The doctrine of interest, both simple & compound explained in a more exact and satisfactory method then [sic] has hitherto been published : discovering the errors of the ordinary tables of rebate for annuities at simple interest, and containing tables for the interest and rebate of money for days, months, and years, both at simple and compound interest, also tables for the forbearance, discomps, and purchase of annulites : as likewise, equation of payments made practicable and useful for all merchants and others : together with divers other useful reflections / ... Sir S. Morland. Morland, Samuel, Sir, 1625-1695. 1679 Approx. 285 KB of XML-encoded text transcribed from 135 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2007-10 (EEBO-TCP Phase 1). A51383 Wing M2778 ESTC R13339 11702013 ocm 11702013 48259 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. A51383) Transcribed from: (Early English Books Online ; image set 48259) Images scanned from microfilm: (Early English books, 1641-1700 ; 542:3) The doctrine of interest, both simple & compound explained in a more exact and satisfactory method then [sic] has hitherto been published : discovering the errors of the ordinary tables of rebate for annuities at simple interest, and containing tables for the interest and rebate of money for days, months, and years, both at simple and compound interest, also tables for the forbearance, discomps, and purchase of annulites : as likewise, equation of payments made practicable and useful for all merchants and others : together with divers other useful reflections / ... Sir S. Morland. Morland, Samuel, Sir, 1625-1695. [56], 203 p. Printed by A. Godbid and J. Playford, and are to be sold by Robert Boulter ..., London : 1679. Includes interest tables. Reproduction of original in Cambridge University Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO. EEBO-TCP aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the Text Encoding Initiative (http://www.tei-c.org). The EEBO-TCP project was divided into two phases. The 25,363 texts created during Phase 1 of the project have been released into the public domain as of 1 January 2015. Anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source. Users should be aware of the process of creating the TCP texts, and therefore of any assumptions that can be made about the data. Text selection was based on the New Cambridge Bibliography of English Literature (NCBEL). If an author (or for an anonymous work, the title) appears in NCBEL, then their works are eligible for inclusion. Selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. In general, first editions of a works in English were prioritized, although there are a number of works in other languages, notably Latin and Welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so. Image sets were sent to external keying companies for transcription and basic encoding. Quality assurance was then carried out by editorial teams in Oxford and Michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet QA standards were returned to the keyers to be redone. After proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. Any remaining illegibles were encoded as s. Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Interest -- Tables. 2005-12 TCP Assigned for keying and markup 2005-12 Aptara Keyed and coded from ProQuest page images 2007-01 Ali Jakobson Sampled and proofread 2007-01 Ali Jakobson Text and markup reviewed and edited 2007-02 pfs Batch review (QC) and XML conversion The Doctrine OF INTEREST , BOTH SIMPLE & COMPOUND : EXPLAINED In a more exact and satisfactory Method then has hitherto been Published . DISCOVERING The Errors of the Ordinary Tables of Rebate for Annuities at Simple Interest . AND CONTAINING Tables for the Interest and Rebate of Money for Days , Months , and Years , both at Simple and Compound Interest : Also Tables for the Forbearance , Discompt , and Purchase of Annuities . AS LIKEWISE , Equation of Payments made Practicable and useful for all Merchants and others . Together with divers others useful Reflections . Humbly Presented to His Most Sacred Majesty , CHARLES II. By Sir S. Morland , Knight and Baronet . Printed at London , by A. Godbid and J. Playford . and are sold by Robert Boulter , at the Turks-Head , over against the Royal-Exchange in Cornhill , 1679. coat of arms or blazon A necessary and useful INTRODUCTION . THe Author of this little Book hopes he has served the Publick somewhat better than other Arithmeticians , who have gone before him , for the following Reasons . 1. In that his Method is more plain and easie than that of other Men ; and those things which they have left intricate and difficult to be understood , are here made evident by clear Demonstrations , obvious to the meanest capacity . 2. In that his Tables are Calculated with greater care , and are much more correct than those that have been Published of late years . For instance , all those Tables in Mr. Newton's Book , Printed 1667 , are full of Errors and mistakes ; and which is very remarkable , the Tables which Mr. Dary has Published as his own , are only transcribed out of Mr. Newton's Book , and that with all the Errors , which are so many , that they must needs mislead and discourage either young or old Practitioners from trusting to , or making use of them . In Mr. Clavel's Tables , which seem to be more correct than the others , there will be found many very considerable Errors . As for instance , if you would know what an Annuity of 600 l. to continue 21 years is worth in ready Money , you will find it there to be but 6058 : 08 : 11.034 , which is too little by 1000 l. also if you would know the Present Worth of 60 l. Annuity for the like time ( in the same Page that resolves the former Question ) you will find but 605 : 16 : 10.703 , which is less than the truth by 100 l. Whereas in this little Book , by the more than ordinary care and diligence of Mr. John Playford , Printer , ( whom I have found the most ingenious and dexterous of any of his Profession , in Printing of Tables , and all sorts of Mathematical Operations ) it is presumed that there will hardly be found one false Figure ; but if there should , the Tables are so framed , that what the one does by Multiplication , the other proves by Division , & vice versâ , whatever the one performs by Division , the other makes out by Multiplication to be a truth . 3. Because the Operations according to the Rules and Tables of this little Book , where the Sums are great , are much more easie , practicable , and satisfactory , than by Mr. Clavel's Tables ; besides that those Tables do not answer very many useful Questions that will daily occur to Men of business . For Example . He has Tables for the Amount of any Sum to 10000 l. for 365 days , but he has no Tables of Rebate for so many days , which is full as useful as the other , not only in Simple , but also Compound Interest ; so that the Practitioner must very often be forced to have recourse to the Logarithms , or other tedious Calculations . 4. Because all the Operations in this Book are performed by DECIMAL ARITHMETICK , which of all other is the most useful and Practicable , when well understood : And in order thereunto , the Author has here given several Examples , and explained the respective Operations in such a manner , that any person who does at all understand the Vulgar Arithmetick , may in one hours time throughly comprehend this . Addition and Subtraction of Decimals . AS for the Operations of Addition and Subtraction , they are the very same with the Vulgar . For Example . To 375.42 Add 49.32 Sum 424.74 . There must only care be had of setting Unites under Unites , and Fractions under Fractions , in their proper Ranks and Files ; as likewise that there be as many places of Fractions in the Total , as are found in either of the Sums , before they are added together . Thus : To 375.42 Add 495.4 Sum 870.82 Or thus : To 375.42 Add 95.03 Sum 470.45 Again , From 870.82 Deduct 495.4 Remainder 375.42 Or , From 470.45 Deduct 95.03 Remainder 375.42 Multiplication of Decimals . Rule . SET one Number over another , ( making only a Full-point to Distinguish between whole Numbers and Fractions ) in the very same manner as is none in ordinary Multiplication , only when the Product is finished , look how many places of Fractions are found , both in the Multiplicand and Multiplicator jointly , just so many must be left in the Product . For Example . To multiply 342.34 by 3.123 , place them thus : Explanation . Because in the Multiplicand there are two places of Fractions , and in the Multiplicator three , in all five ; therefore in the Product there must be also five places of Fractions . Thus , Here the Practitioner is to observe , that the Fraction which is in truth less , is set over the whole Number , which is really greater . But because the Fraction consists of more places , it is set uppermost , though it is a thing indifferent , for if they were set otherwise , the Product would still be the same . For , Which is the very same as before . By the same Reason , The Product must evermore be compleated , as to the number of places that are found in both Multiplicand and Multiplicator . An excellent Method of Contracting a Long Multiplication . 1. IN the first place , wherever a Multiplication consists of above three places , the Author does recommend to all Practitioners ( as a thing which he has sufficiently experienced to be the most safe and easie ) to make use of a Tariffa , or Table of Multiplication for the Multiplicand : And though it may and will seem at the first view to be more tedious , yet it will be found to be the shortest of all other ways whatsoever , being performed by Addition only , and less subject to error ; and not only so , but whereas all other Operations of Multiplication do extreamly distort the Eyes by looking stedfastly upon Figures placed Diagonally , by this Tariffa the Eye looks on them always in a streight Line , and no otherwise . For Example . Suppose the two Sums to be multiplied one by another were 259879.890625 , and 1.1173698 , but the Product to consist of no more than Eleven places . Tariffa for the Multiplicand . 1 259879890625 2 519759781250 3 779639671875 4 1039519562500 5 1299399453125 6 1559279343750 7 1819159234375 8 2079039125000 9 2338919015625 Having made a Tariffa , and placed the Multiplicand and Multiplicator as is before directed , because in the Multiplicand there are twelve places , and eight in the Multiplicator , in all twenty places ; and it is desired to contract them to eleven places . First let a Line be drawn , leaving eight places to the right hand ; and then let all the imaginary places underneath be supplied with Points or Cyphers , decreasing in a Triangular Figure to nothing . Then let the Multiplication be performed as follows . Let the last Figure in the Multiplicator be found in the Margin of the Tariffa , and the Product answering to it subscribed , only let eight places be imaginary according to the number of Points or Cyphers , and let the remaining Figures ( viz. 20790 ) be transcribed on the other side of the Line . Under the next Figure of the Multiplicator ( 9 ) , are seven Points or Cyphers , therefore the Product answering to ( 9 ) in the Margin of the Tariffa being found , viz. 233891 / 9015625 , let the first seven Figures to the right hand be left , and the other six inscribed , as in the Example is better seen . And thus must the Operation be performed , 'till all be finished ; and considering that there are eight several Products , it may be well imagined , that at least ( 2 ) must be carried from the last place ; and therefore ( 2 ) being added to ( 27 ) there must be set down ( 9 ) , and ( 2 ) carried to the next place ; and thus must be wrought the whole Multiplication , and at last it gives the Product , as is here-under exprest , viz. And after this manner may any Multiplication be contracted to any number of places , more or less . Division of Decimals . IN Division of Decimals , the greatest difficulty is to know of what nature the first Figure or Cypher in the Quotient ought to be , for that being once known , all other things are the very same as in the ordinary Operation of Division . And therefore I shall give this General Rule , for the finding of what nature or quality the first Figure or Cypher of any Quotient in a Decimal Operation ought to be . General Rule . The first Figure in the Quotient must and will always be of the same nature and quality with that Figure or Cypher in the Dividend , which at the first Question stands over the place of Vnites in the Divisor . Example 1. Let 7.4944 be given to be divided by 32. By the foregoing Rule , because the Figure ( 4 ) of the Dividend stands over the Unite ( 2 ) of the Divisor , and the Figure ( 4 ) is a Fraction ; therefore the first Figure of the Quotient ( viz. 2 ) must be a Fraction , and have a Point prefixed . And then all the other Figures of the Quotient follow in course , as in the ordinary Method of Division . Tariffa . 1 32 2 64 3 96 4 128 5 160 6 192 7 224 8 256 9 288 Example 2. Direction In this last Example , because the Divisor may not be placed under the first Figure of the Dividend , nor indeed under the second , therefore are two Cyphers put first in the Quotient , but under the third Figure it may be set , and then .0204 is found three times in 0652 , and 40 over ; then bringing down ( 8 ) , and adding it to 40 , makes the Product ( 408 ) , which is just the double of ( 204 ) which gives ( 2 ) for the last Figure of the Quotient . And after this manner may any Division be wrought , without the least difficulty or uncertainty . Example 3. Let. 0006258 be the Dividend , and. 0032 the Divisor . Here must be a remove before the Divisor will come under the the Dividend , which is the occasion of putting one Cypher in the Quotient , before the Figure ( 2 ) . Tariffa . 1 0032 2 0064 3 0096 4 0128 5 0160 6 0192 7 0224 8 0256 9 0288 Again , Let the Numbers in the first Example be given thus : Tariffa . 1 74944 2 149888 3 224832 4 299776 5 374720 6 449664 7 524608 8 599552 9 674496 Explanation of the foregoing Examples . Explanation of the first Example . IN the first Example a less Number 7.4944 is divided by a greater , viz. 32. The young Practitioner will presently object , and demand how this can be , for to divide one Number by another , is to demand how many times that other Number is found in the first ; that is , in this case , how many times 32 is found in 7 Integers , and a Fraction of .4944 . Explanation . The Answer in plain English is this : First , 32 is not found so much as once in 7 , and that is the reason of the Full-point ( . ) in the Quotient , before the Figures of the Fraction , to signifie , that the whole Quotient consists of Decimal Parts . Secondly , the first Figure of the Fraction being ( 2 ) denotes this , namely that 32 comes no nearer , being found in 7.49 , &c. than ●2 / 10 , or two Tenths of once , or one time ; that is to say , it comes no nearer than 2 is to 10. And the second Figure of the Quotient ( 3 ) gives to understand , that 32 comes no nearer , being found in 7.49 , &c. so much as once , or one time , than 23 / 100 , or Twenty three Hundred parts ; that is to say , no nearer than 23 is to 100. And ( 4 ) the third Figure goes farther , and says , that 32 comes no nearer , being found once , or one time , in 7.49 , &c. than 234 / 1000 , or Two Hundred thirty four Thousand parts ; that is to say , no nearer than 234 is to 1000. And the last Figure determining the Question , yet somewhat more exactly ; that is to say , denotes that 32 comes no nearer , being found so much as once in 7.4944 , than 2342 / 10000 ; that is , no nearer than 2342 is to 10000. Explanation of the second Example . IN the second Example likewise a less Number seems to be divided by a greater , viz. .0006528 by .0204 , and also ( in the third Example ) by .0032 ; and an Explanation of one of these may serve for both . And the true meaning is , 1. First .0032 cannot be found once in .0006528 , therefore is a Point prefixed before the Quotient . 2. The first Cypher denotes that .0032 comes not so near , being found once in .0006528 , as 1 / 10 , or one Tenth ; that is , not so near as 1 to 10. 3. The second Cypher tells the Practitioner that it comes not so near as 1 / 100 , or as 1 to 100. 4. The Figure ( 3 ) in the third place , acquaints him , that 32 is no nearer , being found once in .0006528 , than 1 / 1000 , that is , Three parts of a Thousand , or no nearer than 3 is to 1000. And the last Figure in the Quotient , ( viz. 2 ) signifies that 32 is no nearer , being found once in . 0006528 , than 32 / 10000 ; that is to say , no nearer than 32 is to 10000. And this Mystery being once throughly comprehended , and digested by the young Practitioner , there can be no farther difficulty , about a less Number being divided by a greater . 5. In the fifth and last place , by this little Book may be compared together the Operations of Simple and Compound Interest , and so may be discovered how erroneous and extravagant the one is , and how true and rational the other , and only fit to be made use of by all those who deal in matters of Money , or Purchases , which that the Reader may better comprehend , let him consider well the following Animadversions , or Reflections . Reflections upon Simple and Compound Interest . Reflection I. LEt there be proposed an Annuity of 100 l. to be continued 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , or 100 Years , and let it be demanded , what the Present Worth of such an Annuity is for any of the following Terms , at the Rate of 6 per Cent. and that as well according to Simple as Compound Interest ? Answer . An Annuity of 100 l. to continue for 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , or 100 Years , is worth in present Money so many Years Purchase as is hereafter exprest , viz. Number of Years to be continued . Years Purchase , at Simple Interest . Years Purchase at Compound Interest . 10 7.93 7.35 20 14.27 11.46 30 20.03 13.76 40 25.52 15.01 50 30.87 15.72 60 36.13 16.16 70 41.32 16.38 80 46.48 16.50 90 51.60 16.57 100 56.71 16.61 By which Table it is very observable , what a small difference there is at Compound Interest , between the Present Worth of 50 Years , and the Present Worth of 100 Years , ( viz. 00. 89 / 100 ) in comparison with the difference between 50 and 100 Years , at Simple Interest , ( viz. 25 Years Purchase , more by 84 / 100 ) the one not exceeding 16 Years Purchase , more by 61 / 100 ; and the other still increasing as far as almost 57 Years Purchase ; and if continued to a greater Number of Years , would still swell into an extravagant Sum , for the Purchase , Treble , or Quadruple , to the usual Rate of Purchases in the Kingdom where wo live . Reflection II. FOr as much as it is a truth received by all , That the Purchase of an Estate or Revenue for ever , in most places of England , is not worth above 20 Years Purchase , and that to be computed according to Compound Interest , which is made up of so many Geometrical Proportional Numbers : What reason can there be given , why the Present Worth of any Payment , due at any time hereafter , should not be computed by the same proportion , although the Payment be but for a Year , nay , for a Day , or Hour , or Minute to come ? Thus , according to the Rate of Compound Interest , the Purchase of an Estate for 30 years to come , at 4 per Cent. is worth 17 years Purchase , and somewhat more ; at 5 per Cent. is worth 15 years Purchase , more by ●9 / 100 ; at 6 per Cent. is worth 13 years Purchase , more by 76 / 100. The same Estate for 20 years to come , at 4 per Cent. is worth 13 years and a half Purchase , and somewhat more ; and for 10 years to come , is worth above 8 years Purchase ; and for two years to come , is worth one years Purchase , more by 88 / 100 ; and all these Numbers are Calculated as Geometrical Proportionals : Why then should the same Estate , for a Year , or 6 Months to come , be Calculated by any other Proportion ? Or indeed , how can it be rightly Calculated by any other Proportion , without doing wrong to either Buyer or Seller ? Reflection III. COmpound Interest being made up of Geometrical Proportionals , the Debtor ought not really to pay after the Rate of 30 s. a Quarter for 100 l. let out to him at 6 per Cent. because , if 100 l. be put out to Interest , and the Interest come to 1 l. 10 s. the first Quarter , that 101 l. 10 s. by the end of the next Quarter ( keeping to Geometrical Proportion ) will become 103 l. 5 d. 1 q. more by 6● / 100 of a Farthing ; that is to say , 100 l. after this manner , would amount in a years time to 106 l. 2 s. 8 d. 2 q. more by Ninety Hundred parts of a Farthing , as may be seen by the following Calculation . Which in a great Sum is more considerable . For suppose the Crown to be indebted 1 Million , or ( 1000000 l. ) and it were agreed to pay at the Rate of 30 s. for each 100 l. the first Quarter , and it were not paid 'till the Twelve Months end ; the Amount would be as follows . Which at the years end amounts to 60000 l. ( which ought to be the Total Sum of the Interest for 12 Months , at 6 per Cent. ) and over and above the said 60000 l. there is 1363 l. 11 s. which 1 Million according to such an Accompt , if put out for a year , would amount to : So that in effect this is not 6 per Cent. but 6 l. 2 s. 8 d. 2 q. Ninety Hundred parts of a Farthing per Cent. For he who lends Money , if so soon as the first Quarters Interest grows due , and the Creditor pay it not at the just time , ( if he so please ) obliges the Creditor to acknowledge so much Principal , and then it increases as aforesaid . Divers other Reflections of this kind might be made , and applied to the manifold abuses that may be committed , by Selling according to one Rate of Interest , and Buying by another , and so confounding together Simple and Compound Interest , as it makes most for the advantage of the Money Merchant , there being very few so well versed in Numbers as to contradict them . The truth is , it is as great pity that there should be two so different Calculations of Interest , as that there should be so many different Weights and Measures , and those divided and subdivided into so many Heterogeneous Fractions , which must of necessity create to all Dealers innumerable difficulties ; whereas if Coyns , Weights , and Measures , were divided and subdivided by Decimals , all Calculations would be performed with ease and pleasure . For Instance , If a Pound were divided into 10 Shillings , a Shilling into 10 Pence , and a Peny into 10 Farthings , and only a Point to distinguish Integers from Fractions ; then the following Sum would easily be added together , viz. l. s. d. q. 15 9 6 1 39 8 3 0 48 7 2 1 For they might be set down thus : That is to say , the Sum would be 104 Pound , 5 Shillings , 1 Peny , and 2 Farthings . Or if these were Weights , they might be , 104 Pound-weight , 5 Ounces , 1 Dram , 2 Scruples . And after this manner might all Calculations be abbreviated , and made much more practicable than now they are , especially the Operations of Multiplication and Division . For to multiply 48 l. 7 s. 2 d. 2 q. as they are now divided , by 124 , is very troublesom , and requires many Operations , both of Multiplication and Division ; but in a Decimal way , it would be plain and easie by one single Multiplication , viz. That is to say , 6041 Pound , 5 Shillings , 2 Pence , and 8 Farthings . The convenience and expedition would yet be greater in Long and Square Measures ; and all former Accompts by unequal Divisions , might for the present be reconciled and reduced to Decimals , and in a few years utterly forgotten , and become altogether useless . But for as much as a private Person can only give hints of what he conceives to be of publick use and benefit , and that it is a thing wholly in the Power of those who are Law-makers , to inspect and rectifie what they in their great Wisdom shall judge amiss ; the AUTHOR does in all Humility lay by his Pen , and puts a period to his Discourse upon this Subject . The CONTENTS of the First Book . OF Interest in general , Page 1 The Reduction of Shillings , Pence , and Farthings , into Decimal Fractions , 3 The Interest of One Pound for a Year , at any Rate of Simple Interest , from 1 to 12 per Cent. 4 The Interest of One Pound for half a Year , at any Rate of Simple Interest , from 1 to 12 per Cent. 5 The Interest of One Pound for a Quarter of a Year , at any Rate of Simple Interest , from 1 to 12 per Cent. 6 The Interest of One Pound for a Month , at any Rate of Simple Interest , from 1 to 12 per Cent. 7 The Interest of One Pound for a Day , at any Rate of Simple Interest , from 1 to 12 per Cent. 8. The Golden Table of Trigonal Progression , of admirable Vse in all Calculations of the Amount or Present Worth of Annuities , &c. 9 The Number of Days from the beginning of any Month to the end of any other , throughout the Year , 13 The Amount of One Pound forborn any Number of Years under 32 , at 6 per Cent. Simple Interest , Page 16 The Amount of One Pound forborn any Number of equal Months under 25 , at 6 per Cent. Simple Interest , 17 The Amount of One Pound forborn any Number of Days under 366 , at 6 per Cent. Simple Interest , 18 The Present Worth of One Pound due after any Number of Years to come , not exceeding 32 , at 6 per Cent. Simple Interest , 27 The Present Worth of One Pound due after the expiration of any Number of Months under 25 , at 6 per Cent. Simple Interest . 29 The Present Worth of One Pound due after the expiration of any Number of Days under 366 , at 6 per Cent. Simple Interest , 31 The Reduction of Pence and Farthings into Decimal Fractions , to the Hundredth part of a Farthing , 41 The Use of the foregoing Tables , 46 A Comparison between these and Mr. Clavel's Tables , wherein it is proved , that the former are less troublesome , and more exact than the latter , 52 Of Annuities , 59 The Multiplication of any Rates of Interest whatsoever belonging to each Year , for a forborn Annuity , to 100 Years , 60 To find the Amount of any Annuity for any given time , at any Rate of Simple Interest , Page 64 This kind of Interest for Annuities useless and ridiculous , 65 The Errors of the ordinary Rules and Tables for Rebate relating to Annuities , ibid. Mr. Kersey and Dr. Newton both mistaken , 66 Diophantus Alexandrinus his third Proposition concerning Poligonal Numbers considered , 73 A second Reflection on Mr. Kersey's and Dr. Newton's mistake , 87 Equation of Payments rectified , 93 The Amount and Present Worth of an Annuity of 100 l. for 5 Years , at 1 , 2 , 3 , 4 , 5 , 6 , and 10 per Cent. Simple Interest , Page 95 , &c. Observations on these Tables , 102 Equation of Vnequal Payments at times not equidistant , 123 The CONTENTS of the Second Book . COmpound Interest explained , Page 129 A Reflection upon Geometrical Progression , 131 The Amount of One Pound put out to Interest , and forborn any Number of Years under 32 , or Quarters under 125 , at 6 per Cent. Compound Interest , 141 The Amount of One Pound put out to Interest for any Number of Months under 25 , at 6 per Cent. Compound Interest , Page 146 The Amount of One Pound put out to Interest for any Number of Days under 366 , at 6 per Cent. Compound Interest , 147 The Present Worth of One Pound due after any Number of Years under 32 , or Quarters under 125 , at 6 per Cent. Compound Interest , 157 The Present Worth of One Pound due after the expiration of any Number of Months under 25 , at 6 per Cent. Compound Interest , 162 The Present Worth of One Pound due after the expiration of any Number of Days under 366 , at 6 per Cent. Compound Interest , 163 The Present Worth of One Pound Annuity , to continue any Number of Years under 32 , and payable by yearly Payments , at 5 , 6 , 7 , 8 , 9 , and 10 per Cent. Compound Interest , 173 What Annuity to continue any Number of Years under 32 , and payable by yearly Payments , One Pound will Purchase , at 5 , 6 , 7 , 8 , 9 , and 10 per Cent. Compound Interest , 179 The Present Worth of any Lease or Annuity , for 21 , 31 , 41 , 51 , 61 , 71 , 81 , or 91 Years ; as likewise the Present Worth of the Fee-Simple , at 5 , 6 , 8 , and 10 per Cent. Compound Interest , 184 The several Vses of the foregoing Tables , 186 THE DOCTRINE OF SIMPLE INTEREST EXPLAINED By a New and Exact Method , And the Errors of the Ordinary Rules and Tables of Rebate discovered and rectified . CHAP. I. INterest is either Simple , or Compound . 1. Simple Interest , is the Increase which arises from the Principal only , at 4 , 5 , 6 , 7 , &c. per Cent. 2. Compound Interest , is the Increase which arises from the Principal , and also from the Interest thereof . Thus , if 100 l. be lent at Simple Interest for Two Years , at 6 per Cent. the Increase thereof is 12 l. But if at Compound Interest , it gives 6 l. for the first Year , and 6 l. for the second Year , together with the Interest of the first 6 l. for the second Year . That is to say : To which adding the Principal ( viz. 100 l. ) the Amount of both Principal and Compound Interest , for Two years , is 112.36 l. which by the Table of Reduction in the following Page is 112 l. 7 s. 2 d. 1 q. more by .000626 parts of a Pound . The Doctrine of Simple Interest is plainly and clearly set forth in the following Propositions . But that the Practitioner may meet with no difficulty in the respective Operations , he will here find made ready to his hand Seven short ( but very significant ) Tables . TABLE I. Reduction of Shillings , Pence , and Farthings , into Decimal Fractions . Shillings . Decimals .   Pence . Decimals .       11 .0458333 19 .95   10 .0416666 18 .9   9 .0375 17 .85   8 . 03●3333 16 .8   7 .0291666 15 .75   6 .025 14 .7   5 .0208333 13 .65   4 .0166666 12 .6   3 .0125 11 .55   2 .0083333 10 .5   1 .0041666 9 .45       8 .4   Farth . Decimals . 7 .35       6 .3   3 .003125 5 .25   2 .0020833 4 .2   1 .0010416 3 .15   ½ .0005208 2 .1   ¼ .0002604 1 .05   1 / 8 .0001302 TABLE II. The INTEREST of One Pound for One Year . From 1 to 12 per Cent. Rates per Cent. Interest . 1 .01 2 .02 3 .03 4 .04 5 .05 6 .06 7 .07 8 .08 9 .09 10 .10 11 .11 12 .12 TABLE III. The INTEREST of One Pound for One Half-Year . From 1 to 12 per Cent. Rates per Cent. Interest . 1 .005 2 .01 3 .015 4 .02 5 .025 6 .03 7 .035 8 .04 9 .045 10 .05 11 .055 12 .06 TABLE IV. The INTEREST of One Pound for One Quarter . From 1 to 12 per Cent. Rates per Cent. Interest . 1 .0025 2 .005 3 .0075 4 .01 5 .0125 6 .015 7 .0175 8 .02 9 .0225 10 .025 11 .0275 12 .03 TABLE V. The INTEREST of One Pound for One Month. From 1 to 12 per Cent. Rates per Cent. Interest . 1 .0008333 2 .0016666 3 .0025 4 .00333 5 .00416 6 .005 7 .00583 8 .00666 9 .0075 10 .08333 11 .09166 12 .1 TABLE VI. The INTEREST of One Pound for One Day . From 1 to 12 per Cent. Rates per Cent. Interest . 1 .00002739726 2 .00005479452 3 .00008219178 4 .00010958904 5 .00013698630 6 .00016438356 7 .00019178082 8 .00021917808 9 .00024657534 10 .00027397260 11 .00030136986 12 .00032876712 TABLE VII . THE GOLDEN TABLE OF Trigonal Progression . Of excellent use , in all Calculations of the Amount or present Worth of Annuities , &c. Arith. Prog. Trigonal Prog. 1 1 2 3 3 6 4 10 5 15 6 21 7 28 8 36 9 45 10 55 11 66 12 78 13 91 14 105 15 120 16 136 17 153 18 171 Arith. Prog. Trigonal Prog. 19 190 20 210 21 231 22 253 23 276 24 300 25 325 26 351 27 378 28 406 29 435 30 465 31 496 32 528 33 561 34 595 35 630 36 666 37 703 38 741 39 780 40 820 41 861 42 903 43 946 44 990 45 1035 46 1081 47 1128 48 1176 49 1225 50 1275 51 1326 52 1378 53 1431 54 1485 55 1540 56 1596 57 1653 58 1711 59 1770 60 1830 Arith. Prog. Trigonal Prog. 61 1891 62 1953 63 2016 64 2080 65 2145 66 2211 67 2278 68 2346 69 2415 70 2485 71 2556 72 2628 73 2701 74 2775 75 2850 76 2926 77 3003 78 3081 79 3160 80 3240 81 3321 82 3403 83 3486 84 3570 85 3655 86 3741 87 3828 88 3916 89 4005 90 4095 91 4186 92 4278 93 4371 94 4465 95 4560 96 4656 97 4753 98 4851 99 4950 100 5050 101 5151 102 5253 Arith. Prog. Trigonal Prog. 103 5356 104 5460 105 5565 106 5671 107 5778 108 5886 109 5995 110 6105 111 6216 112 6328 113 6441 114 6555 115 6670 116 6786 117 6903 118 7021 119 7140 120 7260 121 7381 122 7503 123 7626 124 7750 TABLE VIII . A TABLE SHEWING The Number of Days from the Beginning of any Month to the End of any other . JAnuary , 31. February , 59. March , 90. April , 120. May , 151. June , 181. July , 212. August , 243. September , 273. October , 304. November , 334. December , 365. February , 28. March , 59. April , 89. May , 120. June , 150. July , 181. August , 212. September , 242. October , 273. Novemb. 303. Decemb. 334. Jan. 365. March , 31. April , 61. May , 92. June , 122. July , 153. August , 184. September , 214. Octob. 245. Novemb. 275. Decemb. 306. Jan. 337. Febr. 365. April , 30. May , 61. June , 91. July , 122. August , 153. September , 183. October , 214. November , 244. December , 275. January , 306. February 334. March , 365. May , 31. June , 61. July , 92. August , 123. September , 153. October , 184. November , 214. December , 245. Jan. 276. February , 304. March , 335. April , 365. June , 30. July , 61. August , 92. September , 122. October , 153. Novemb. 183. December , 214. January , 245. Febr. 273. March , 304. April , 334. May , 365. July , 31. August , 62. September , 92. October , 123. Novemb. 153. Decemb. 184. January , 215. February , 243. March , 274. April , 304. May , 335. June , 365. August , 31. September , 61. Octob. 92. November , 122. December , 153. January , 184. February , 212. March , 243. April , 273. May , 304. June , 334. July , 365. September , 30. October , 61. November , 91. December , 122. January , 153. February , 181. March , 212. April , 242. May , 273. June , 303. July , 334. Aug. 365. October , 31. November , 61. Decemb. 92. January , 123. February , 151. March , 182. April , 212. May , 243. June , 273. July , 304. August , 335. September , 365. November , 30. December , 61. January , 92. February , 120. March , 151. April , 181. May , 212. June , 242. July , 273. Aug. 304. Sept. 334. Octob. 365. December , 31. January , 62. February , 90. March , 121. April , 151. May , 182. June , 212. July , 243. August , 274. September , 304. October , 335. Novemb. 365. Note , That in every Leap-year , February has 29 Days , and then you must allow a Day more than is here computed for that Month. TABLE IX . The Amount of One Pound , put out to Interest , and forborn any Number of Years under 32. At the Rate of 6 per Cent. Simple Interest ; And that Interest payable Yearly . Years . Amount . 1 1.06 2 1.12 3 1.18 4 1.24 5 1.30 6 1.36 7 1.42 8 1.48 9 1.54 10 1.60 11 1.66 12 1.72 13 1.78 14 1.84 15 1.90 16 1.96 17 2.02 18 2.08 19 2.14 20 2.20 21 2.26 22 2.32 23 2.38 24 2.44 25 2.50 26 2.56 27 2.62 28 2.68 29 2.74 30 2.80 31 2.86 TABLE X. The AMOUNT of One Pound , for any Number of equal Months under 25. At the Rate of 6 per Cent. Simple Interest . Months . Amount . 1 1.005 2 1.010 3 1.015 4 1.020 5 1.025 6 1.030 7 1.035 8 1.040 9 1.045 10 1.050 11 1.055 12 1.060 13 1.065 14 1 070 15 1.075 16 1.080 17 1.085 18 1.090 19 1.095 20 1.100 21 1.105 22 1.110 23 1.115 24 1.120 TABLE XI . The AMOUNT of One Pound , for any Number of Days under 366. At the Rate of 6 per Cent. Simple Interest . Days . Amount . 1 1.000164383 2 1.000328767 3 1.000493150 4 1.000657534 5 1.000821917 6 1.000986301 7 1.001150684 8 1.001315068 9 1.001479452 10 1.001643835 11 1.001808219 12 1.001972602 13 1.002136986 14 1.002301369 15 1.002465753 16 1.002630136 17 1.002794520 18 1.002958904 19 1.003123287 20 1.003287671 21 1.003452054 22 1.003616438 23 1.003780821 24 1.003945205 25 1.004109589 26 1.004273972 27 1.004438356 28 1.004602739 29 1.004767123 30 1.004931506 31 1.005095890 32 1.005260273 Days . Amount . 33 1.005424657 34 1.005589041 35 1.005753424 36 1.005917808 37 1.006082191 38 1.006246575 39 1.006410958 40 1.006575342 41 1.006739725 42 1.006904109 43 1.007068493 44 1.007232876 45 1.007397260 46 1.007561643 47 1.007726027 48 1.007890410 49 1.008054794 50 1.008219178 51 1.008383561 52 1.008547945 53 1.008712328 54 1.008876712 55 1.009041095 56 1.009205479 57 1.009369862 58 1.009534246 59 1.009698630 60 1.009863013 61 1.010027397 62 1.010191780 63 1.010356164 64 1.010520547 65 1.010684931 66 1.010849314 67 1.011013698 68 1.011178082 69 1.011342465 70 1.011506849 71 1.011671232 72 1.011835616 73 1.011999999 74 1.012164383 75 1.012328767 76 1.012493150 Days . Amount . 77 1.012657534 78 1.012821917 79 1.012986301 80 1.013150684 81 1.013315068 82 1.013479451 83 1.013643835 84 1.013808219 85 1.013972602 86 1.014136986 87 1.014301369 88 1.014465753 89 1.014630136 90 1.014794520 91 1.014958903 92 1.015123287 93 1.015287671 94 1.015452054 95 1.015616438 96 1.015780821 97 1.015945205 98 1.016109588 99 1.016273972 100 1.016438356 101 1.016602739 102 1.016767123 103 1.016931506 104 1.017095890 105 1.017260273 106 1.017424657 107 1.017589040 108 1.017753424 109 1.017917808 110 1.018082191 111 1.018246575 112 1.018410958 113 1.018575342 114 1.018739725 115 1.018904109 116 1.019068492 117 1.019232876 118 1.019397260 119 1.019561643 120 1.019726027 Day . Amount . 121 1.019890410 122 1.020054794 123 1.020219177 124 1.020383561 125 1.020547945 126 1.020712328 127 1.020876712 128 1.021041095 129 1.021205479 130 1.021369862 131 1.021534246 132 1.021698629 133 1.021863013 134 1.022027397 135 1.022191780 136 1.022356164 137 1.022520547 138 1.022684931 139 1.022849314 140 1.023013698 141 1.023178081 142 1.023342465 143 1.023506849 144 1.023671232 145 1.023835616 146 1.023999999 147 1.024164383 148 1.024328766 149 1.024493150 150 1.024657534 151 1.024821917 152 1.024986301 153 1.025150684 154 1.025315068 155 1.025479451 156 1.025643835 157 1.025808218 158 1.025972602 159 1.026136986 160 1.026301369 161 1.026465753 162 1.026630136 163 1.026794520 164 1.026958903 Day . Amount . 165 1.027123287 166 1.027287670 167 1.027452054 168 1.027616438 169 1.027780821 170 1.027945205 171 1.028109588 172 1.028273972 173 1.028438355 174 1.028602739 175 1.028767123 176 1.028931506 177 1.029095890 178 1.029260273 179 1.029424657 180 1.029589040 181 1.029753424 182 1.029917807 183 1.030082191 184 1.030246575 185 1.030410958 186 1.030575342 187 1.030739725 188 1.030904109 189 1.031068492 190 1.031232876 191 1.031397259 192 1.031561643 193 1.031726027 194 1.031890410 195 1.032054794 196 1.032219177 197 1.032383561 198 1.032547944 199 1.032712328 200 1.032876712 201 1.033041095 202 1.033205479 203 1.033369862 204 1.033534246 205 1.033698629 206 1.033863013 207 1.034027396 208 1.034191780 Day . Amount . 209 1.034356164 210 1.034520547 211 1.034684931 212 1.034849314 213 1.035013698 214 1.035178081 215 1.035342465 216 1.035506848 217 1.035671232 218 1.035835616 219 1.036000000 220 1.036164383 221 1.036328766 222 1.036493150 223 1.036657533 224 1.036821917 225 1.036986301 226 1.037150684 227 1.037315068 228 1.037479451 229 1.037643835 230 1.037808218 231 1.037972602 232 1.038136985 233 1.038301369 234 1.038465753 235 1.038630136 236 1.038794520 237 1.038958903 238 1.039123287 239 1.039287670 240 1.039452054 241 1.039616437 242 1.039780821 243 1.039945205 244 1.040109588 245 1.040273972 246 1.040438355 247 1.040602739 248 1.040767122 249 1.040931506 250 1.041095890 251 1.041260273 252 1.041424657 Day . Amount . 253 1.041589040 254 1.041753424 255 1.041917807 256 1.042082191 257 1.042246574 258 1.042410958 259 1.042575342 260 1.042739725 261 1.042904109 262 1.043068492 263 1.043232876 264 1.043397259 265 1.043561643 266 1.043726026 267 1.043890410 268 1.044054794 269 1.044219177 270 1.044383561 271 1.044547944 272 1.044712328 273 1.044876711 274 1.045041095 275 1.045205479 276 1.045369862 277 1.045534246 278 1.045698629 279 1.045863013 280 1.046027396 281 1.046191780 282 1.046356163 283 1.046520547 284 1.046684931 285 1.046849314 286 1.047013698 287 1.047178081 288 1.047342465 289 1.047506848 290 1.047671232 291 1.047835615 292 1.048000000 293 1.048164383 294 1.048328766 295 1.048493150 296 1.048657533 Day . Amount . 297 1.048821917 298 1.048986300 299 1.049150684 300 1.049315068 301 1.049479451 302 1.049643835 303 1.049808218 304 1.049972602 305 1.050136985 306 1.050301369 307 1.050465752 308 1.050630136 309 1.050794520 310 1.050958903 311 1.051123287 312 1.051287670 313 1.051452054 314 1.051616437 315 1.051780821 316 1.051945204 317 1.052109588 318 1.052273972 319 1.052438355 320 1.052602739 321 1.052767122 322 1.052931506 323 1.053095889 324 1.053260273 325 1.053424657 326 1.053589040 327 1.053753424 328 1.053917807 329 1.054082191 330 1.054246574 331 1.054410958 332 1.054575341 333 1.054739725 334 1.054904109 335 1.055068492 336 1.055232876 337 1.055397259 338 1.055561643 339 1.055726026 340 1 . 0558●●41● Day . Amount . 341 1.056054793 342 1.056219177 343 1.056383561 344 1.056547944 345 1.056712328 346 1.056876711 347 1.057041095 348 1.057205478 349 1.057369862 350 1.057534246 351 1.057698629 352 1.057863013 353 1.058027396 354 1.058191780 355 1.058356163 356 1.058520547 357 1.058684930 358 1.058849314 359 1.059013698 360 1.059178081 361 1.059342465 362 1.059506848 363 1.059671232 364 1.059835615 365 1.060000000 TABLE XII . The PRESENT WORTH of One Pound , due after any Number of Years to come , under 32. At the Rate of 6 per Cent. Simple Interest . Years to come . Present Worth. 1 .94339622 2 .89285714 3 .84745762 4 .80645161 5 .76923076 6 .73529411 7 .70422535 8 .67567567 9 .64935064 10 .62500000 11 .60240963 12 .58139534 Years to come . Present Worth. 13 .56179775 14 .54347826 15 .52631578 16 .51020408 17 .49504950 18 .48076923 19 .46728971 20 .45454545 21 .44247787 22 .43103448 23 .42016806 24 .40983606 25 .40000000 26 .39062500 27 .38167939 28 .37313432 29 .36496350 30 .35714285 31 .34965034 TABLE XIII . The PRESENT WORTH of Due Pound , due after the expiration of any Number of Months under 25. At the Rate of 6 per Cent. Simple Interest . Months to come . Present Worth. 1 .99502487 2 .99009900 3 .98522167 4 .98039215 5 .97560975 6 .97087378 7 .96618357 8 .96153846 9 .95693779 10 .95238095 11 .94786729 12 .94339622 Months to come . Present Worth. 13 .93896713 14 .93457943 15 .93023255 16 .92594444 17 .92165898 18 .91743119 19 .91324200 20 .90909090 21 .90497737 22 .90090090 23 .89686098 24 .89285714 TABLE XIV . The PRESENT WORTH of One Pound , due after the expiration of any Number of Days under 366. At the Rate of 6 per Cent. Simple Interest . Days to come . Present Worth. 1 .99983564 2 .99967134 3 .99950709 4 .99934290 5 .99917876 6 .99901467 7 .99885064 8 .99868666 9 .99852273 10 .99835886 11 .99819504 12 .99803128 13 .99786757 14 .99770391 15 .99754031 16 .99737676 17 .99721326 18 .99704982 19 .99688643 20 .99672310 Days to come . Present Worth. 21 .99655982 22 .99639659 23 .99623341 24 .99607029 25 .99590723 26 .99574421 27 .99558125 28 .99541835 29 .99525549 30 .99509269 31 .99492994 32 .99476725 33 .99460460 34 .99444201 35 .99427948 36 .99411700 37 . 993954●7 38 .99379220 39 .99362987 40 .99346761 41 .99330539 42 .99314323 43 .99298112 44 .99281906 45 .99265706 46 .99249511 47 .99233321 48 .99217136 49 .99200957 50 .99184782 51 .99168614 52 .99152450 53 .99136292 54 .99120139 55 .99103991 56 .99087848 57 .99071711 58 .99055579 Days to come . Present Worth. 59 .99039453 60 .99023331 61 .99007215 62 .98991104 63 .98975000 64 .98958899 65 .98942804 66 .98926714 67 .98910629 68 .98894549 69 .98878475 70 .98862406 71 .98846341 72 .98830283 73 .98814230 74 .98798181 75 .98782138 76 .98766100 77 .98750067 78 .98734040 79 .98718018 80 .98702001 81 .98685989 82 .98669983 83 .98653980 84 .98637984 85 .98621993 86 .98606008 87 .98590027 88 .98574052 89 .98558081 90 .98542116 91 .98526156 92 .98510202 93 .98494253 94 .98478309 95 .98462369 96 .98446435 Days to come . Present Worth. 97 .98430506 98 .98414582 99 .98398663 100 .98382749 101 .98366841 102 .98350937 103 .98335039 104 .98319146 105 .98303258 106 .98287376 107 .98271498 108 .98255626 109 .98239758 110 .98223896 111 .98208039 112 .98192188 113 .98176340 114 .98160498 115 .98144662 116 .98128830 117 .98113004 118 .98097183 119 .98081367 120 .98065556 121 .98049750 122 .98033949 123 .98018152 124 .98002361 125 .97986576 126 .97970795 127 .97955020 128 .97939250 129 .97923485 130 .97907725 131 .97891970 132 .97876221 133 .97860473 134 .97844734 Days to come . Present Worth. 135 .97828999 136 .97813269 137 .97797545 138 .97781825 139 .97766111 140 .97750401 141 .97734697 142 .97718997 143 .97703304 144 .97687614 145 .97671930 146 .97656250 147 .97640576 148 .97624906 149 .97609242 150 .97593583 151 .97577928 152 .97562279 153 .97546635 154 .97530996 155 .97515362 156 .97499732 157 .97484108 158 .97468489 159 .97452875 160 .97437266 161 .97421662 162 .97406063 163 .97390470 164 .97374880 165 .97359296 166 .97343717 167 .97328143 168 .97312574 169 .97297009 170 .97281450 171 .97265896 172 .97250346 Days to come . Present Worth. 173 .97234803 174 .97219263 175 .97203729 176 .97188199 177 .97172675 178 .97157155 179 .97141640 180 .97126131 181 .97110626 182 .97095127 183 .97079631 184 .97064141 185 .97048656 186 .97033177 187 .97017702 188 .97002232 189 .96986767 190 .96971307 191 .96955852 192 .96940401 193 .96924959 194 .96909518 195 .96894083 196 .96878652 197 .96863227 198 .96847806 199 .96832387 200 .96816976 201 .96801570 202 .96786169 203 .96770773 204 .96755382 205 .96739995 206 .96724614 207 .96709237 208 .96693865 209 .96678498 210 .96663136 Days to come . Present Worth. 211 .96647778 212 .96632426 213 .96617079 214 .96601737 215 .96586399 216 .96571066 217 .96555738 218 .96540415 219 .96525096 220 .96509783 221 .96494475 222 .96479171 223 .96463871 224 .96448578 225 .96433289 226 .96418004 227 .96402725 228 .96387451 229 .96372181 230 .96356916 231 .96341656 232 .96326401 233 .96311151 234 .96295906 235 .96280665 236 .96265429 237 .96250198 238 .96234972 239 .96219750 240 .96204533 241 .96189322 242 .96174114 243 .96158912 244 .96143715 245 .96128522 246 .96113334 247 .96098151 248 .96082973 Days to come . Present Worth. 249 .96067800 250 .96052631 251 .96037467 252 .96022309 253 .96007154 254 .95992004 255 .95976860 256 .95961720 257 .95946585 258 .95931454 259 .95916329 260 .95901208 261 .95886092 262 .95870981 263 .95855875 264 .95840773 265 .95825676 266 .95810584 267 .95795497 268 .95780414 269 .95765335 270 .95750262 271 .95735194 272 .95720130 273 .95705071 274 .95690016 275 .95674967 276 .95659922 277 .95644882 278 .95629847 279 .95614816 280 .95599790 281 .95584769 282 .95569753 283 .95554742 284 .95539735 285 .95524732 286 .95509735 Days to come . Present Worth. 287 .95494742 288 .95479753 289 .95464770 290 .95449791 291 .95434817 292 .95419847 293 .95404882 294 .95389923 295 .95374967 296 .95360016 297 .95345070 298 .95330129 299 .95315192 300 .95300261 301 .95285334 302 .95270411 303 .95255493 304 .95240580 305 .95225672 306 .95210768 307 .95195869 308 .95180974 309 .95166084 310 .95151199 311 .95136318 312 .95121442 313 .95106573 314 .95091706 315 .95076844 316 .95061986 317 .95047134 318 .95032285 319 .95017442 320 .95002603 321 .94987769 322 .94972939 323 .94958114 324 .94943204 Days to come . Present Worth. 325 .94928478 326 .94913667 327 .94898861 328 .94884059 329 .94869262 330 .94854470 331 .94839682 332 .94824898 333 . 9481●120 334 .94795346 335 .94780576 336 .94765811 337 .94751051 338 .94736295 339 .94721544 340 .94706798 341 .94692056 342 .94677319 343 .94662587 344 .94647858 345 .94633135 346 .94618416 347 .94603701 348 .94588991 349 .94574286 350 .94559585 351 .94544889 352 .94530198 353 .94515512 354 .94500829 355 .94486151 356 .94471478 357 .94456809 358 .94442145 359 .94427485 360 .94412830 361 .94398179 362 .94383533 363 .94368897 364 .94354254 365 .94339622 TABLE XV. A most useful TABLE for Reduction of Pence and Farthings into DECIMAL FRACTIONS , to the Hundredth part of a Farthing . Farthings . Dectmal Fractions . 1 .0010416 2 .0020833 3 .0031250 Pence & Farthings . Decimal Fractions . ( 1 ) .0041666 1 .0052083 2 .0062500 3 .0072916 ( 2 ) .0083333 1 .0093750 2 .0104166 3 .0114583 ( 3 ) .0125000 1 .0135416 2 .0145833 3 .0156250 Pence & Farthings . Decimal Fractions . ( 4 ) .0166666 1 .0177708 2 .0187500 3 .0197916 ( 5 ) .0208333 1 .0218750 2 .0229166 3 .0239583 ( 6 ) .0250000 1 .0260416 2 .0270833 3 .0281250 ( 7 ) .0291666 1 .0302083 2 .0312500 3 .0322916 ( 8 ) .0333333 1 .0343750 2 .0354166 3 .0364583 ( 9 ) .0375000 1 .0385416 2 .0395833 3 .0406250 ( 10 ) .0416666 1 .0427082 2 .0437500 3 .0447916 ( 11 ) .0458333 1 .0468750 2 .0479166 3 .0489583 DECIMAL FRACTIONS for every Hundredth part of a Farthing . Hundred Parts . Decimal Fractions . 1 .000010416 2 .000020833 3 .000031249 4 .000041666 5 .000052083 6 .000062499 7 .000072916 8 .000083333 9 .000093749 10 .000104166 11 .000114583 12 .000124999 13 .000135416 14 .000145833 15 .000156249 16 .000166666 17 .000177083 18 .000187499 19 .000197916 20 .000208333 21 .000218749 22 .000229166 23 .000239583 24 .000249999 25 .000260416 26 .000270833 27 .000281249 28 .000291666 29 .000302083 30 .000312499 Hundred Parts . Decimal Fractions . 31 .000322916 32 .000333333 33 .000343749 34 .000354166 35 .000364583 36 .000374999 37 .000385416 38 .000395833 39 .000406249 40 .000416666 41 .000427083 42 .000437499 43 .000447916 44 .000458333 45 .000468749 46 .000479166 47 .000489583 48 .000499999 49 .000510416 50 .000520833 51 .000531249 52 .000541666 53 .000552083 54 .000562499 55 .000572916 56 .000583333 57 .000593749 58 .000604166 59 .000614583 60 .000624999 61 .000635416 62 .000645833 63 .000656249 64 .000666666 65 .000677083 66 .000687499 67 .000697916 68 .000708333 Hundred Parts . Decimal Fractions . 69 .000718749 70 .000729166 71 .000739583 72 .000749999 73 .000760416 74 .000770833 75 .000781249 76 .000791666 77 .000802083 78 .000812499 79 .000822916 80 .000833333 81 .000843749 82 .000854166 83 .000864583 84 .000874999 85 .000885416 86 .000895833 87 .000906249 88 .000916666 89 .000927083 90 .000937499 91 .000947916 92 .000958333 93 .000968749 94 .000979166 95 .000989583 96 .000999999 97 .001010416 98 .001020833 99 .001031249 The Use of the foregoing TABLES . BEcause the usual Rate of Interest is 6 per Cent. there are Tables calculated for the more ready dispatch of Questions relating either to the Amount , or Present Worth of any Sum ; but for any other Rate from ( 1 ) to ( 12 ) the method will be very plain and practicable . I shall begin with some Examples at 6 per Cent. Example 1. What is the Amount of 540 l. in seven Years , at 6 per Cent. Simple Interest ? Rule . See for 7 years in the Margin of Table IX . and against it you find 1.42 , the Amount of 1 l. in 7 years ; multiply 540 by 1.42 , and the Product is the Answer . Example 2. What is the Amount of 540 l. in fifteen Months , at 6 per Cent. Simple Interest ? Rule . Find 15 Months in the Margin of Table X. and against it is 1.075 ; by that multiply 540 , and the Product is the Answer . Example 3. What is the Amount of 540 l. in 279 Days , at 6 per Cent. Simple Interest ? Rule . Find 279 Days in the Margin of Table XI . and against it is 1.0458 , ( you may take more or less of the Fraction , according as you desire to be more or less exact ; ) then multiply 1.0458 by 540 , and the Product is the Answer . Example 4. What is the Present Worth of 766.8 l. at the end of 7 Years , at 6 për Cent. Simple Interest ? Rule . Find 7 Years in Table XII . and against it is .704225 ; then multiply that by the given Number 766.8 , and the Product is the Answer . Which is within 26 Hundred Parts of a Farthing of the truth , and is a sufficient Proof of the first Example . Example 5. What is the present Worth of 580.5 l. due after 15 Months , at 6 per Cent. Simple Interest ? Rule . Find 15 Months in Table XIII . and against it is .93023 , &c. this being multiplied by 580.5 , is an Answer . Which is within one Farthing of the truth , and may be made within one Hundredth part of a Farthing of the truth , and is a clear Proof of the second Example . And after this manner may any Question of this kind be easily and exactly resolved , and where the Sums are very great , the Operation will not be so tedious as that of working by Mr. Clavel's Tables . For a Proof of which , I shall here insert two Examples , one of the Amount , and the other of the present Worth of a considerable Sum. Example 6. Suppose the King borrows of some Bankers 259879 l. 17s . 9d . 3q . for a year and 349 Days ; what will be the Amount of Principal and Interest at the expiration of a Year and 349 Days , allowing them 6 per Cent ? The Operation by Mr. Clavel's Tables . In Mr. Clavel's Tables I can find no more of this Sum at one time than 10000 l. therefore I seek the Interest of that , and find the Interest of 10000 l. The odd Money I reduce into Decimal Parts of a Pound , by the Decimal Table in Mr. Russel's Appendix to Mr. Clavel , thus , Then because 200000 is twenty times 10000 , I must multiply this Fraction and whole Number by 20 , to find the Interest of 200000 l. for a Year , and 349 Days : and also multiply the said whole Number and Fraction by 5 , for the Interest of 50000 , ( there being five times 10000 contained in it ) for the like time . Example . The Interest of the remaining part of the aforesaid Sum , viz. 9879 l. 17 s. ( omitting the 9 d. 3 q. ) is to be found in this manner : Reduce the Decimal Fractions of the Interest of 200000 , and 50000 , into Shillings and Pence , and then is the The Answer ( without considering the Interest of 9 d. 3 q. which is not to be found by Mr. Clavel's Tables ) is 290381 l. 19 s. 2 d. 3 q. very near . The Operation according to the Rules of this little Book is performed by Simple Addition , thus ; The given Sum reduced by Table I. is 259879.890625 The Amount of 1. l. for 365 and 349 Days , viz. 714 Days , is 1.1173698 Tariffa for the Multiplicand . 1 259879890625 2 519759781250 3 779639671875 4 1039519562500 5 1299399453125 6 1559279343750 7 1819159234375 8 2079039125000 9 2338919015625 The Multiplication contracted , as is directed in the Introduction to this little Book . This Product , viz. 290381 l. 18 s. 9 d. 3 q. more by 86 / 100 of a Farthing , is the Answer . Example 7. Suppose there will be due after 349 Days , upon the several Branches of the King's Revenue , the Sum of 290381.94139 l. ( or 18 s. 9 d. 3 q. more by 36 / 100 of a Farthing ; ) and His Majesty have occasion to convert this into ready Money , allowing the Advancers 6 per Cent. what is the present Worth of that Sum ? or what must those persons advance in ready Money for the Premises ? Though it be the truest , and most exact way of all other , to Calculate either the Amount or present Worth of Money by Days , yet there is no help at all by Mr. Clavel's Tables to answer this Question . But by this little Book , The Rule is , l. Multiply the given sum 290381.94139 by the Present Worth of 1 l. due at the end of 349 Days ( which you will find in Table XIV . ) .94574 , and the Product is an Answer to the Question . Tariffa for the Multiplicand . 1 29038194139 2 58076388278 3 87114582417 4 116152776556 5 145190970695 6 174229164834 7 203267358973 8 232305553112 9 261343747251 The Multiplication contracted , as in the Introduction is directed . After the same manner are resolved any Questions , concerning either the Amount , or present Worth of any Sum , either for Years , Months , or Days . The next thing I shall Treat of is Annuities at Simple Interest , which shall be the Subject of the following Chapter . CHAP. II. Of Annuities at Simple Interest . THe increase of Annuities is by Multiplication of the respective Rates of Interest , according to a Trigonal Progression , which may be better seen by comparing the Golden Table of Trigonal Progression in Chap. 1. with the following Table of Trigonal Increase , or Addition of ( 6 ) the Rate of Interest per Cent. and after it short Rules , which will hold for finding the Amount , or Present Worth of any Annuity , for any number of years , at any Rate of Simple Interest whatsoever . TABLE . The Multiplication of any Rates of Interest whatsoever belonging to each Year , for a Forborn Annuity to 100 Years . This Table is Composed from the Golden Trigonal Table . Years . A Trigonal Increase , or Addition of Rates of Interest . 1 0 2 1 3 3 4 6 5 10 6 15 7 21 8 28 9 36 10 45 11 55 12 66 13 78 14 91 15 105 16 120 Years . A Trigonal Increase , or Addition of Rates of Interest . 17 136 18 153 19 171 20 190 21 210 22 231 23 253 24 276 25 300 26 325 27 351 28 378 29 406 30 435 31 465 32 496 33 528 34 561 35 595 36 630 37 666 38 703 39 741 40 780 41 820 42 861 43 903 44 946 45 990 46 1035 47 1081 48 1128 49 1176 50 1225 Years . A Trigonal Increase , or Addition of Rates of Interest . 51 1275 52 1326 53 1378 54 1431 55 1485 56 1540 57 1596 58 1653 59 1711 60 1770 61 1830 62 1891 63 1953 64 2016 65 2080 66 2145 67 2211 68 2278 69 2346 70 2415 71 2485 72 2556 73 2628 74 2701 75 2775 76 2850 77 2926 78 3003 79 3081 80 3160 81 3240 82 3321 83 3403 84 3486 Years . A Trigonal Increase , or Addition of Rates of Interest . 85 3570 86 3655 87 3741 88 3828 89 3916 90 4005 91 4095 92 4186 93 4278 94 4371 95 4465 96 4560 97 4656 98 4753 99 4851 100 4950 PROP. I. To find the Amount of any Annuity , for any given time , and at any Rate of Simple Interest . General Rule . TO the Sum of the Annual , half-yearly , Quarterly , or Monthly Payments , add the Product of the Annual , Half-Yearly , Quarterly , of Monthly Rate , multiplied by the Number in the foregoing Table , answering to the Number of Years , Half-Years , Quarters , or Months , in the Margin , that the Annuity is to continue ; and the Total Sum is the true Amount of that Annuity . Example 1. What is the true Amount of an Annuity of 100 l. in five Years ? The Number in the foregoing Table answering to 5 in the Margin , is — 10 That multiplied by 6 ( the Annual Interest of 100 l. ) makes — 60 To which add the five Annual Payments , viz. — 500 The whole Amount is — 560 Example 2. What is the Amount of an Annuity of 62 l. in four Years ? The Number in the foregoing Table answering to 4 in the Margin , is — 6 That multiplied by 3.72 ( the Annual Interest of 62 l. makes 22.32 To which adding the 4 Annual Payments , viz. 4 times 62 l. — 248.00 The whole Amount is — 270.32 PROP. II. To know the Present Worth of any Annuity for any given Time , at any Rate , accompting Simple Interest . FOr as much as the Present Worth of an Annuity is in effect , and must be imagined , a Principal , and the whole Amount of the Annuity as the Amount of the said Principal or Present Worth , in so long a time as the Annuity is continued , The Proportion is , As the Amount of 1 l. for any time , Is to 1 l. So is the Amount of an Annuity , To the Present Worth. Therefore the Rule is , Divide the Amount of the Annuity by the Amount of 1 l. in the given Time , and the Quotient is an Answer . Example 1. What is the Present Worth of an Annuity of 62 l. for four Years ? The Amount of 62 l. per Annum for four years by the foregoing Rules is found to be 270.32 , and the Amount of 1 l. forborn four years , by Table IX . is found to be 1.24 ; wherefore I divide 270.32 by 1.24 , thus : The Quotient 218 l. is the Answer . Example 2. What is the Present Worth of an Annuity of 100 l. to continue 100 Years ? The Amount of 100 l. Annuity for 100 years is 39700 l. the Amount of 1 l. put out to Interest for 100 years is 7 l. wherefore divide 39700 by 7 , and the Quotient is the Answer . For Proof of this , let 5671.4 be put out to Interest for 100 years , at 6 per Cent. Wherefore the Operation is exact and just , though at the same time it is a certain Argument , that the said Annuity to continue 100 years at Simple Interest , would be valued at above 56 years Purchase ; for dividing 5671 by 100 ( that is to say , cutting off the two last Figures ) the remaining Figures shew it to be 56 years Purchase , over and above the Fraction of .71 . After the same Method , The Amount of 100 l. Annuity in 50 years is 12350 l. the Amount of 1 l. put out to Interest at 6 per Cent. for 50 years is 4 l. wherefore dividing the said Amount by ( 4 ) , 4 ) 12350 ( 3087.5 The Quotient , or Present Worth is 3087.5 , which is above 30 years Purchase . From whence it is clear and manifest , that all Calculations of Annuities at Simple Interest are absolutely useless and ridiculous : For the truth is , all Present Worths or Purchases , either of Annuities , or Principal Sums , due at any time hereafter , ought to be considered in a Geometrical Proportion , from a Purchase for ever , ( or to the end of the World ) according to the several and respective Rates of Compound Interest . And if this be a truth as to Present Worths , it will be also a truth as to the Amounts , ( as has been sufficiently explained in the Introduction to this Book . ) And consequently , all Calculations , according to Simple Interest , ought wholly to be laid aside as erroneous and useless . CHAP. III. The ERRORS of the ordinary Rules and Tables of Rebate , relating to Annuities , according to the Rate of Simple Interest , discovered and rectified . ALthough all Tables of Rebate for Annuities at Simple Interest , ought to be wholly rejected as most ridiculous and useless , for the Reasons laid down in the foregoing Chapter , yet I do think it here seasonable , and indeed necessary ▪ to animadvert upon the ordinary Rules relating to the present worth of such Annuities , which have been Composed by the respective Authors upon great mistakes , and for want of due reflection upon Arithmetical and Geometrical Progressions . I shall mention only two Examples . The first is a Rule laid down by Mr. John Kersey , in his Appendix , bound up with Mr. Wingate's Arithmetick , Chap. 5. Pag. 378. Printed 1678. which is the very same with that made use of by Dr. Newton , in his Scale of Interest , pag. 20. When it is required to find the present worth of an Annuity , by Rebating or Discompting at a given Rate of Simple Interest , the Operation will be as in the following Example , viz. How much present Money is equivalent to an Annuity of 100 l. per Annum , to continue 5 Years , Rebate being made at the rate of 6 per Cent ? Answer 425 l. 18s . 9d . 2q . very near : Thus , For , saith he , it is manifest that there must be computed the present worth of 100 l. due at the first Years end . Also the present worth of 100 l. due at the second Years end , and in like manner for the third , fourth , and fifth Years . All which present Worths being added together , the Aggregate or Sum will be the total present worth of the Annuity , that is , 425 l. 18s . 9d . 2q . very near . I must confess I cannot but wonder how such gross mistakes should pass through the hands of so many Learned and Ingenious Artists . For this very Example I find Published by the same Mr. Kersey , in the year 1650. and since that time , owned and made use of by several others . But for the right understanding of the truth of this , and all other Questions of this kind . It is necessary to request the Reader to contemplate with me a few things . 1. What is due of an Annuity that is not paid , at the end of the first , second , third , fourth , and fifth years , at Simple Interest . 1. At the first years end .   l. At the first years end there is due the just sum of 100 II. At the second years end . At the second years end , there is due ,   l. 1. For the first year 100 2. For the second year 100 Sum 200 And besides this , For the Interest of the 100 l. due at the first years end , and detained during the whole second year — 6 l. III. At the third years end . At the third years end , there is due ,   l. 1. For the first year 100 2. For the second year 100 3. For the third year 100 Sum 300 Besides this , there is due ,   l. 1. For the Interest of the first 100 l. for the second year 6 2. For the Interest of the first 100 l. for the third year 6 3. For the Interest of the second 100 l. for the third year 6 Sum 18 IV. At the fourth years end .   l. 1. For the first year 100 2. For the second year 100 3. For the third year 100 4. For the fourth year 100 Sum 400 Besides this ,   l. 1. For the Interest of the first 100 l. for the second year 6 2. The Interest of the first 100 l. for the third year 6 3. The Interest of the first 100 l. for the fourth year 6 4. The Interest of the second 100 l. for the third year 6 5. The Interest of the second 100 l. for the fourth year 6 6. The Interest of the third 100 l. for the fourth year 6 Sum 36 V. At the fifth years end .   l. 1. For the first year 100 2. For the second year 100 3. For the third year 100 4. For the fourth year 100 5. For the fifth year 100 Sum 500 Besides this ,   l. 1. For the Interest of the first 100 l. for the second year 6 2. For the Interest of the first 100 l. for the third year 6 3. For the Interest of the first 100 l. for the fourth year 6 4. For the Interest of the first 100 l. for the fifth year 6   24 5. For the Interest of the second 100 l. for the third year 6 6. For the Interest of the second 100 l. for the fourth year 6 7. For the Interest of the second 100 l. for the fifth year 6 8. For the Interest of the third 100 l. for the fourth year 6 9. For the Interest of the third 100 l. for the fifth year 6 10. For the Interest of the fourth 100 l. for the fifth year 6   36   24 Sum Total of the Interest for the five years 60 To which adding the five Annual payments , viz. 500 The whole Amount of the Annuity of 100 l. forborn five years , is 560 In the next place , I desire the Ingenious Reader to consider well the third Prop. of Diophantus Alexandrinus , concerning Peligonal Numbers . ΔΙΟΦΑΝΤΟΥ ΑΛΕΞΑΝΔΡΕΩΣ ΠΕΡΙ ΠΟΛΙΤΟΝΩΝ ΑΡΙΘΜΩΝ . PROP. III. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 &c. It Numbers ( how many soever they be ) exceed one another by an equal Internal , then the Internal between the greatest and the least , is Multipler of that equal Internal , according to the multitude of Numbers propounded , less by one . For Example . Let there be five given Terms , A , B , C , D , E , and let G be the common Interval or Difference . To apply which , let A = 100 G = 6 Number of Terms . Then is ,     1 A = A 1 A = 100 2 B = A+G 2 B = 100+6 3 C = A+G+G 3 100+6+6 4 D = A+G+G+G 4 D = 100+6+6+6 5 E = A+G+G+G+G 5 E = 100+6+6+6+6 That is to say , the greatest Term is equal to the least , and as many Differences as there are more Terms besides the least . So here E is equal to 100 , and 4 Differences , or 4 times 6. And the Sums of those Numbers are the true Amount of an Annuity at Simple Interest ; thus , The Annual Rents , together with the Annual Interests The Sums of Annual Rents , & Annual Interests , for therespective Years . The Number of Annual Interests , or Differences , that are contained in every respective Sum , besides the Annual Rents . 1 100 1 100   2 100+6 2 206 = 1 3 100+6+6 3 318 = 3 4 100+6+6+6 4 436 = 6 5 100+6+6+6+6 5 560 = 10 Therefore the true Amount of an Annuity of 100 l. is as follows . Year .   Amounts 1 100 100 2 100+100+6 206 3 100+100+100+6+6+6 318 4 100+100+100+100+6+6+6+6+6+6 436 5 100+100+100+100+100+6+6+6+6+6+6+6+6+6+6 560 Consequently the Proportion is not as Mr. Kersey makes it , save only for the first year . But the true proportion holds thus , viz.       Amounts . Present worths . At the 1 years end As 106 to 100 ∷ So 100 to 94.33962 2 As 112 to 100 ∷ So 206 to 183.92856 3 As 118 to 100 ∷ So 318 to 269.49152 4 As 124 to 100 ∷ So 436 to 351.61290 5 As 130 to 100 ∷ So 560 to 430.76923 Now therefore to perfect the Demonstration , 1. The present worth of the first year is 94.33962 2. Because 183.92856 is the present worth of the two first years , therefore if the present worth of the first year ( Viz. 94.33962 ) be deducted out of it , it must needs leave the present worth of the second year , viz. 89.58894 3. Because 269.49152 is the present worth of the three first years , therefore deducting out of it 183.92856 , ( viz. the present worth of the two first years ) it leaves the present worth of the third year , viz. 85.56296 4. So deducting 269.49152 out of 351.61290 , there remains the present worth of the fourth year 82.12138 5. And 430.76923 less by 351.61290 , is the present worth of the last year , viz. 79.15633 Total Sum of all the present worths 430.76923 To conclude , it is evident from the two last Calculations , and that by clear Demonstration , That ,   l. 1. The Amount of the first year is 100 2. The Amount of the two first years is 206 3. The Amount of the three first years is 318 4. The Amount of the four first years is 436 5. The Amount of all five years is 560 As likewise , That the present Worth , 1. Of the first year is 94.33962 2. Of the two first years is 183.92856 3. Of the three first years is 269.49152 4. Of the four first years is 351.61290 5. Of all five years is 430.76923 And lastly , it is evident and plain , That the present Worth , 1. Of the first year is 94.33962 2. Of the second year is 89.58894 3. Of the third year is 85.56296 4. Of the fourth year is 82.12138 5. Of the fifth year is 79.15633 Total Sum of the present worths 430.76923 Whereas Mr. Kersey makes the Total of the present worths but 425.93933 , which is a very great mistake ; as are all his particular present worths , ( the first only excepted ) which he could not well Calculate amiss . Besides , if 425.93933 be put out for five years , it will amount to no more than 553.714109 . Whereas 430.76923 in five years , at 6 per Cent. amounts to 560 l. which is the true Amount of 100 l. per Annum for five years , as has been sufficiently Demonstrated , and agrees exactly with the foregoing Rule : So that Mr. Kersey in this Example falls short of the truth , as to the present worth , no less than 4.8353 , that is , 4 l. 16 s. 8 d. 1 q. more by 96 / 100 of a Farthing . Which Error , if it be so considerable in an Annuity of 100 l. per Annum , what would it be in an Annuity of 100000 paid per Annum ? No less than 4835 l. 6 s. 2 Example . A second Example I have borrowed from Mr. Dary , who has truly detected the Error of it , although he has not sufficiently explained the Reason of the Error ; and therefore the Reader will find it here more strictly examined and refuted by a plain Demonstration obvious to the meanest capacity . The Example is this : What is the present worth of an Annual Rent of 62 l. to be enjoyed four Years to come , allowing the Purchaser 6 per Cent. Simple Interest ? The usual Method , says Mr. Dary , is thus : Now let the Error of this Operation be traced from the beginning . 1. The Annual Interest of 62 l. per Annum , is 3.72 ; wherefore by the foregoing Prop. of Diophantus Alexandrinus , pag. 73. The Amount of Therefore the true Amount of an Annuity of 62 l. at each years end , is as follows . At the end of the Wherefore the true Calculation of the present Worths is as follows , viz. At the end of the Now therefore , 1. The present worth of the first year is — 58.490 2. The present worth of the two first years less by the present worth of the first , that is , from 114.035 deducting 58.490 , the present worth of the second year is — 55.545 3. The present worth of the three first years ( the present worth of the two first being deducted ) that is , from 167.084 deducting 114.035 , the remainder of the present worth of the third year is — 53.049 4. Deducting from 218 the present worth of all four years , 167.084 the present worth of the first three years , the remainder ( viz. 50.916 ) is the present worth of the fourth year — 50.916 So then , The present worth , Whereas the usual way of Rebate makes it not above 216.390 , which is less than the truth by 1.610 , which is ● l. 12 s. 2 d. 2 q. ferè . And if 216.390 be put out at Interest at 6 per Cent. for four years , it will amount to no more than 268.3236 , which is less than the true Amount of 218 l. viz. 270.32 by 1.9964 , which being reduced , is 1 l. 19 s. 11 d. more by 11 / 100 parts of a Farthing . All which may serve as a sufficient caution against such erroneous Tables and Calculations . A second Reflection upon that Example of Mr. John Kersey . I Must confess that the present Worth of 100 l. payable a year hence is 94.33962 ; and that the present worth of a single 100 l. payable two years hence , is as he has put it 89.28571 ; and the present worth of another bare 100 l. payable three years hence is 84.74576 ; and so to the end . And the Total of those present worths is as he has put it , viz. And this is part of that very Table which I have calculated ( being the twelfth Table of the first Chapter of this first Book ) for The present worth of One Pound after any Number of Years under 32. But reason tells me , that in this Calculation there is no consideration had of the Forbearance of Interest ; for certain it is , if the first 100 l. had been paid at the first years end , it might have been put out to Interest , and at the five years end would have given an increase of four times 6 l. or 24 l. at Simple Interest ; and so the second 100 l. would have increased in the three last years three times 6 l. or 18 l. That is to say ,   l. The first 100 l. would increase in the four last years 24 The second 100 l. would increase in the three last years 18 The third 100 l. would increase in the two last years 12 The fourth 100 l. would increase in the last year 6 The whole increase 60 Therefore there would be due , if all were forborn ,   l. 1. At the first years end 100 2. At the second years end 100+06 3. At the third years end 100+12 4. At the fourth years end 100+18 5. At the fifth years end 100+24   500+60 Now to Calculate the present worth of any , or all of these Sums , let it be considered by what proportion the Calculation ought to be made . For Example . Suppose the Annuity to be forborn only two years , and it be required to give the present worth of the two first years . Whatsoever the Answer is , all will agree , that the Sum which is given in to be the present worth of those two years , being put out to interest , must amount to ●●● at the end of two years . Therefore I say , As 112 to 100 ∷ So 206 to 183.92856 . If this be a true Answer , then that Sum , viz. 183.92856 , being put out to Interest at 6 per Cent. for two years , must amount to 206. By the former Rules . Now this Total Sum wants but ● / 100000 of 206. For , But now take the Sum of Mr. Kersey's two years present Worths , viz. Let therefore this Sum be put out to Interest for two years . Wherefore as before , Which is less than 206 ( the true Amount of an Annuity of 100 l. for two years ) by .3396304 ; which though it be but 6 s. 8 d. and somewhat more in two years time , yet were the Sum greater , or the time longer , it would prove a very considerable Error . Wherefore I conclude , that Mr. Kersey's Calculations are erroneous as to Annuities , and mine exact : And there needs no further Illustrations or Demonstrations about it . The next thing to be Treated of in course , is touching the Equation of several Payments , and reducing them into one entire Payment at a certain time , so as there may be no loss either to Creditor or Debtor . CHAP. IV. Equation of Payments Rectified , and made Practicable for all Merchants , and others . EQuation of Payments is by all agreed to be the reducing of several Payments into one entire Payment , at such a time , as neither Creditor or Debtor may be a loser by it , they being both agreed , the one to pay , and the other to receive , the said entire Payment at the appointed time . Now of the Books that I have met with , and the Men I have discoursed with , about Equation of Payments at Simple Interest , some have adventured to give Rules for it , others have endeavoured to shew that such Rules are erroneous , and some of the most Learned of them have concluded the thing to be absolutely impracticable and impossible ; and so left the poor Merchants to agree as they please about it . The truth is , they have been , and are all of them , mistaken about the present Worths of Annuities at Simple Interest , and that mistake has begot many others . The method that I shall therefore take , shall be , First , to expose to the Readers view both the true Amount and present Worth of an Annuity of 100 l. for five years , at several Rates of Interest . And from thence frame , and give a general Rule for the reducing of several equal Payments due at equi-distant times , to one entire Payment . And after that , another Rule for reducing of unequal Payments at several times not equidistant , to one entire Payment at a certain time , so as neither he who pays , nor he who receives it , shall be any loser by it . The Tables of the Amounts and present Worths of an Annuity of 100 l. for five years ( at different Rates of Interest ) do here follow in their order . TABLE I. At ( 1 ) per Cent. Simple Interest . Years . The Amount of 100 l. 〈◊〉 1 , 2 , 3 , 4 , or 5 Years . Amount of Annual Payments at the end of 1 , 2 , 3 , 4 or 5 Years . The present Worth of the first year , the first two years , the first three years , the first four years , or all the five years . The particular present Worths of the first , second , third , fourth , or fifth year The Amount of the Total present Worth of all the five Payments , at the end of the first , second , third , fourth , or fifth year . 1 101 100 99.00990 99.00990 490.57142 2 102 201 197.05882 98.04892 495.42856 3 103 303 294.17475 97.11593 500.28570 4 104 406 390.38461 96.20986 505.14285 5 105 510 485.71428 95.32967 509.99999         485.71428   TABLE II. At ( 2 ) per Cent. Simple Interest . Years . The Amount of 100 l. in 1 , 2 , 3 , 4 , or 5 Years . Amount of Annual Payments , at the end of 1 , 2 , 3 , 4 or 5 Years . The present Works of the first two years , the first three years , the first four years , or all the five years . The particular present Works of the first , second , third fourth , or fifth year The Amount of the Total present Worth of all the five Payments , at the end of the first , second , third , fourth , or fifth year . 1 102 100 98.03921 98.03921 482.18181 2 104 202 194.23076 96.19155 491.63635 3 106 306 288.67924 94.44848 501.09089 4 108 412 381.48148 92.80224 510.54543 5 110 520 472.72727 91.24579 519.99999         472.72727 TABLE III. At ( 3 ) per Cent. Simple Interest . Years . The Amount of 100 l. in 1 , 2 , 3 , 4 , or 5 Years . Amount of Annual Payments , at the end of 1 , 2 , 3 , 4 or 5 Years . The present Worth of the first year , the first two years , the first three years , the first four years , or all the five years . The particular present Worths of the first , second , third , fourth , or fifth year . The Amount of the Total present Worth of all the five Payments , at the end of the first , second , third , fourth , or fifth Year . 1 103 100 . 97.08737 97.08737 474.69564 2 106 203 191.50943 94.42206 488.52173 3 109 309 283.48623 91.97680 502.34782 4 112 418 373.21428 89.72805 516.17390 5 115 530 460.86956 87.65528 529.99999         460.86956   TABLE IV. At ( 4 ) per Cent. Simple Interest . Years . The Amount of 100 l. in 1 , 2 , 3 , 4 , or 5 Years . Amount of Annual Payments , at the end of 1 , 2 , 3 , 4 or 5 Years . The present Worth of the first year , the first two years , the first three years , the first four years , or all the five years . The particular present Worths of the first , second , third , fourth , or fifth year . The Amount of the Total present Worth of all the five Payments , at the end of the first , second , third , fourth , or fifth year . 1 104 100 96.15384 96.15384 468.00000 2 108 204 188.88888 92.73504 486.00000 3 112 312 278.57142 89.68254 504.00000 4 116 424 365.51724 86.94582 522.00000 5 120 540 450.00000 84.48276 540.00000         450.00000   TABLE V. At ( 5 ) per Cent. Simple Interest . Years . The Amount of 100 l. in 1 , 2 , 3 , 4 , or 5 Years . Amount of Annual Payments , at the end of 1 , 2 , 3 , 4 or 5 Years . The present Worth of the first year , the first two years , the first three years , the first four years , or all the five years . The particular present Worths of the first , second , third , fourth , or fifth year . The Amount of the Total present Worth of all the five Payments , at the end of the first , second , third , fourth , or fifth year . 1 105 100 95.23809 95.23809 461.99999 2 110 205 186.36363 91.12554 483.99999 3 115 315 273.91304 87.54941 505.99999 4 120 430 358.33333 84.42029 527.99999 5 125 550 439.99999 81.66666 549.99999         439.99999   TABLE VI. At ( 6 ) per Cent. Simple Interest . Years . The Amount of 100 l. in 1 , 2 , 3 , 4 , or 5 Years . Amount of Annual Payments , at the end of 1 , 2 , 3 , 4 , or 5 Years . The present worth of the first year , the first two years , the first three years , the first four years , or all the five years . The particular present Worths of the first , second , third , fourth , or fifth year . The Amount of the Total present Worth of all the five Payments , at the end of the first , second , third , fourth , or fifth year . 1 106 100 94.33962 94.33962 456.61538 2 112 206 183.92856 89.58894 482.46153 3 118 318 269.49152 85.56296 508.30769 4 124 436 351.61290 82.12138 534.15384 5 130 560 430.76923 79.15633 559.99999         430.76923   TABLE VII . At ( 10 ) per Cent. Simple Interest . Years . The Amount of 100 l. in 1 , 2 , 3 , 4 , or 5 Years . Amount of Annual Payments , at the end of 1 , 2 , 3 , 4 , or 5 Years . The present worth of the first year , the first two years , the first three years , the first four years , or all the five years . The particular present Worths of the first , second , third , fourth , or fifth year . The Amount of the Total present Worth of all the five Payments , at the end of the first , second , third , fourth , or fifth year . 1 110 100 90.90909 90.90909 440.00000 2 120 210 175.00000 84.09090 480.00000 3 130 330 253.84615 78.84615 520.00000 4 140 460 328.57142 74.72527 560.00000 5 150 600 400.00000 71.42858 600.00000         399.99999   Observations upon the foregoing Tables . 1. IT is observable , That as the Rate of Interest increases , the present Worth decreases ; that is to say , For the present Worth of an Annuity of 100 l. for five years , At 1 per Cent. is 485.71428 2 472.72727 3 460.86956 4 450.00000 5 439.99999 &c. 6 430.76923 10 399.99999 &c. 2. It is no less observable , That an Annuity of 100 l. increases by a Trigonal Progression of the respective Rates . But the Present Worth increases by an Unitarian Addition of the Rate to the Principal for each year respectively . And these two ways are very different the one from the other , as may be seen by comparing them together , as follows in the Example of an Annunity of 100 l. at 6 perCent . Simple Interest .       Amount of the Annuity for each of the five years . Amount of the present Worth of the Amuity for each of the five years . At the 1 years end 100 456.61538 2 206 482.46153 3 318 508.30769 4 436 534.15384 5 560 559.99999 &c. And yet how different soever they are at their first setting out , and by the way , yet the further they go , the nearer they come together , and at last agree to an insensible difference , and such as may be diminished in insinitum , either to the Hundredth , or Theirsandth , or any less part of a Farthing whatsoever can be desired . 3. As a consequence of the foregoing Observation : If A. be to pay B. 100 l. per Annum for five years , and they agree that the 500 l. shall be paid at one entire Payment , they must be sure to pitch upon such a time , as that the said 500 l. being put out to Interest from that time to the end of five years , may be equal to the whole Amount of those five Annual Payments . For Example . Let the Rate of Interest be 6 per Cent. per Annum , and the Time of paying the said 500 l. be at the end of three years , and so there are two years to come . If the said 500 l. for two years , at 6 per Cent. will amount to 560 l. ( which is the whole Amount of the Annuity at the five years end ) the Time is right , if not , it is a false Time. But the Annual Interest of 500 l. is 30 l. therefore in two years it is 60 l. and that added to 500 , makes 560 l. And therefore it was a just time to pay the said 500 l. at one entire Payment . For so B. has at the five years end , the whole effect of his Annuity improved to the utmost , at 6 per Cent. Simple Interest . And B. having paid nothing before of the Annuity , and being obliged to pay nothing of it afterwards ; but having enjoyed it for three years ( which is the best part of it ) already , and being to enjoy it two years more ; the 500 l. he now pays , is only as a Purchase of the Amount of the whole Annuity , which will be due at the five years end , viz. 560 l. and so gives the Present Worth of 560 l. from the three years end to the 5 years end ; and so he pays not a Farthing more than the true worth of it . And for that true worth of it , to the end of the 5 years he has enjoyed , and must enjoy the said Annuity it self to the end of the said five years . And so neither A. nor B. have the least wrong or loss , neither the one by paying , nor the other by receiving , this one entire Payment of 500 l. at the three years end ; and if either or both should sell their concerns , it would be the same thing . 4. It is observable that the present Worth of the said Annuity , at any Rate of Interest , does at the three years end exceed the Aggregate of the said five Sums , ( or 500 l. ) For Example . The present Worth of an Annuity of 100 l. per Annum to continue five years , does at the end of three years , Amount , At 1 per Cent. to 500.28570 2 501.09089 3 502.34782 4 504.00000 5 505.99999 6 508.30769 10 520.00000 5. It being as evident from this last Observation , That the present Worth of the whole Annuity being put out to Interest for three years , will at any Rate of Interest , exceed the Aggregate of all the five Payments , and the greater the Rate of Interest is , the greater is that Excess . For Example . At ( 1 ) per Cent. the Excess is but . 28570 ; at ( 2 ) per Cent. the Excess is somewhat more , viz. 1.09089 ; at ( 3 ) per Cent. it is 2.34782 ; at ( 10 ) per Cent. it is 20 l. and at ( 15 ) per Cent. it would be much more . And it being likewise evident by all the foregoing Tables , whatever the Rate of Interest be , That 500 l. more by the Interest of 500 l. for two years , is equal to the whole Amount of the Annuity of 500 l. for five years . That is to say , The Interest of 500 l. for two years , At ( 1 ) per Cent. is 10 l. which added to 500 l. makes the Amount 510 l. At ( 2 ) per Cent. is 20 l. which added to 500 l. makes the Amount 520 l. At ( 3 ) per Cent. is 30 l. which added to 500 l. makes the Amount 530 l. At ( 4 ) per Cent. is 40 l. which added to 500 l. makes the Amount 540 l. At ( 5 ) per Cent. is 50 l. which added to 500 l. makes the Amount 550 l. At ( 6 ) per Cent. is 60 l. which added to 500 l. makes the Amount 560 l. And lastly , it being sufficiently evident by the third Observation , That if the 500 l. be paid at one entire Payment , at the end of three years , or , which is all one , two years before the Annuity be at an end ; neither Creditor nor Debtor can have the least wrong , or suffer the least loss . It may therefore be safely concluded , That it is practicable and possible to give a good and true Rule for Equation of several Payments ; and likewise , that it is no way necessary ( as some very Learned Artists would needs have perswaded me ) to try that Rule by this Mark , viz. That the present Worth of the said 500 l. at the three Years end , must be the present Worth of the whole Amount of the said Annuity . For by what has been already proved , the present Worth of the whole Annuity at any Rate of Interest , will at the three years end exceed the said 500 l. Now therefore I shall proceed to give two General Rules . The first , for Equation of several equal Payments at equi-distant times . The second , for Equation of several unequal Payments at several times not equi-distant . 1. General Rule . For Equation of any given Number of equal Payments due at equi-distant Times . Rule . Out of the whole Amount of the Annuity , of Monthly Payment , deduct the Aggregate of the several Payments , and the Remainder , if Annual , multiply by 365 ; if Monthly , by 30.416 ; then divide the Product by the Annual , or Monthly Interest of the said Aggregate , and the Quotient is the number of Days , before the end or Term of the Annuity , or Monthly Payment , when the said Aggregate may be paid without loss to either Creditor or Debtor . 1. Example . Let the Annuity be 100 l. The time it is to continue five years . The whole Amount of the Annuity 560 The Aggregate of the several Payments 500 The Annual Interest of the Aggregate ( viz. 500 ) 30 Now suppose A. be obliged to pay to B. 100 l. per Annum for five years , but they both agree that A. shall pay to B. the Aggregate 500 l. at one entire Payment . And then the Question is , at what time the said 500 l. is to be paid ? Therefore as the Rule directs , That divided by 30 , gives a Quotient , which is the true number of Days before the end of the said Annuity , when the said 500 l. is to be paid , viz. 730 Those 730 Days divided by 365 , gives a Quotient of two years . So then the true time of paying the said Aggregate of several payments , ( viz. 500 l. ) is two years before the end of five years ; that is at the end of the third year . 2. Example . A. is to pay B. 62 l. per Annum for four years ; but they agree that A. shall pay the Aggregate of the several Sums , ( viz. 248 ) at one entire Payment . If the Annual Payment be 62.00 The Amount of that in four years , at 6 per Cent. will be 270.32 The Aggregate of the several Payments , or four times 62 , is 248.00 The Annual Interest of the said Aggregate 14.88 Wherefore , This 22.32 is first to be multiplied by 365 , which is 8146.80 . That Product 8146.80 being divided by the Annual Interest of the Aggregate 248 , viz. 14.88 , gives a Quotient of 547.5 Days . This Quotient 547 Days and ● / 10 , or a half , is a true Answer to the Question ; that is to say , 547 Days and a half , or one Year , and 172 Days and a half , before the end of four years , is the just time to pay the said Aggregate , or 248 , at one entire Payment ; so as neither he who pays it , nor he who receives it , may be a loser . But that all things may be exposed clearly to the Readers view , I shall here repeat the thing , and set down the whole Operation . 1. The Annual Payment for four years is 62.00 l. 2. The whole Amount of this in four years is 270.32 l. 3. The Aggregate , or four years Payments , that is , four times 62 l. is 248.00 l. 4. The Annual Interest of the said Aggregate is 14.88 l. Now the Question is , what is the true time for paying the said Aggregate , or 248 , at one entire Payment ? To Answer this , I proceed according to the aforesaid Rule . 2. I multiply this Remainder by 365 , thus , 3. This Product I divide by the Annual Interest of the Aggregate , viz. 14.88 . Tariffa for the Divisor . 1 1488 2 2976 3 4464 4 5952 5 7440 6 8928 7 10416 8 11904 9 13392 10 14880 And the Quotient 547.5 , is an Answer to the Question . That is to say , ( as before ) one year , and 172 days , and a half , before the end of four years . Or , which is the same thing , two years , and 192 days and a half , after the Agreement , must the 248 l. be paid at one entire Payment ; and for the Reasons aforesaid , there is no loss to either A. or B. For Proof of this , Let 248 l. be put out to Interest at 6 per Cent. for one year , and 172 days and a half , that is , 547.5 days ( which is the Quotient , or the time given from the Payment thereof to the end of the Annuity ) and if it make up the whole Amount of the Annuity , viz. 270.32 , the Operation is right . Tariffa for the Multiplicand . 1 00016438356 2 00032876712 3 00049315068 4 00065753424 5 00082191780 6 00098630136 7 00115068492 8 00131506848 9 00147945204 10 00164383560 For Example . Then I make the following Tariffa , and proceed to multiply the foregoing Product by 248. Tariffa for the Multiplicand . 1 089999999100 2 179999998200 3 269999997300 4 359999996400 5 449999995500 6 539999994600 7 629999993700 8 719999992800 9 809999991900 Wanting but 1 / 10000 ( which is not the Hundredth part of a Farthing ) of the true Amount of the whole Annuity , viz. 270.32 , and therefore the Operation is just . And thus may any Question of this nature be resolved , to a Day , and parts of a Day ; for if both these last Questions had been made for Months , the same Rule must have been observed . For Example . If A. is to pay to B. 100 l. per Month for five Months , when may he pay the 500 l. at one entire Payment , at the Rate of 6 per Cent ? The Payments being Monthly , Therefore , In pursuance of the aforesaid General Rule . Let therefore 5 be multiplied by 30.416 , or the true number of Days that are in one equal Month. And let that ( 152.080 ) be divided by 2.500 , or the Monthly Interest of 500 l. at 6 per Cent. Tariffa for the Divisor . 1 25 2 50 3 75 4 100 5 125 6 150 7 175 8 200 9 225 The Quotient ( 60.832 ) is a true Answer in Days . That is to say , 60 Days and 832 / 1000 of a Day , ( which makes two equal Months ) before the end of five Months ; or ( which is all one ) three Months after the agreement , or after the first day , when the said Debt was growing due , is the just time of paying the 500 l. at one entire Payment . For Proof of this . If 500 l. be put out to Interest at 6 per Cent. for two equal Months , or 60.832 Days , and does give 5 l. it makes the 500 l. become 505 l. which is the full Amount of those five Months Payments , and is a just Answer to the Question . But , And the Operation is exact . And this I take to be sufficient for the Resolution of any Question of this nature . I shall proceed in the next place to Discourse about unequal Payments , at times not equi-distant . A brief Discourse concerning the Equation of unequal Payments at Times not equi-distant . For Example . A Merchant owes 500 l. to be paid at three several unequal Payments , viz. at the end of four Months 300 l. at the end of six Months 100 l. and at the end of twelve Months 100 l. but the Debtor agrees with the Creditor to discharge the Debt ( viz. 500 l. ) at one entire Payment . The Question is , at what time this 500 l. may be paid , without damage or prejudice to either Creditor or Debtor ? The General Rule is this . First find the true Amount of each of the Sums , from the first day of the Agreement , to the last day of Payment , as supposing them to be forborn to the last . Then out of that deduct the Aggregate of the respective Payments , and multiply the Remainder , if Annual , by 365 ; if Monthly , by 30.4166 ; and the Product divide by the Annual or Monthly Interest of the said Aggregate , and the Quotient is the Number of Days from the last Day of Payment , accompting backwards . The Operation is as follows . First , the length of Time from the day of the Agreement , to the last day of Payment , is just twelve Months . So then , 1. In the first place , 300 l. payable after 4 Months , and being forborn to the end of 12 Months , has 8 Months Interest to accompt for , viz. 12.000 2. In the second place , 100 l. payable after 6 Months , and being forborn to the end of 12 Months , has 6 Months Interest to accompt for , viz. 3.000 3. To these Sums adding the 500.000 The whole Amount is 515.000 Then , And the Proportion is this . If 2.5 be the Interest of 500 l. for one Month , how many Months Interest will 15 make ? Wherefore divide 15 by 2.5 , and the Quotient is the Answer to the Question . Tariffa for the Divisor . 1 25 2 50 3 75 4 100 5 125 6 150 7 175 8 200 9 225 The Operation is this . That is to say , if the said 500 l. be paid six Months before the end of twelve Months or ( which is all one ) at the end of six Months , there will be no loss or damage either to Creditor or Debtor . For Proof of this , 1. In the first place , 300 l. was due at 4 Months end , and being continued 2 Months longer , the Interest thereof for 2 Months is 3 l. the whole Amount is 303 l. 2. In the next place , 100 l. paid at 6 Months end , is the time it was due 100 l. 3. In the last place , the other 100 l. paid 6 Months before the time , there must be an abatement made of 3 l. 97 l. Total Sum 500 l. So that in the first Sum there is an increase of 3 l. and in the last there is a decrease of 3 l. which are to be set one against the other ; and the whole Amount is the Aggregate of the respective Sums , and being paid at the end of 6 Months makes the Equation just 500 l. After this manner may any Number of unequal Sums payable at any Number of Times not equi-distant , be Equated , and a time set for the Payment of the Aggregate ; and not only so , but if the Debtor A. owe to B. 100 l. per Annum , or per Month , for any Number of Years or Months , and A. and B. agreeing together that it shall be in the power of A. to pay to B. the true value of his Pretensions at the end of any of the Years or Months , it is very Practicable , for the present Worth of the whole Amount at the end of any of the Years or Months resolves the doubt , and is an Answer the Question . For Example . A. owes to B. 100 l. per Annum for five years , and they agree that A. shall buy it off at the end of any of the four years , for at the end of 5 years nothing less than the whole 560 l. will pay the Dein . 1. The present worth of 560 l. ( or the whole Amount ) at the first years end , is 451.6129 2. The present worth of 560 l. at the second years end , is 474.5762 3. The present worth of 560 l. at the third years end , is 500 , 0000 4. The present worth of 560 l. at the fourth years end , is 528.3018 5. The present worth of 560 l. at the fifth years end , is 560 , 0000 Thus I have as briefly as the nature of the thing would permit , explained the Doctrine of Simple Interest , as likewise that of Annuities , and Equation of several Payments at Simple Interest , which is of excellent use for 6 12 , or 18 Months , because the difference between Simple and Compound Interest is not material in so short a time . But for as much as the business of An●uities , or Purchases , for any considerable Number of years , does most properly and truly belong to the Doctrine of Compound Interest , I shall make that the Subject of the following Book . THE DOCTRINE OF COMPOUND INTEREST . LIB . II. CHAP. I. The Doctrine of Compound Interest explained . Compound Interest , or Interest upon Interest , increases not only from the Principal , but also from the Interest , in the manner hereafter exprest . That is to say , If 100 l. be put out to Interest at 6 per Cent. 1. The first year there will be due 106.0000 2. The second year that ( 106 l. ) is made a Principal , and being put out for a year , becomes 112.3600 3. The third year that ( 112.360 ) is made a Principal , and being put out to Interest , amounts to 119.1016 And so in infinitum . So that the respective Amounts for each respective year , are so many Geometrical Proportional Numbers . For , As 100 to 106 , for the first year ∷ So 106 to 112.36 , for the second year . Again , As 106 to 112.36 ∷ So 112.36 to 119.1016 , for the third year . Item , As 112.36 to 119.1016 ∷ So 119.1016 to 126.247016 , for the fourth year . &c. But to the end , that the Ingenious Practitioner may have entire satisfaction in the business of Interest upon Interest , it will be necessary to make some Reflection upon Geometrical Proportion and Progression . Reflection upon Geometrical Progression . If Numbers ( how many soever they be ) contain the one the other by an equal Ratio , then the greatest of those Numbers is Multipler of the Powers of the Denomination of that equal Ratio multiplied by the least , according to the multitude of the given Numbers less by one . Let the given Numbers be 2 , 6 , 18 , 54. Then by the Hypothesis , the first multiplied by ( 3 ) is equal to the second ; and the second multiplied by ( 3 ) is equal to the third ; and so in infinitum . That is to say , First Term 2 = 2 — — — — 2 Second 6 = 2 into 3 — — — 6 Third 18 = 2 into 3 into 3 — — 18 Fourth 54 = 2 into 3 into 3 into 3 — 54 That is to say in a Symbolical way , Let there be any Number of Proportionals , A , B , C , D , E , F , G , and the Ratio R. First Term A = A Second B = A into R Third C = A into R into R Fourth D = A into R into R into R Fifth E = A into R into R into R into R Sixth F = A into R into R into R into R into R Seventh G = A into R into R into R into R into R into R To apply this to the present purpose , let the first Geometrical Term be ( 1. ) the Ratio ( . 06 )   First Power . Second Power . Third Power . Fourth Power . Fifth Power . Sixth Power . Geometrical Proportional Numbers . A = 100             100.000 B = 100 into 1.06           106.000 C = 100 into 1.06 into 1.06         112.360 D = 100 into 1.06 into 1.06 into 1.06       119.101 E = 100 into 1.06 into 1.06 into 1.06 into 1.06     126.247 F = 100 into 1.06 into 1.06 into 1.06 into 1.06 into 1.06   133.823 G = 100 into 1.06 into 1.06 into 1.06 into 1.06 into 1.06 into 1.06 141.851 The Geometrical Numbers at length are these that follow , though there is no necessity of making use of them all , the difference being indiscernable . Years . Amount at 6 per Cent. Compound Interest . 1 1.06 2 1.1236 3 1.191016 4 1.26247696 5 1.3382255776 6 1.418519112256 7 1.50363025899136 8 1.5938480745308416 9 1.689478959002692096 10 1.79084769654285362176 NOw for as much as these Geometrical Proportional Numbers swell into a great Number of places , and the Multiplications become tedious , it has been look'd upon as impracticable to find them out by any way , but by the help of the Logarithms . But I shall endeavour to shew a way how it may be very practicable to find out any of these Numbers , for any year under 32 , without much trouble or difficulty . For Example . Let it be demanded to give the Amount of I l. in eight years , at 6 per Cent. Compound Interest , not having any help of a Table . The Operation is thus . First , I Square 1.06 , which is 1.1236 , and the Product is the Amount in two years . Secondly , I Square 1.1236 , and that gives me the Proportional Number answering to ( 4 ) in the Margin , viz. 1.26247 , &c. Thirdly , I Square 1.26247 , and that gives me the Proportional Number answering to ( 8 ) in the Margin , which was the thing proposed , viz. 1.59384 , &c. Now if it had been demanded to find the Proportional Number answering to ( 16 ) in the Margin , it is the Square of 1.59384 , &c. And the Square of 1.59384 gives the Proportional Number answering to ( 32 ) in the Margin . Thus far the Method is clear for all even Numbers ; but for the odd Numbers , The Rule is this : Having found the Proportional Number answering to the greater half of the given Number in the Margin , Square it , and divide it by the least and first Proportional Number , and the Quotient is the Number desired . For Example . Let it be demanded to find the Proportional Number answering to ( 3 ) in the Margin , and let ( 1.06 ) be the least Proportional Number . Having found 1.1236 to be the Proportional Number answering to ( 2 ) in the Margin , which ( 2 ) is the greater half of ( 3 ) , I Square 1.1236 , and it gives 1.26247 , which I divide by 1.06 , and the Quotient ( 1.1910 , &c. ) is the Proportional Number desired . Again , Let it be demanded to find the Proportional Number answering to ( 5 ) in the Margin . Having found the Proportional Number answering to ( 3 ) in the Margin , ( which 3 is the greater half of 5 ) viz. 1.1910 ; the Square thereof , viz. 1.4185 , being divided by 1.06 , the Quotient is the Number desired , viz. 1.3382 . Thus ( 9 ) is the greater half of ( 17 ) , and therefore the Proportional Number answering to ( 9 ) in the Margin being Squared , and that Square divided by 1.06 , gives 3.700 , for the Proportional Number answering to ( 17 ) in the Margin . And so may the Proportional Number of any odd Number in the Margin be found out , without the help of Logarithms . But for as much as exact Tables truly Calculated are most ready for use , I have with no small Pains and Charge ( not Transcribed other Mens Tables and Errors , but ) carefully and exactly Calculated several Tables of my own ; by the help of which , may easily and readily be found out either the Amount , or Present Worth of any Sum , at any Rate of Compound Interest , and the like for Annuities and Purchases , after the same manner , and in the same method as I have done in the first Book of this small Treatise , for the Amount and Present Worth of either Principal Sums , or Annuities , at Simple Interest . TABLE I. The AMOUNT of One Pound put out to Interest , and forborn any Number of Years under 32 , or Quarters under 125. At the Rate of 6 per Cent. Compound Interest . Years and Quarters . Amount . ( 0 ) 1.000000 1 1.014674 2 1.029563 3 1.044670 ( 1 ) 1.060000 1 1.075554 2 1.091336 3 1.107351 ( 2 ) 1.123600 1 1.140087 2 1.156817 3 1.173792 ( 3 ) 1.191016 1 1.208493 2 1.226226 3 1.244219 ( 4 ) 1.262477 1 1.281002 2 1.299799 3 1.318872 ( 5 ) 1.338225 1 1.357862 2 1.377787 3 1.398005 ( 6 ) 1.418519 1 1.439334 2 1.460455 3 1.481885 ( 7 ) 1.503630 1 1.525694 2 1.548082 3 1.570798 ( 8 ) 1.593848 1 1.617236 2 1.640967 3 1.665046 ( 9 ) 1.689479 1 1.714270 2 1.739425 3 1.764949 ( 10 ) 1.790847 1 1.817126 2 1.843790 3 1.870846 ( 11 ) 1.898298 1 1.926154 2 1.954418 3 1.983096 ( 12 ) 2.012196 1 2.041723 2 2.071683 3 2.102082 ( 13 ) 2.132928 1 2.164226 2 2.195984 3 2.228207 ( 14 ) 2.260904 1 2.294080 2 2.327743 3 2.361900 ( 15 ) 2.396558 1 2.431725 2 2.467407 3 2.503614 ( 16 ) 2.540351 1 2.577628 2 2.615452 3 2.653831 ( 17 ) 2.692773 1 2.732286 2 2.772379 3 2.813061 ( 18 ) 2.854339 1 2.896223 2 2.938722 3 2.981844 ( 19 ) 3.025599 1 3.069996 2 3.115045 3 3.160755 ( 20 ) 3.207135 1 3.254196 2 3.301948 3 3.350400 ( 21 ) 3.399564 1 3.449448 2 3.500065 3 3.551424 ( 22 ) 3.603537 1 3.656415 2 3.710069 3 3.764509 ( 23 ) 3.819749 1 3.875800 2 3.932673 3 3.990380 ( 24 ) 4.048934 1 4.108348 2 4.168633 3 4.229803 ( 25 ) 4.291870 1 4.354849 2 4.418751 3 4.483591 ( 26 ) 4.549383 1 4.616139 2 4.683876 3 4.752607 ( 27 ) 4.822346 1 4.893108 2 4.964909 3 5.037763 ( 28 ) 5.111686 1 5.186695 2 5.262803 3 5.340029 ( 29 ) 5.418388 1 5.497896 2 5.578571 3 5.660431 ( 30 ) 5.743491 1 5.827770 2 5.913284 3 6.000054 ( 31 ) 6.088101 TABLE II. The AMOUNT of Due Pound , put out to Interest for any Number of Months under 25. At the Rate of 6 per Cent. Compound Interest . Months . Amount . 1 1.004867 2 1.009758 3 1.014673 4 1.019612 5 1.024575 6 1.029562 7 1.034574 8 1.039610 9 1.044670 10 1.049755 11 1.054865 12 1.060000 13 1.065159 14 1.070344 15 1.075554 16 1.080789 17 1.086050 18 1.091337 19 1.096649 20 1.101987 21 1.107351 22 1.112741 23 1.118158 24 1.123600 TABLE III. The AMOUNT of Due Pound , put out to Interest for any Number of Days under 366. At the Rate of 6 per Cent. Compound Interest . Days . Amount . 1 1.000160 2 1.000319 3 1.000479 4 1.000639 5 1.000798 6 1.000958 7 1.001118 8 1.001278 9 1.001438 10 1.001598 11 1.001757 12 1.001917 13 1.002077 14 1.002237 15 1.002397 16 1.002557 17 1.002717 18 1.002878 19 1.003038 20 1.003198 21 1.003358 22 1.003518 23 1.003678 24 1.003839 25 1.003998 26 1.004159 27 1.004320 28 1.004480 29 1.004640 30 1.004801 31 1.004961 32 1.005121 33 1.005282 34 1.005442 35 1.005603 36 1.005764 37 1.005924 38 1.006085 39 1.006245 40 1.006406 41 1.006567 42 1.006727 43 1.006888 44 1.007049 45 1.007209 46 1.007370 47 1.007531 48 1.007692 49 1.007853 50 1.008014 51 1.008175 52 1.008336 53 1.008497 54 1.008658 55 1.008818 56 1.008980 57 1.009141 58 1.009302 59 1.009463 60 1.009624 61 1.009786 62 1.009947 63 1.010108 64 1.010269 65 1.010431 66 1.010592 67 1.010753 68 1.010915 69 1.011076 70 1.011237 71 1.011398 72 1.011560 73 1.011722 74 1.011883 75 1.012045 76 1.012207 77 1.012368 78 1.012530 79 1.012691 80 1.012853 81 1.013015 82 1.013177 83 1.013338 84 1.013500 85 1.013662 86 1.013824 87 1.013986 88 1.014147 89 1.014309 90 1.014471 91 1.014633 92 1.014795 93 1.014957 94 1.015119 95 1.015281 96 1.015443 97 1.015605 98 1.015768 99 1.015930 100 1.016093 101 1.016254 103 1.016417 103 1.016579 104 1.016741 105 1.016903 106 1.017066 107 1.017228 108 1.017391 109 1.017553 110 1.017715 111 1.017878 112 1.018040 113 1.018203 114 1.018365 115 1.018528 116 1.018691 117 1.018853 118 1.019016 119 1.019179 120 1.019341 121 1.019504 122 1.019667 123 1.019830 124 1.019992 125 1.020155 126 1.020318 127 1.020481 128 1.020644 129 1.020807 130 1.020970 131 1.021133 132 1.021296 133 1.021459 134 1.021622 135 1.021785 136 1.021948 137 1.022112 138 1.022275 139 1.022438 140 1.022601 141 1.022765 142 1.022928 143 1.023091 144 1.023254 145 1.023418 146 1.023581 147 1.023745 148 1.023908 149 1.024072 150 1.024235 151 1.024399 152 1.024562 153 1.024726 154 1.024889 155 1.025053 156 1.025217 157 1.025380 158 1.025544 159 1.025708 160 1.025871 161 1.026035 162 1.026199 163 1.026363 164 1.026527 165 1.026691 166 1.026855 167 1.027018 168 1.027182 169 1.027346 170 1.027510 171 1.027675 172 1.027839 173 1.028003 174 1.028167 175 1.028331 176 1.028495 177 1.028659 178 1.028824 179 1.028988 180 1.029152 181 1.029316 182 1.029481 183 1.029645 184 1.029809 185 1.029974 186 1.030138 187 1.030302 188 1.030467 189 1.030632 190 1.030796 191 1.030961 192 1.031126 193 1.031290 194 1.031455 195 1.031619 196 1.031784 197 1.031949 198 1.032114 199 1.032278 200 1.032443 201 1.032608 202 1.032773 203 1.032938 204 1.033103 205 1.033268 206 1.033433 207 1.033598 208 1.033763 209 1.033928 210 1.034098 211 1.034258 212 1.034423 213 1.034588 214 1.034753 215 1.034919 216 1.035084 217 1.035249 218 1.035414 219 1.035580 220 1.035745 221 1.035910 222 1.036076 223 1.036241 224 1.036407 225 1.036572 226 1.036737 227 1.036903 228 1.037069 229 1.037234 230 1.037400 231 1.037565 232 1.037731 233 1.037897 234 1.038062 235 1.038228 236 1.038394 237 1.038560 238 1.038725 239 1.038891 240 1.039057 241 1.039223 242 1.039389 243 1.039555 244 1.039721 245 1.039887 246 1.040053 247 1.040219 248 1.040385 249 1.040551 250 1.040717 251 1.040883 252 1.041050 253 1.041216 254 1.041382 255 1.041548 256 1.041715 257 1.041881 258 1.042047 259 1.042214 260 1.042380 261 1.042546 262 1.042713 263 1.042879 264 1.043046 265 1.043212 266 1.043379 267 1.043545 268 1.043712 269 1.043879 270 1.044045 271 1.044212 272 1.044379 273 1.044545 274 1.044712 275 1.044879 276 1.045046 277 1.045213 278 1.045380 279 1.045546 280 1.045713 281 1.045880 282 1.046047 283 1.046214 284 1.046381 285 1.046548 286 1.046715 287 1.046883 288 1.047050 289 1.047217 290 1.047384 291 1.047551 292 1.047719 293 1.047886 294 1.048053 295 1.048220 296 1.048388 297 1.048555 298 1.048723 299 1.048890 300 1.049057 301 1.049225 302 1.049393 303 1.049560 304 1.049728 305 1.049895 306 1.050063 307 1.050230 308 1.050398 309 1.050566 310 1.050734 311 1.050901 312 1.051069 313 1.051237 314 1.051405 315 1.051573 316 1.051741 317 1.051908 318 1.053076 319 1.052244 320 1.052412 321 1.052580 322 1.052748 323 1.052916 324 1.053084 325 1.053253 326 1.053421 327 1.053589 328 1.053757 329 1.053925 330 1.054094 331 1.054262 332 1.054430 333 1.054599 334 1.054767 335 1.054935 336 1.055104 337 1.055272 338 1.055441 339 1.055609 340 1.055778 341 1.055946 342 1.056115 343 1.056284 344 1.056452 345 1.056621 346 1.056790 347 1.056958 348 1.057127 349 1.057296 350 1.057465 351 1.057633 352 1.057802 353 1.057971 354 1.058140 355 1.058309 356 1.058478 357 1.058647 358 1.058816 359 1.058985 360 1.059154 Days . Amount . 361 1.059323 362 1.059492 363 1.059661 364 1.059830 365 1.060000 TABLE IV. The PRESENT WORTH of Due Pound , due after any Number of Years under 32 , or Numters under 125. At the Rate of 6 per Cent. Compound Interest . Years and Quarters . Present Worth. ( 0 ) .0000000 1 .9855383 2 .9712858 3 .9572394 ( 1 ) .9433962 1 .9297531 2 .9163074 3 .9030560 ( 2 ) .8899964 1 .8771256 2 .8644409 3 .8519397 ( 3 ) .8396193 1 .8274770 2 .8155103 3 .8037167 ( 4 ) .7920936 1 .7806387 2 .7693493 3 .7582233 ( 5 ) .7472581 1 .7364516 2 .7258013 3 .7153050 ( 6 ) .7049605 1 .6947656 2 .6847182 3 .6748160 ( 7 ) .6650571 1 .6554393 2 .6459606 3 .6366189 ( 8 ) .6274123 1 .6183389 2 .6093967 3 .6005839 ( 9 ) .5918984 1 .5833386 2 .5749026 3 .5665885 ( 10 ) .5583947 1 .5503194 2 .5423609 3 .5345175 ( 11 ) .5267875 1 .5191693 2 .5116612 3 .5042618 ( 12 ) .4969693 1 .4897823 2 .4826993 3 .4757187 ( 13 ) .4688390 1 .4620588 2 .4553767 3 .4487912 ( 14 ) .4423009 1 .4359045 2 .4296006 3 .4233879 ( 15 ) .4172650 1 .4112307 2 .4052836 3 .3994226 ( 16 ) .3936463 1 .3879535 2 .3823430 3 .3768137 ( 17 ) .3713644 1 .3659939 2 .3607010 3 .3554847 ( 18 ) .3503438 1 .3452772 2 .3402839 3 .3353629 ( 19 ) .3305130 1 .3257332 2 .3210226 3 .3163801 ( 20 ) .3118047 1 .3072955 2 .3028515 3 .2984718 ( 21 ) .2941554 1 .2899014 2 .2857089 3 .2815771 ( 22 ) .2775051 1 .2734919 2 .2695367 3 .2656388 ( 23 ) .2617972 1 .2580112 2 .2542799 3 .2506026 ( 24 ) .2469785 1 .2434068 2 .2398867 3 .2364176 ( 25 ) .2329986 1 .2296291 2 .2263082 3 .2230355 ( 26 ) .2198100 1 .2166312 2 .2134983 3 .2104108 ( 27 ) .2073679 1 .2043690 2 .2014135 3 .1985008 ( 28 ) .1956301 1 .1928010 2 .1900128 3 .1872648 ( 29 ) .1845567 1 .1818877 2 .1792573 3 .1766649 ( 30 ) .1741101 1 .1715924 2 .1691113 3 .1666663 ( 31 ) .1642569 TABLE V. The PRESENT WORTH of One Pound , due after the expiration of any Number of Months under 25. At the Rate of 6 per Cent. Compound Interest . Months . Present worth . 1 .9951560 2 .9903355 3 .9855383 4 .9807644 5 .9760136 6 .9712858 7 .9665810 8 .9618988 9 .9572394 10 .9526026 11 .9479884 12 .9433962 13 .9388264 14 .9342788 15 .9297531 16 .9252494 17 .9207676 18 .9163074 19 .9118689 20 .9074518 21 .9030561 22 .8986817 23 .8943285 24 .8899964 TABLE VI. The PRESENT WORTH of One Pound , due after the expiration of any Number of Days under 366. At the Rate of 6 per Cent. Compound Interest . Days . Present Worth. 1 .9998404 2 .9996808 3 .9995212 4 .9993616 5 .9992021 6 .9990426 7 .9988831 8 .9987237 9 .9985643 10 .9984048 11 .9982455 12 .9980861 13 .9979268 14 .9977675 15 .9976083 16 .9974490 17 .9972898 18 .9971306 19 .9969714 20 .9968123 21 .9966532 22 .9964941 23 .9963350 24 .9961759 25 .9960169 26 .9958579 27 .9956990 28 .9955400 29 .9953810 30 .9952222 31 .9950633 32 .9949045 33 .9947457 34 .9945869 35 .9944282 36 .9942694 37 .9941107 38 .9939520 39 .9937934 40 .9936347 41 .9934760 42 .9933175 43 .9931590 44 .9930004 45 .9928419 46 .9926834 47 .9925250 48 .9924665 49 .9923081 50 .9921497 51 .9919914 52 .9917330 53 .9915747 54 .9914165 55 .9912582 56 .9911000 57 .9909418 58 .9907836 59 .9906254 60 .9904673 61 .9903092 62 .9901511 63 .9899930 64 .9898350 65 .9896769 66 .9895190 67 .9893611 68 .9892031 69 .9890452 70 .9888874 71 .9887297 72 .9885718 73 .9884139 74 .9882561 75 .9880983 76 .9879406 77 .9877829 78 .9876252 79 .9874676 80 .9873100 81 .9871523 82 .9869948 83 .9868372 84 .9866797 85 .9865222 86 .9863647 87 .9862073 88 .9860498 89 .9858924 90 .9857350 91 .9855777 92 .9854204 93 .9852631 94 .9851058 95 .9849486 96 .9847913 97 .9846341 98 .9844770 99 .9843198 100 .9841627 101 .9840056 103 .9838485 103 .9836914 104 .9835344 105 .9833774 106 .9832204 107 .9830635 108 .9829066 109 .9827497 110 .9825928 111 .9824359 112 .9822791 113 .9821223 114 .9819656 115 .9818088 116 .9816521 117 .9814954 118 .9813387 119 .9811821 120 .9810254 121 .9808688 122 .9807123 123 .9805557 124 .9803992 125 .9802427 126 .9800862 127 .9799298 128 .9797733 129 .9796169 130 .9794606 131 .9793042 132 .9791479 133 .9789916 134 .9788353 135 .9786791 136 .9785228 137 .9783666 138 .9782105 139 .9780543 140 .9778982 141 .9777421 142 .9775860 143 .9774300 144 .9772739 145 .9771179 146 .9769620 147 .9768060 148 .9766500 149 .9764942 150 .9763383 151 .9761824 152 .9760266 153 .9758708 154 .9757150 155 .9755593 156 .9754036 157 .9752479 158 .9750922 159 .9749366 160 .9747809 161 .9746253 162 .9744697 163 .9743142 164 .9741587 165 .9730032 166 .9738477 167 .9736922 168 .9735368 169 .9733814 170 .9732260 171 .9730707 172 .9729154 173 .9727600 174 .9726047 175 .9724495 176 .9722942 177 .9721390 178 .9719839 179 .9718287 180 .9716736 181 .9715185 182 .9713634 183 .9712084 184 .9710534 185 .9708983 186 .9707433 187 .9705884 188 .9704334 189 .9702785 190 .9701236 191 .9699688 192 .9698140 193 .9696591 194 .9695044 195 .9693496 196 .9691949 197 .9690400 198 .9688954 199 .9687308 200 .9685762 201 .9684216 202 .9682670 203 .9681124 204 .9679579 205 .9678033 206 .9676489 207 .9674944 208 .9673400 209 .9671855 210 .9670311 211 .9668768 212 .9667224 213 .9665681 214 .9664138 215 .9662596 216 .9661053 217 .9659511 218 .9657969 219 .9656428 220 .9654886 221 .9653345 222 .9651804 223 .9650263 224 .9648723 225 .9647183 226 .9645643 227 .9644103 228 .9642563 229 .9641024 230 .9639485 231 .9637946 232 .9636408 233 .9634870 234 .9633332 235 .9631794 236 .9630256 237 .9628719 238 .9627182 239 .9625645 240 .9624109 241 .9622573 242 .9621037 243 .9619500 244 .9617965 245 .9616430 246 .9614895 247 .9613360 248 .9611825 249 .9610291 250 .9608757 251 .9607223 252 .9605690 253 .9604157 254 .9602623 255 .9601091 256 .9599558 257 .9598026 258 .9596494 259 .9594962 260 .9593430 261 .9591799 262 .9590369 263 .9588837 264 .9587306 265 .9585775 266 .9584245 267 .9582715 268 .9581185 269 .9579656 270 .9578127 271 .9576598 272 .9575069 273 .9573541 274 .9572013 275 .9570485 276 .9568957 277 .9567430 278 .9565902 279 .9564375 280 .9562849 281 .9561322 282 .9559796 283 .9558270 284 .9556744 285 .9555219 286 .9553693 287 .9552168 288 .9550644 289 .9549119 290 .9547595 291 .9546071 292 .9544547 293 .9543023 294 .9541500 295 .9539977 296 .9538454 297 .9536932 298 .9535409 299 .9533887 300 .9532365 301 .9530843 302 .9529322 303 .9527800 304 .9526280 305 .9524759 306 .9523239 307 .9521719 308 .9520199 309 .9518679 310 .9517160 311 .9515640 312 .9514121 313 .9512603 314 .9511084 315 .9509566 316 .9508048 317 .9506530 318 .9505013 319 .9503495 320 .9501978 321 .9500462 322 .9498945 323 .9497429 324 .9495913 325 .9494397 326 .9492881 327 .9491366 328 .9489851 329 .9488336 330 .9486822 331 .9485307 332 .9483793 333 .9482279 334 .9480766 335 .9479251 336 .9477739 337 .9476226 338 .9474713 339 .9473201 340 .9471689 341 .9470177 342 .9468665 343 .9467154 344 .9465642 345 .9464131 346 .9462621 347 .9461110 348 .9459600 349 .9458090 350 .9456580 351 .9455071 352 .9453561 353 .9452052 354 .9450543 355 .9449035 356 .9447526 357 .9446018 358 .9444511 359 .9443003 360 .9441495 361 .9439988 362 .9438481 363 .9436975 364 .9435468 365 .9433962 TABLE VII . The PRESENT WORTH of One Pound Annuity , to continue any Number of Years under 32 , and payable by Yearly Payments , at 5 , 6 , 7 , 8 , 9 , and 10 per Cent. Compound Interest . The PRESENT WORTH of One Pound Annuity , Comp. Int. At Years . 5 per Cent. 6 per Cent. 7 per Cent. 1 0.95238 0.94339 0.93457 2 1.85941 1.83339 1.80801 3 2.72324 2.67301 2.62431 4 3.54595 3.46510 3.38721 5 4.32947 4.21236 4.10019 6 5.07569 4.91732 4.76653 7 5.78637 5.58238 5.38928 8 6.46321 6.20979 5.97129 9 7.10782 6.80169 6.51523 10 7.72173 7.36008 7.02358 11 8.30641 7.88687 7.49867 12 8.86325 8.38384 7.94268 13 9.39357 8.85268 8.35765 14 9.89864 9.29498 8.74546 15 10.37965 9.71224 9.10791 16 10.83776 10.10589 9.44664 17 11.27406 10.47725 9.76322 18 11.68958 10.82760 10.05908 19 12.08532 11.15811 10.33559 20 12.46220 11.46992 10.59401 21 12.82115 11.76407 10.83552 22 13.16300 12.04158 11.06124 23 13.48857 12.30337 11.27218 24 13.79864 12.55035 11.46933 25 14.09394 12.78335 11.65358 26 14.37518 13.00316 11.82577 27 14.64303 13.21053 11.98671 28 14.89812 13.40616 12.13711 29 15.14107 13.59072 12.27767 30 15.37245 13.76483 12.40904 31 15.59283 13.92908 12.53187 Years . 8 per Cent. 9 per Cent. 10 per Cent. 1 0.92592 0.91743 0.90909 2 1.78326 1.75911 1.73553 3 2.57709 2.53129 2.48685 4 3.31212 3.23971 3.16986 5 3.99270 3.88965 3.79078 6 4.62287 4.48591 4.35526 7 5.20636 5.03295 4.86841 8 5.74663 5.53481 5.33492 9 6.24688 5.99524 5.75902 10 6.71008 6.41765 6.14456 11 7.13896 6.80519 6.49506 12 7.53607 7.16072 6.81369 13 7.90377 7.48690 7.10335 14 8.24423 7.78614 7.36668 15 8.55947 8.06068 7.60608 16 8.85136 8.31255 7.82371 17 9.12163 8.54363 8.02155 18 9.37188 8.75562 8.20141 19 9.60359 8.95011 8.36492 20 9.81814 9.12854 8.51356 21 10.01680 9.29224 8.64869 22 10.20074 9.44242 8.77154 23 10.37105 9.58020 8.88322 24 10.52875 9.70661 8.98474 25 10.67477 9.82258 9.07704 26 10.80997 9.92897 9.16094 27 10.93516 10.02658 9 . 237●● 28 11.05107 10.11613 9.30656 29 11.15840 10.19828 9.36960 30 11.25778 10.27365 9.42691 31 11.34981 10.34284 9.47901 TABLE VIII . Shewing what Annuity , to continue any Number of Years under 32 , and payable by Yearly Payments , One Pound will Purchase , at the Rate of 5 , 6 , 7 , 8 , 9 , and 10 per Cent. Compound Interest . A TABLE shewing what Annuity One Pound will Purchase , at several Rates of Comp. Int. Years . 5 per Cent. 6 per Cent. 7 per Cent. 1 1.05000 1.06000 1.07000 2 .53780 .54543 .55309 3 .36720 .37411 .38105 4 .28209 .28859 .29519 5 .23097 .23739 .24389 6 .19701 .20336 .20979 7 .17281 .17913 .18555 8 .15472 .16103 .16746 9 .14069 .14702 .15348 10 .12950 .13586 .14237 11 .12038 .12679 .13335 12 .11282 .11927 .12590 13 .10645 .11296 .11965 14 .10102 .10758 .11434 15 .09634 .10296 .10979 16 .09226 .09895 .10585 17 .08869 .09544 .10242 18 .08554 .09235 .09941 19 .08274 .08962 .09675 20 .08024 .08718 .09439 21 .07799 .08500 .09228 22 .07597 .08304 .09040 23 .07413 .08127 .08871 24 .07247 .07967 .08718 25 .07095 .07822 .08581 26 .06956 .07690 .08456 27 .06829 .07569 .08342 28 .06712 .07459 .08239 29 .06604 .07357 .08144 30 .06506 .07264 .08058 31 .06418 .07181 .07983 Years . 8 per Cent. 9 per Cent. 10 per Cent. 1 1.08000 1.09000 1.10000 2 .56076 .56846 .57619 3 .38803 .39505 .40211 4 .30192 .30866 .31547 5 .25045 .25709 .26379 6 .21631 .22291 .22960 7 .19207 .19869 .20545 8 .17401 .18067 .18744 9 .16007 .16679 .17364 10 .14902 .15582 .16274 11 .14007 .14694 .15396 12 .13269 .13965 .14676 13 .12652 .13356 .14077 14 .12129 .12843 .13574 15 .11682 .12405 .13147 16 .11298 .12029 .12781 17 .10962 .11704 .12466 18 .10670 .11421 .12192 19 .10412 .11173 .11954 20 .10184 .10954 .11745 21 .09983 .10761 .11562 22 .09803 .10590 .11400 23 .09642 .10438 .11257 24 .09497 .10302 .11126 25 .09367 .10180 .11016 26 .09250 .10071 .10915 27 .09144 .09973 .10825 28 .09048 .09885 .10745 29 .08961 .09805 .10672 30 .08882 .09733 .10607 31 .08814 .09670 .10550 TABLE IX . The PRESENT WORTH of any Lease , or Annuity , for 21 , 31 , 41 , 51 , 61 , 71 , 81 , or 91 Years ; as likewise the PRESENT WORTH of the see Simple . At 5 , 6 , 8 , and 10 per Cent. Compound Interest . Years to be Purchased . At 5 per Cent. The Purchase of Freehold Land. At 6 Per Cent. The Purchase of Copyhold Land , or Leases of Land.   Years . Qua. Mo. Years . Qua. Mo. 21 12 3 1 11 3 0 31 15 3 1 13 3 2 41 17 0 1 15 0 1 51 18 1 0 15 3 0 61 18 3 2 16 2 0 71 19 1 1 16 1 2 81 19 2 1 16 2 0 91 19 3 0 16 2 0 Fee Simple . 20 0 0 16 2 2 Years to be Purchased . At 8 per Cent. The Purchase of very good Houses . At 10 per Cent. The Purchase of Leases of ordinary Houses .   Years . Qua. Mo. Years . Qua. Mo. 21 10 0 0 8 3 2 31 11 1 1 9 3 0 41 11 3 2 9 3 2 51 12 1 0 9 3 2 61 12 1 1 10 0 0 71 12 1 2 10 0 0 81 12 2 0 10 0 0 91 12 2 0 10 0 0 Fee Simple . 12 2 0 10 0 0 CHAP. II. The Use of the preceding TABLES of Compound Interest . HAving with all imaginable care framed and calculated divers Tables relating to Compound Interest , it will be needful to apply the same to Use and Practice . The Use of the first TABLE , shown in Two Examples . Example I. Suppose it be demanded to give the Amount of 136 l. 15 s. 6 d. being forborn 20 years , at 6 per Cent. Compound Interest . Direction . The given Sum must first be reduced by the first Table of the first Book , and made 136.775 , and then multiplied by the Number in the first Table answering to ( 20 ) , in the Margin of pag. 144 , viz. 3.20713 . Tariffa for the Multiplicand . 1 136775 2 273550 3 410325 4 547100 5 683875 6 820650 7 957425 8 1094200 9 1230975 The Operation may be contracted , according to the Rule in the Introduction . The Product 438.6552 is the true Answer , and being reduced by the fifteenth Table of the first Book , makes 438 l. 13 s. 1 d. 1 q. Example 2. Suppose it be demanded to give the Amount of the aforesaid Sum in 20 Years and 3 Quarters . Direction . Multiply the aforesaid Sum of 136.775 , by the Number which answers to 20 Years and 3 Quarters , viz. 3.35040 . The Operation may be contracted as before . The Product 458.2509 is the Answer to the Question , and being reduced by the fifteenth Table of the first Book , makes 458 l. 5 s. more by 22 / 100 ( or Ninety Two Hundred parts ) of a Farthing . The use of the second TABLE . Suppose it be desired to know the Amount of 42 l. in 7 Months , at 6 per Cent. Compound Interest . Direction . Seek the Number in Table II. answering to ( 7 ) in the Margin , viz. 1.034574 , and multiply it by 42 , and the Product is the Answer . Which Product being reduced is 43 l. 9 s. 2 q. more by 2 / 100 ( or Two Hundred parts ) of a Farthing . The use of the third TABLE . Suppose it be demanded to find the Amount of 42 l. in 104 Days . Direction . Find the Number in Table III. answering to ( 104 ) in the Margin , viz. 1.016741 , and multiply that by 42 , and the Product is the Answer . Which Product being reduced , makes 42 l. 14 s. 2 q. more by 99 / 100 ( or Ninety Nine Hundred parts ) of a Farthing . The use of the fourth TABLE . Let it be demanded to find the Present Worth of 438 l. 13 s. 1 d. 1 q. due and payable after the expiration of 20 Years , at 6 per Cent. Compound Interest . Direction . The given Sum being converted into a Decimal Number is 438.65520 , then find the Number in Table IV. answering to ( 20 ) in the Margin , viz. .3118047 , and multiplying one by another , the Product is the Answer . Tariffa for the Multiplicand . 1 4386552 2 8773104 3 13159656 4 17546208 5 21932760 6 26319312 7 30705864 8 35092416 9 39478968 And the Operation may be contracted thus to seven places , by the Rule in the Introduction . The Product 136.7747 is a manifest Proof of the truth of the Operation in the first Example of the use of the first Table , pag. 186. there being not so much as the Hundredth part of a Farthing difference . The Use of the fifth TABLE . Let the Present Worth of 43.452108 l. due after the expiration of 7 Months , be sought , according to the Rate of 6 per Cent. Compound Interest . Tariffa for the Multiplicand . 1 43452108 2 86904216 3 130356324 4 173808432 5 217260540 6 260712648 7 304164756 8 347616864 9 391068972 The Operation contracted by the Rule in the Introduction . Which Product 41.99997 is a clear and manifest Proof of the truth of the Operation in the use of the second Table , pag. 190. The Use of the sixth TABLE . Let it be required to find the Present Worth of 42.703122 l. after the end of 104 Days , at the Rate of 6 per Cent. Compound Interest . Tariffa for the Multiplicand . 1 42703122 2 85406244 3 128109366 4 170812488 5 213515610 6 256218732 7 298921854 8 341624976 9 384328098 The Operation contracted by the Rule in the Introduction . Which Product 41.99997 , is a clear and manifest Proof of the truth of the Operation in the Example , calculated to shew the use of the third Table , pag. 191. The Use of the seventh and ninth TABLES . Let it be required to find the Present Worth of an Annuity of 56 l. to continue 21 years , and payable by yearly payments , at the Rate of 6 per Cent. Compound Interest . Direction . First find the Number in Table VII . answering to ( 21 ) in the Margin , which is 11.76407 ; then multiply it by 56 , without a Tariffa , because there are but two places in the Multiplicator , and the Product is the Answer . Which Product being reduced by the fifteenth Table of the first Book , makes 657 l. 15 s. 9 d. more by 40 / 100 ( or Forty Hundred parts ) of a Farthing , which agrees with Mr. Kersey's Example , in his Appendix to Mr. Wingate's Arithmetick , pag. 412. only this Calculation is more exact than his , and somewhat nearer to the truth . The Use of the eighth TABLE . Let it be demanded what Annuity , to continue 14 years , and payable by yearly payments , will 320 l. buy , allowing 6 per Cent. Compound Interest . Direction . Seek in the Margin of Table VIII . the Number ( 14 ) , and the Number answering to it , under the Title of 6 per Cent. viz. .10758 ; which multiply by 320 , and the Product is the Answer . Which Product 34.42560 , being reduced by the fifteenth Table of the first Book , makes 34 l. 8 s. 6 d. more by 57 / 100 ( or Fifty Seven Hundred parts ) of a Farthing . A farther Use of the seventh TABLE . To convert a present Sum or Fine into an Annual Rent ; or on the contrary , to bring down an Annual Rent by a present Sum or Fine . Example 1. A Landlord Le ts a Lease of a House and Land for 21 years , and is to have 100 l. for that Lease , and a yearly payment of 30 l. what Fine or present Money must the Tenant give , to bring down the Rent from 30 l. to 10 l. per Annum , allowing 6 per Cent. Compound Interest ? Direction . First find the difference of 10 l. and 30 l. which is 20 l. then find by Table VII . pag. 176. what an Annuity of 1 l. to continue 21 years , is worth in present Money , which is 11.76407 l. then multiply 11.76407 by 20 , and the Product gives the Present Worth of 20 l. per Annum for 21 years . The Operation . Which Product 235.28140 being reduced , is 235 l. 5 s. 7 d. 2 q. more by 14 / 100 ( or Fourteen Hundred parts ) of a Farthing . Example 2. A Landlord demands a Fine , or present Sum , for a Lease of 127 l. per Annum , to continue 7 years ; what is the Sum , allowing 6 per Cent. Compound Interest ? Direction . Find by Table VII . the Present Worth of 1 l. Annuity for 7 years , at 6 per Cent. viz. 5.58238 , which multiply by 127 , and the Product is the Answer . The Operation . Which Product 708.96226 being reduced by the fifteenth Table of the first Book , makes the just Sum of 708 l. 19 s. 2 d. 3 q. more by 77 / 100 ( or Seventy Seven Hundred parts ) of a Farthing . And here the Reader is desired to take notice of a printed Sheet sold in Westminster-Hall , Entituled , “ A President for Purchasers , &c. Or Anatocisme ( commonly called Compound Interest ) made easie , &c. Computedly W. Leybourn . The principal Table in this Sheet is printed from a Copper Plate , but so full of gross Errors and mistakes , that it is not sit to be used : For In this last Example , that Table makes tho Sum but 706 l. 16 s. 6 d. which the Table of this Book makes 708 l. 19 s. 2 d. 3 q. and more , ( which Sum agrees with Mr. Clavel's Tables ) . But in very many places there is no less than 4 , 5 , 6 , and 10 Pound mistaken , which must needs deceive all those , who do in the least rely upon , or give any credit to it . So that the AUTHOR of this little Book hopes , That the manifold Errors in the Calculations of other Writers , will occasion a more kind acceptance of his more than ordinary care and diligence in all the foregoing Tables ; if not , — Redit Labor actus in Orbem . 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . FINIS . ADVERTISEMENT . THere is newly Printed a Vade Mecum , or Necessary Companion ; containing , 1. Sir Samuel Morland ' s Perpetual Almanack , in Copper Plates , with many useful Tables proper thereto . 2. The Computation of Years , comparing the Years of each King's Reign from the Conquest with the Years of Christ . 3. Reduction of Weights and Measures . 4. The ready Casting up any Number of Farthings , Half-pence , Pence , Shillings , Nobles and Marks ; with Sir Samuel Morland ' s New Table for Guinneys . 5. The Interest and Rebate of Money , the Forbearance , Discompt , and Purchase of Annuities , at 6 per Cent. 6. The Rates of Post-Letters , both Inland and Outland , with the Times for sending or receiving them ; also the Post-Stages , shewing the Length of each Stage , and the Distance of each Post-Town from London . 7. The Rates or Fares of Coach-men , Carr-men , and Water-men . And are sold by R. Northcot , Bookseller , either next St. Peter's Alley in Cornhill , or at the Anchor and Mariner on Fish-Street-Hill ; by John Playford , Printer , near the Blew-Anchor Inn in Little Britain ; and by Charles Blount , Bookseller , at the Black-Raven in the Strand , near the Savoy .