rule is : G ) AA p+A dpp+⅓ ddppp ( = S Or , in other terms : To the Square of the Side multiplied by the Perpendicular , add the Fact of one Side in the Increment multiplied by the Square of the Perpendicular , more ⅓ of the q. of the Increment in the Cube of the Perpendicular , and the Aggregate divided by the Polygons respective Divisor , the Quote will be the Solidity . And further it is to be well observed , if your Frustum of a Pyramid stand upon its greater Base , the Rule then is thus varied : G ) : BB p − B dpp+⅓ ddppp ( = S That is to say : From the Square of a Side at the Base multiplied by the Perpendicular , subduct the Rectangle of one of those Sides in the Decrement multiplied by the Square of the Perpendicular , more one third of the Square of the Decrement in the Cube of the Perpendicular , and that Remainder divided by the Divisor proper to the form of the Base , the Quote is equal to the Solidity . Note also , that p may be put for a part of the Perpendicular , and the Answer will be the Content of that part required . G ) or the Divisors for these 8 Regular Polygons , and the Cone . For Cubick Inches . For Ale Gallons . Trigon 2.30940 Trigon 651.2000 Tetragon   Tetragon 282.0000 Pentagon .58123 Pentagon 157.2600 Hexagon .38497 Hexagon 108.5400 Heptagon .27513 Heptagon 77.5867 Octogon .20710 Octogon 58.4022 Nonagon .16176 Nonogon 45.6163 Decagon .12997 Decagon 36.6515 Cone 1.27324 Cone 359.0500 If your Tunn be the Frustum of a Cone : Let A or B be the Diameter at the Base , and d the Increment or Decrement of any two Diameters between A and B , at one Inch distance of the Perpendicular , and the Divisor as per Table . I shall only give you some Examples of the three first , and the Cones Frustum , which I think will be sufficient to inform any ingenious Practitioner how to perform the rest . The Trigon . Admit a Tunn be in the form of an equilateral Triangle , the Dimensions being , A = 126 Inches , the length of a Side above , B = 108 Inches , the length of a Side below , P = 60 Inches , the Perpendicular , Q. The Content in Ale Gallons ? The Coefficients are found , according to the precedent directions , thus : These three divided by the Divisor for Ale Gallons , viz. 651.2 , are as followeth : The Answer = 1201.9020 Ale Gallons . A Tunn of the same Dimensions standing upon its greater Base , the Coefficients are thus found : Being divided by the same Divisor with the former , they are : And are thus used : The Answer as before = 1201.9080 Ale Gallons . Now if each of these Tunns have 30 Inches of the Perpendicular wet , how much do they contain ? The First . The Answer = 537.84750 Ale Gallons . The Second . The Answer = 663.9630 Ale Gallons . Proof . The difference being less than a Pint. The Tetragon or Square Pyramid . There is a Tunn in the form of the Frustum of a Square Pyramid , A = 144 Inches , the length of a Side above , B = 108 Inches , the length of a Side below , P = 60 Inches , the Perpendicular , Q. The Content in Gallons ? The Coefficients being found by the former Rule and Example , viz. P ) A − B ( = d. ⅓ of dd = the first Coefficient .12 dB = the second Coefficient 64.8 BB = the third Coefficient 11664 These three being divided by 282 the Cubick Inches in the Ale Gallon , are equal to The Answer = 3400.8960 Ale Gallons . If the Tunn stand upon its greater Base , the Coefficients then are P ) B − A ( = − d , and the ⅓ d d = the first , A d = the second , and A A = the third , which divided by 282 the number of Cubick Inches in an Ale Gallon , they do appear to be = +.00042553 And if the Content be required The Answer = 3400,855680 If 40 Inches of the Perpendicular be wet in the first Tunn , and 20 in the latter , and it be demanded what they contain in Ale Gallons . The first . The second . Proof . The Pentagonal Pyramid . A Tunn in the form of the Frustum of a Pyramid , whose Bases are in the form of a Pentagon , A = 144 Inches , the length of each Side above , B = 108 Inches , the length of each Side below , P = 60 Inches , the Perpendicular , Q. The Content in Ale Gallons ? The three Coefficients found in the last Example , viz. For Ale Gall. The Answer = 6099.46920 Ale Gall. Let another Frustum of a Pyramid of the same Bases and Altitude , stand upon its greater Base , and the Content in Ale Gallons be demanded . The Coefficients so found and divided as before directed , are as followeth : The Answer = 6099.45120 If 50 Inches of the Perpendicular in the first Tunn be wet , and 10 Inches in the last , what is the Content in Ale Gallons ? The first . The last . Proof . A Tunn in the form of a Frustum of a Cone , the Bases being alike and alike situate , as in the precedent Examples , the Dimensions being , A = 144 Inches , the diameter above , B = 108 Inches , the diameter below , P = 60 Inches , the Perpendicular , Q. The Content in Ale Gallons ? p ) A − B ( = d. These three divided severally by the Divisor proper to a Cone , as in the Table mentioned , viz. 359.05 they quote : The Answer = 2671.0440 Ale Gall. And that the young Gauger may not be obliged to Dray-men to repleat the horizon with liquor , of such Tunns whose Bases are not posited parallel thereto , ( as indeed most are not being made with a Drip or Fall ) let him take this Example , a b c d e f g h m x , a Cone or Pyramid . Having the length of each Line in this Diagram , and the Content of the whole Cone or Pyramid in Cubick Inches , Gallons , or Barrels , &c. the Quantity of the Hoof h c b d is found by this Analogy : As the Cube of the Line a c , to the whole Solidity : So is the Cube of a Geometrical Mean between a c and a h , to the Content of the Cone or Pyramid cut off : Which subducted from the whole , the remainder is the Content of the Hoof. Some of the Lines being given , the rest are to be found . Example . There is a Tunn taken as the Frustum of a Cone , c d = 144 Inches , the greater Base , e f = 108 Inches , the lesser Base , g b = 60 Inches , the depth . Admitting the Base were raised 3 Inches , as the Line c h , it is then necessary to take another Diameter between c h and e f to find c d. To find a b the Cones Axe . As 18 , the semi . diff . of Diam . co . ar . 8,744728 to 72 , the semi-diam . at the Base : 1,857332 So is 60 , the Tuns depth , 1,778151 to 240 , the Cones Axe . Log. of the Sum of their q square . The Logarithm of ac 250.567 2,398924 The Logarithm of ab 247.567 2,393692 A Geometrical Mean 2,396308 To find the Content of the whole Cone . The Square of 144 4,316724 ⅓ of ab = 80 1,903090 359 &c. complement arithm . 7,444842 Content in Ale Gallons 4620 ferè 3,664656 The Line a c , the Line a h , and the Content being known , to find the Content of the Fall c b d h. As the Cube of ac , co.ar. 2,803228 to the whole Content : 3,664656 So is the Cube of the Mean , 7,188924 to the Content of the Cone cut off . 3,656808 82.6 the Content of the Fall. Or thus : As the Cube of ac , co . ar . 2,803228 to the whole Cones Solidity : 3,664656 So is the Cube of ah , 7,181076 to the lesser Cones Solidity . 3,648960 The whole Cone = 4620 The Frustum mxhcbd = 164 The being ½ the Fall or Hoof = 82 Some Practical RULES & EXAMPLES FOR CASK-GAVGING . Some Practical RULES & EXAMPLES FOR CASK-GAVGING . THe Corner-stone in the whole Fabrick of Cask-Gauging , as full , was long since laid by Mr. Oughtred , taking a Cask to be the Frustum of a Spheroid , under which capacity they are generally received , though indeed there have been , and daily are found some Cask differing in form , and really are more Parabolical than Spheroidal , I shall therefore lay down a plain Method for the performance of the Work ( viz. finding their Content ) under these four Considerations : As Spheroidal , As The Frustum of a Parabolical Spindle , As The Frustum of a Parabolical Conoid , As The Frustums of two Cones abutting upon one common Base . These severally , with and without a Table of Area's of Circles . And forasmuch as the Dimensions must be the first thing known , before the Content can be found , I shall therefore shew the young Tyro , how by some of the Dimensions to find the rest , if any obstruction prohibit the taking of all . The Boung-diameter , and Head-diameter , and Diagonal , to find the Casks length . First subduct the semi-difference of Diameters from the Boung-diameter , and Square the Remainder , which Square subduct from the Square of the Diagonal , and the Remainder is the Square of the Casks semi-length . Example . Let BD be the Boung-diameter = 29 Inches , HE be the Head-diameter = 23 Inches , BE be the Diagonal = 35.3836 > Inches , SD the semi-difference = 3 Inches : Q. the Length = LT ? The Square of BE = 1252 Square of BD − SD = 676 Sqaure of semi-length = 576 √ 24 48 = LT This very Quest. was intended by Mr. Smith , p. 176. but through a Mistake it was left out . The Boung-diameter , Diagonal , and Length , to find the Head-diameter . The Rule . From the quadrupled Square of the Diagonal subduct the Square of the Length , ( which done ) the Square Root of the Remainder is equal to the Sum of the Boung-diameter and one Head-diameter . Example . The Square of Diagonal = 5008 Square of the Length = 2304 Remainder = 2704 √ = 52 Subduct the Boung-diameter = 29 Head-diameter = 23 The Head-diameter , Boung diameter , and the Length , to find the Diagonal . The Rule . To the Square of the Semi-length add the Square of the Boung diameter , less the Semi-difference of Diameters , and the Square Root of their Sum is equal to the Diagonal . Example . Square of Semi-length = 576 Square of 26 = 676 Square of Diagonal = 1252 √ 35.3836 > A Cask taken as the Frustum of a Spheroid , cut with two Plane Parallels , each Plane bisecting the Axis at right Angles , B the Boung-diameter = 29 Inches , H E the Head-diameter = 23 Inches , L T the Length = 48 Inches : Q. The Content in Wine Gallons ? The Rule . To the doubled Square of the Boung-diameter add the Square of the Head-diameter , their Aggregate multiply by the Length , and to the Product add the tenth part of it self , more one third of that tenth part , and from the Sum cut off as many places toward the right hand as were in the Multiplicand . Example . The doubled Square of 29 = 1682 The Square of 23 = 529 The Aggregate = 2211 The Answer = 120.2784 Wine Gallons . Another way . The q square . of the Boung-diameter = 841 The q square . of the Head-diameter = 529 The Sum of their Squares = 1370 The Semi-sum = 685 The Semi-diff . of Squares = 156 Their Aggregate = 2211 The Length = 48 16788 8844 The Product = 106128 The tenth part = 106128 The part or ⅓ of = 35376 The Answer in Wine Gall. 120.2784 The same Cask being taken as the Frustum of a Parabolical Spindle , the Content may be thus found . q square . of the Boung-diameter = 841 q square . of the Head-diameter = 529 Their Aggregate = 1370 The Semi-sum = 685 The tenth part of the Diff. = 312 The Length = 48 The Product = 1001376 Tenth of the Product = 1001376 ⅓ of The Answer in Wine Gall. 113.48928 If taken as the Frustum of a Parabolical Conoid , cut as before mentioned , the Content may be found as in this Example . q square . of the Boung-diameter = 841 q square . of the Head-diameter = 529 The Sum = 1370 The Product = 98640 Tenth part of the Product = 98640 The Answer in Wine Gall. 111.7920 If a Cask of the same Dimensions be taken as the middle Frustum of two Cones abutting upon one common Base , cut with two Planes parallel , and each bisecting the Axis at Right Angles , the Content in Wine Gallons may be found as in this Example . q square . of the Boung-diameter = 841 q square . of the Head-diameter = 529 The Sum = 1370 The Semi-sum = 685 Semi q square . of diff . of Diam . = 18 The Answer in Wine Gall. 110.8128 For finding the Capacity of these , or any other Vessels , it is convenient to have always in readiness a Table of Area's of Circles in Wine and Ale Gallons : I think it unnecessary to swell this intended small Volume with them , there being two lately Printed , exactly Calculated to every tenth part and quarter of an Inch , and also a Table of Area's of Segments of a Circle , by my good Friend Mr. Iohn Smith , in his Book of Gauging , to whom in gratitude I am obliged to render my hearty acknowledgment for many favours and kind assistances in these Studies ; yet that you may be able to find any Area of a Circle upon demand , in Wine or Ale Gallons , without a Table , take this Rule . Divide the q. of the Diameter by 294.1 for Wine , and by 359.05 for Ale Gallons , and the Quotient exhibits the Area . Or , saith Mr. Smith , Multiply the q. of the Diameter by .0034 for Wine , and by .0027851 for Ale Gallons , and the Product exhibits the Area in such Gallons . As in these Examples . The Diameter of a Circle = 21.7 : Q. The Circles Area in Wine Gallons ? The Diameter of a Circle = 26.8 : Q. The Area in Ale Gallons ? For finding the Capacity of a Cask , taken as Spheroidal , by a Table of Area's of Circles in Gallons . Example . A Casks Boung-diameter = 29 Inches , Head-diameter = 23 , and the Length = 48 Inches : Q The Content in Wine Gallons ? ⅓ of the Area of the Boung ⊙ = .9531 ⅓ of the Area of the Head ⊙ = .5996 Their Sum = 1.5527 Semi-sum = .7763 Semi ∽ = .1768 Area of Mean Circle = 2.5058 The Length = 48 The Answer in Wine Gall. 120.2784 Another way . ⅓ of the Area of Boung ⊙ = 1.9062 ⅓ of the Area of Head ⊙ = .5996 The Area of Mean ⊙ = 2.5058 The Length = 48 The Answer = 120.2784 That is , 120 Gallons , 1 Quart , and ¼ of a Pint , ferè . To find the solid Content of a Cask , when taken as the middle Frustum of a Parabolical Spindle , &c. The Dimensions as before . ⅓ of the Area of Boung ⊙ = .9531 ⅓ of the Area of the Head ⊙ = .5996 Their Sum = 1.5527 Semi-sum = .7763 Tenth of the Difference = .03535 Area of Mean ⊙ = 2.36435 The Length = 48 The Answer in Wine Gall. 113.48880 That is , 113 Gallons , and almost 2 Quarts . And as the Frustum of a Parabolical Conoid , the Capacity is thus found : ⅓ of the Area of the Boung ⊙ = .9531 ⅓ of the Area of the Head ⊙ = .5996 Their Sum = 1.5527 Semi-sum = .7763 Area of the Mean ⊙ = 2.3290 The Length = 48 The Answer in Wine Gall. 111.792 If a Cask be taken as the middle Frustum of two Cones , abutting upon one common Base , &c. The Dimensions as before . ⅓ of the Area of the Boung ⊙ = .9531 ⅓ of the Area of the Head ⊙ = .5996 Their Sum = 1.5527 The Semi-sum = .7763 of Area of 6 the ∽ of Diam . .0204 Area of Mean ⊙ = 2.3086 The Length = 48 The Answer in Wine Gall. 110.8128 The Ullage , or Wants in a Cask , may be found under these two Considerations : 1. A Cask standing on the Head , with the Diameters parallel to the Horizon . 2. A Cask lying with the Axe parallel to the Horizon . Prop. I. In a Cask standing on the Head , with the Diameters parallel to the Horizon , some Liquor remaining , to find how many Wine Gallons it is . Here are these five things necessary to be known : 1. The Diameter at the Boung . 2. The Diameter at the Head. 3. The Length of the Cask . 4. The Depth of the Liquor . 5. The Diameter of the Liquors superficies . Example . The Diameter o p is thus found , first find the Axis of the whole Spheroid e f , thus ; from the Square of half the Boung-diameter ( n h ) subduct the Square of half the Diameter at the Head , and extract the Square Root of the Remainder : Then by the Rule of Proportion , say , As that q √ , is to n h , the Semi-boung-diameter : So is n i , the Casks Semi-length , to e n half the Axis sought . Diameter of the Liquors Superficies 27.4 Having found the Diameter of the Liquors Superficies : Then , to ⅔ of the Area of that Circle = 1.70173 Add ⅓ of Area of the Head Circle = .59953 The Depth of Liquor = 11.6 The Answer in Wine Gallons = 26.694616 Which subducted from the whole Content , leaves the Ullage or Wants . Prop. II. A Cask lying with its Axe parallel to the Horizon , and having some Liquor remaining in it , to find the Content of the said Liquor in Gallons . Let the Dimensions be as before . In this Proposition there is five Requisites attending : h g the Diameter at the Boung = 29. a b the Diameter at the Head = 23. i k the Length = 48. s g the Depth of Liquor = 11.6 . And the Content of the whole Cask in Gallons . Then by the help of a Table of Area's of Segments of a Circle , whose Area is Unity , and the Radius divided in the Ratio of 1.0000 Parts , say by the Rule of Proportion : versed Sine or Arrow of Segment . Then seeking in the Table you will find .4000 , and right against it under the Title Area you will find .37353 . Then say : whole Content 1.0000 . .37353 ∷ 120.2784 . 44.9276 the Liquor remaining . The Inversion of the Question , viz. To find the Liquor wanting . As 29. 17.4 ∷ 1.0000 . .6 Again , the Ullage . As 1.0000 . .62647 ∷ 120.2784 . 75.3509 The Liquor remaining = 44.9276 Which together make 120.2785 the Casks whole Capacity . FINIS . ERRATA . PAg. 3. l. 4. r. in the third . P. 5. l. 22. r. 20 : 13. P. 7. l. 17. r. 42 : 3 : 08 : 15 ; l. 19. r. 33 : 0 : 05 : 04. P. 9. l. 1. r. borrow of the. P. 13. l. 14. r. 5 s. the bushel . P 16. l. 13. r. 2 times . P. 19. l. 1. r. 5 from 6. P. 31. l. 19. dele always . P. 39. l. 22. r. in the decimal Fractions of both Factors . P. 50. l. 15. r. solid measure . P. 51. l. 15. r. 272.25 . P. 52. l. 15. r. .0041667 . P. 67. l. 14. dele as before . P. 82. l. 14. r. + a. P. 188. l. ult . r. ½ being . THe Rules herein mentioned , and other Mathematical Arts , are taught by the Author , viz. Arithmetick , Vulgar , Decimal , and Logarithmetical ; the Doctrine of Triangles , Plain and Spherical ; the Use of the Globes , Quadrant , Sector , and other Mathematical Instruments ; Fair Writing , and Merchants Accompts , by way of Debitor and Creditor ; also the Art of Short Writing .