ends of the Fractions upon which they are respectively scituate I find the third term to fall upon the 33 Fraction and then observing eight Letters or Intervals to be intercepted betwixt the first and second termes accounting as many from the second towards the third I find the Chesseman of the third term to be likely to fall upon the said thirty third Fraction in the one and twentieth Interval signed by C and therefore draw that Chesseman back to the twenty perpendicular upon the same thirty third Fraction this done and I observing one digit at the first term and four at the second I double those four and adde them to the one all which amounting to nine I advance the Chesseman of the last term accordingly setting it in the middle of the one and twentieth Interval then finding also at the first term two minimes and four at the second I likewise double the four and adde them to the 2 all which amount to 10 according to which summe I advance the Chesseman of the third term ten minimes farther and so at last I finde the said third term to fix upon the 33 Fraction at four digits and four minimes of the 21 Interval which is the term required And if at any time in working questions of this kinde you happen to descend below or ascend above the side-rank or otherwise overshoot the table either on the right hand or left you are in such cases to use the Rules aforegoing but still doubling the digits and minimes of the second term as in the premised example In like manner may you also if you please discover a fourth term to those three known and so consequently a fift sixt seventh c. in infinitum VII A point upon any one of the transversals being given to find halfe the distance betwixt that point and the beginning or left end of that transversal follow this direction Take half the Alphabets half the letters half the digits and half the minimes intercepted betwixt the beginning of that line and the point given and so shall you have your desire So if the point c upon the chief transversal were propounded half the distance betwixt the beginning or left end thereof and that point will be found at two digits and two minimes of the letter E in the second Alphabet viz. at the point S for in this case there being three Alphabets and two letters intercepted betwixt the beginning of that traversal and the letter wherein the point given is scituate I take one Alphabet and three letters for the three Alphabets and one letter more for the two odde letters then for the four digits I take two digits and for the four minimes two minimes all which being accounted from the beginning of that transversal will fall at S the point required the same may likewise be acted upon any of the repeated Alphabets and transversals in the body of the table VIII Vpon any one of the Transversals to discover the third part of the distance betwixt the beginning or left end thereof and any point thereupon propounded this is the Rule Take the third part of the distance in Alphabets letters digits and minimes and so shall you attain the point or term required So the point c upon the chief transversal being again propounded the point t will be third part of the distance inquired For in liew of the three Alphabets I take one for the third part of the two odde letters I take four digits for the four other digits I take one digit and two minimes And for the four last minines I take one minime and somewhat more by which meanes t will be found at last the point songht for Thus likewise you may be practised upon the repeated Alphabets and transversals IX A Fraction of the Scale of Numbers being given to finde upon the side rank of Alphabets the half distance betwixt it and the first Fraction including the first Fraction for one proceed in this mnnnner first having placed a plain Chesseman without a point at the right end of the Fraction given observe whether the number of the Fraction next above it bâ— even or odde if even then take half the summe thereof and place another plain Chesseman at the right end of the Fraction next under that half summe but if the number be odde neglecting the odde Fraction proceed with the even number as before nnd so you shall accomplish your desire Example Let the Fraction signed at the right end thereof by 21 be given and let the half distance betwixt it and the first Fraction be demanded Here the number above it is 20 whereof the half is 10 wherefore I taking a Chesseman place it at the right end of the Fraction signed by 11 which is the half distance demanded And if the 22 Fraction were propounded the half distance would still remain the same Howbeit in that case the odde Fraction signed by 21 would remain over and besides the two moities which neverthelesse will produce no errour in the use of the table as shall appear hereafter X. A Fraction of the Scale of Numbers being propounded to discover upon the side-rank of Alphabets the third part of the distance betwixt it and the first Fraction including the first Fraction for one use this Rule Having placed a Chesseman at the right end of the Fraction given as before observe whether the number of the Fraction placed next above it may be divided into three even parts if so then take the third part thereof and place another Chesseman at the right end of the Fraction next under that third part but if that number will not admit such an equall division then neglecting the odde Fraction or Fractions so remaining proceed with the numbers which do so equally divide themselves as before and so you shall discover the third part you look for Example Let the 22 Fraction be given and the third part of the distances required Here the number next above it is 21 whereof the third part is 7 wherefore finding 7 amongst the numbers placed at the right ends of the Fractions I place another Chessman at the right end of the eighth Fraction which denotes the third part required Howbeit the 23 Fraction being given an odde Fraction will remain over and above the number which so equally divides it self into three parrs as aforesaid and if the 24 Fraction were propounded two such odde Fractions would remain which neverthelesse causeth no inconvenience in the practice of this Instruments as shall be manifested in the proper place CHAP. IV. The Application of the Rule of Proportion WE have done with Numeration Application infues which teacheth the use of this Instrument for the easie and ready resolution of divers Propositions in Arithmetick and Geometry as followeth Prop. 1. To three numbers given to finde a fourn in a direct proportion This is termed the Rule of Three or more usually the Golden Rule because it is of greatest use in Arithmetick and Geometry
accordingly as in the first example of the last Rule at last I discover the fourth term required to fall upon the Transversal of the 24 Fraction at 4 digits and 4 Minimes of the said 30 Interval viz. at the point k so likewise if k were the first term h the second and g the third working towards the left hand f would be found to be the fourth c. V. Having three points given upon three severall transversals to discover the transversall upon which the fourth term will fall and also the point of that transversall where that fourth term will beare like distance from the third point that the second beares from the first Observe this direction having placed as before three Chessemen at the three given points place likewise three other plain Chessemen upon the side rank of Alphabets at the right ends of the fractions or Transversals whereupon the points given are scituate respectively This done by the second Rule of this Chapter find upon the said side-rank a fourth term to the three given which will lead you to the Transversal upon which the fourth term required is to be found then proceeding according to the directions of the last Rule you will discover the fourth point or term you look for Example If f g and h the points of the first example of the last Rule be given viz. upon the 5 19 and 10 Fractions as before In this Case I place a plain Chesseman at the right end of the fift Fraction another at the same end of the tenth Fraction and a third at the like end of the nineteenth Fraction and in working downwards discover upon that side-rank of Alphabets by the second Rule of this Chapter a fourth term correspondent to the other three given terms which fourth term leads me to the 24 Fraction and transversal upon which the fourth term in question is scituate And therefore proceeding thereupon as in the first example of the last Rule you will find the fourth term required in this example to fall upon the transversal of that 24 Fraction at four digits and four minimes of the 30 Interval viz. at the point k as before In like manner if k were the first term h the second and g the third in mounting upwards upon the side rank and proceeding upon the Table towards the left hand as I did before towards the right the fourth term will in that case be found to fall upon the transversal of the fift Fraction at four digits and four minimes of the third Interval signed by C viz. at the point f. So if g be the first term h the second and k the third the Fraction or transversal of the fourth term being found upon the side rank and I guiding my work upon the Table towards the right hand the fourth term will fall upon the transversal of the 15 Fraction at 4 digits and 4 minimes of the 3 Interval signed by C viz. at the point l Howbeit you are not to take that for the true point but because in that case you go beyond the Table towards the right hand and for that the right end of the 15 Fraction is conceived to joyn with the left end of the 16 Fraction according to the directions of the 9 16 and 25 Rules of the first Chapter you are to take 4 digits and 4 minimes of the transversal next under it in the same Interval and so the true point required will be in that case found to reside upon the transversal of the 16 Fraction at 4 digits and 4 minimes of the said third Interval viz. at the point m. In like manner if the three termes propounded were h g and f and a fourth term be required answerable unto them In that case the proper Fraction or Transversal of that 4th term being discovered upon the side rank I proceeding towards the left hand the 4th term will fall upon the Transversal of the 14 Fraction at 4 digits and 4 Minimes of the 30 Interval viz. at the point n. Howbeit as in the last aforegoing example you are not to take that for the true point but in that case because you go beyond the Table towards the left hand and for that the left end of the 14 Fraction is conceived to joyn with the right end of the 13 Fraction according to the said 9 16 and 25 Rules of the first Chapter you are instead thereof to take 4 digits and 4 Minimes of the transversal next above it in the same interval and so true point required will be found to rest upon the transversal of the 13 Fraction at 4 digits and 4 minimes of the said 30 Interval viz. at the point p. And here give me leave once for all to insert this direction that in the motion of a Chesseman upon the Table when you are constrained to over-shoot the table either on the right or left hand take the Fraction next to it either above or below it viz. if on the right hand then the Fraction below it but if on the left hand then that above it as in the two last premised examples you finde it practised Again if f be the first term g the second and k the third the fourth term will fall upon the third Fraction at 4 digits and 4 minimes of the third Interval viz. at the point q and in that case you do not onely fall off at the lower end of the side-rank taking it again at the top but likewise overshoot the table upon the right hand and take it again upon the left and in that respect take not the fraction whereunto you are directed by the fourth term found in the side-rank but take the next under it On the other side if k were the first term h the second and f the third the fourth term will reside upon the 27 Fraction at 4 digits and 4 minimes of the 30 Interval viz. at the point r. And in that case also you do not onely mount off at the top of the side rank taking it again at the lower end but likewise over-shoot the Table upon the left hand and take it again upon the right and in that regard also take not the Fraction unto which you are directed by the fourth term found in the side rank but take the next above it according to the direction of the afore-going examples VI. After the same manner may you also discover a third term to two termes propounded save onely that in regard the second term doth in a sort in that case represent the two middle termes you are to double the digits and minimes of the second term and then adde them to the digits and minimes of the first term to the end you may understand by that summe how far to advance the Chesseman of the last term For example Let f be the first term and g the second and let a third term be desired here Chessemen being placed at the terms given and likewise upon the side rank at the
Ludus Mathematicus OR THE MATHEMATICAL GAME Explaining the description construction and use of the Numericall Table of Proportion By help whereof and of certain Chessmen fitted for that purpose any Proposition Arithmetical or Geometrical without any Calculation at all or use of Pen may be readily and with delight resolved when the term required exceeds not 100000. By E. W. Omne tulit punctum qui miscuit utile dulci. LONDON Printed by R. W. Leybourn and are to be sold by Philemon Stephens at the Gilded Lion in Paul's Church yard M DC LIV. THE PREFACE THis Instrument I at first intended for my own private use delight not conceiving it worthy to see the light but being since informed by others well verst in the Mathematicks and finding also by experience that it may prove usefull for others and Bonum quò communas eò melius I have permitted it to launch into the Ocean of censure Howbeit I present it chiefly to such as in some competent manner have already acquainetd themselves with the modern use of Arithmetick I mean by Logarithms Decimals and Scales for they may be able immediately to apprehend the use thereof and that with some pleasure and delight To other Arithmeticians not acquainted with that kinde of Artificiall Arithmetick it may at first seeme somewhat more difficult But unto such as are not at all verst in Arithetick I may object Plato's Inscription placed over the doore of his Academie concerning Geometry including also Arithmetick Nemo Geometriae ignarus huc ingreditor It professeth to render you the term required in any question propounded when it will not amount to above 100000 that is when it exceeds not five figures or places and that it will cleerly do especially towards the beginning of the Scale when the term or terms out of which the Question is to be produced are rationall numbers viz. when the term required to be extracted from them will be precisely a whole number without a fraction attending it but when the term or terms given are irrational numbers which will produce a mixt number consisting of a whole part together with a fraction in that case it will represent unto you only the whole part thereof without the broken part or fraction which defect neverthelesse will occasion no inconvenience in the practise of this Instrument the broken part of a number of such an extent being not considerable in Questions of ordinary practise as is well known to all Artists This advertisement I have thought fit to premise lest it might seeme to promise more than it can perform and so cause the Practitioner to be frustrated of his expectation THE CONTENTS CHAP. I. The Definition description and construction of the Numericall Table of Proportion p. 1. CHAP. II. Numeration upon the Scale of Numbers p. 15. CHAP. III. Numeration upon the Alphabets transversals p. 24 CHAP. IV. Application of the Table in the resolution of these Propositions following p. 46 1. To three numbers given to finde a fourth in a direct proportion ibid. 2. To three numbers given to finde a fourth in an inversed proportion p. 52 3. One number being given to be multiplyed by another to finde the Product p. 54 4. One number being given to be divided by another to find the Quotient p. 59 5. Two numbers being given to find a third Geometrically proportionall unto them and to three a fourth and to four a fift c. p. 63 6. To extract the square-root of any number given under 10000000000. p. 64 7. To extract the Cube-root of any number given under 1000000000000000. p. 69 ERRATA PAge 4. line 23. for 100 read 107. p. 9. l. 22. f. here r. them p. 13. l. 15. f. intervall r. internall p. 16. l. 24. f. brought r. brings p. 42. l. 18 19. f. Thus likewise you may r. This likewise may p. 46. l. 1. f. Rule r. Table p. 47. l. 27. f. k r. k which THE MATHEMATICAL Game CHAP. I The Definition Description and Construction of the Numericall Table of Proportion I. A Table of Proportion is an Instrument framed by Logarithms and invented for the more easie resolving of Arithmeticall and Geometricall Operations In Naturall or Vulgar Arithmetick the Propositions are resolved by using the Numbers themselves as if 4 were given to be multiplyed by 2 we say two times four makes 8 the Product In Artificiall Arithmetick if rhe same Question were propounded insteed of 4 and 2 we take their Logarithms so if the Logarithm of 4 being 0,602060 be added to the Logarithm of 2 being 0,301030 their summe is 0,903090 which being found in the Table of Logarithms is the Logarithm of 8 the Product as before Howbeit here in the use of this Instrument we need not Multiply or Divide Adde ar Substract which for the most part perplex and discourage the Practitioner but by the motion of certain Chesse-men fitted for that purpose we perform with pleasure and delight the hardest Propositions of Arithmetick and Geometry without charging the minde or memory with any thing which may seeme burthensome or distastfull II. This Instrument is twofold Numericall or Trigonometricall III. The Numericall Table of Proportion is an Instrument by help whereof and of certain moveable Chesse-men all Questions Arithmeticall and Geometricall performed by Multiplication Division or the Golden Rule and not Trigonometricall together with mean Proportionals the Extraction of the Roots of all Square-numbers under 11 places and of all Cube-numbers under 16 places as well in mixt and broken as in whole numbers when the term required exceeds not 100000 are with great â—ase and exactnesse Resolved For we â—ntend not here to meddle with any questions that are performed by the Doctrine of Triangles referring them â—o be handled in the use of the Table of Proportion Trigonometricall IV. Of the Numericall Table of Proâ—â—â—tion these things offer themselves to 〈◊〉 considered viz. The Description and â—onstruction or the Vse V. For the more plain describing of â—â—is Instrument it may be said to consist 〈◊〉 two parts viz. The Body of the Taâ—â—e it self and substantiall part or the Appendants and Circumstantiall part thereof VI. The Bodie of the Table it self is a Scale of unequall parts broken off into Fractions and hereafter for distinction sake called the Scale of Numbers This Scale is nothing else but a line of Numbers broken off into 36 fractions or equal parts Now what a line of Numbers is hath been heretofore taught by Mr. Gunter in his Book of the Crosse-staffe and is well enough known to all modern Artists VII A Fraction of the Scale of Numbers is an equall part of the same Scale consisting of Lines Spaces and Divisions So this Scale is broken of or divided into thirty six of those equal parts or fractions numbred at their right ends by 1 2 3 c. to 36 of which the part signed at the left end thereoâ— by 100 is the first Fraction that signe by 100 is the second c.
VIII Each of these Fractions consists of three lines and two spaces so the pricked line which you finde place under each Fraction is not to be take as any part thereof but hath another use as shall be declared in the proper place IX These Fractions together with their Lines and Spaces must be understood to joyn respectively one to another in such sort that the whole Scale of Numbers may be conceived to be one entire and continued Line For Example The right end of the first Fraction marked by 1 A. must be conceived to joyn with the left end of the second Fraction signed by 107 and the right end of the second Fraction marked by 2 B. must be understood to joyn with the left end of the third Fraction noted by 114 And so consequently of the rest in their order so that the whole Scale of Numbers beginning at the left end of the first Fraction signed by 100 and ending at the right end of the last Fraction noted by 36 F. must be conceived to be one intire and continued line as aforesaid And therefore by farther consequence in mounting up wards the lest end of the last Fraction signed by 939 must be also conceived to joyne with the right end of that above it signed by 35 E. and so of the rest in ascending upwards untill you mount to the beginning of the Scale X. The intire Scale of Numbers is first divided into a thousand unequall parts which are hereafter called Hundreds and distinguished by having three figures placed at the beginning of each of them so 100 at the beginning of the Scale are the figures of the first Hundred 101 of the second Hundred 102 of the third Hundred 103 of the fourth Hundred c. XI Each of these Hundreds are again sub-divided into ten other unequall parts hereafter called Tenths and each Tenth also supposed to be again divided into ten other parts called Vnits For the distances between the Tenths being small they will not admit any reall division of the same Tenths into ten other parts And therefore you are to suppose them to be so divided and hereafter when you shall have occasion to use those parts you are to guesse at them as to direct your eye to the middle of them when you are to take five of these Units and somewhat beyond the middle when six of them are propounded c. Howbeit because at the beginning of the Scale of Numbers the distance of the Tenths are so large that you cannot readily in manner aforesaid guesse at the Units comprehended betwixt them I have caused that distance upon the first fix Fractions to be divided into five parts each part representing two Units and from thence upon the six Fractions next after following into two parts each part representing five Units In the mean time distinguishing the Tenths comprehended betwixt every two hundreds by sharp points rising from the middle line of the Scale into the uppermost space thereof and upon all the rest of the Scale leaving the Units to be guessed at as aforesaid XII To describe the Hundreds and Tenths upon the Scale of Numbers Having first prepared a Scale of 100 equall parts containing in length the hundred part of the whole intended Scale of Numbers which Scale of equall parts must be supposed to be divided into 1000 equall parts the distance betwixt each hundred part thereof being supposed to be divided into ten parts repair to the Table of Logarithmes and therein observing the first five figures of the Logarithme of 1001 besides the Characteristique or Index viz. 00043 take with your compasses the distance from the beginning of your Scale of equall parts to the said 43 this done if you applie that extent of the compasses towards the right hand from the beginning of your intended Scale of Numbers the moveable point of the compasses will fall upon the first tenth of that Scale In like manner by the first five figures of the Logarithme of 1002 besides the Index viz. 00086 you may mark out the second tenth of the same Scale and so consequently all the rest in their due order Example If it were propounded to make a Scale of Numbers equall to this whereof we treat this Scale being intirely taken together as one continued Scale according to the ninth Rule aforegoing it conteins in length 75 feet which amount to 900 Inches whereof the hundred part is nine Inches wherefore having prepared a Scale nine Inches long as is above directed I take off with my compasses the parts 43 which extent being applied from the beginning of the Scale of Numbers towards the right hand the moveable point will fall upon the first tenth of the first hundred of that Scale just under the letter Z so likewise if I again take off upon the Scale of equall parts the figures 86 and apply them from the beginning of the Scale of Numbers as before that extent will mark out the second tenth of the same Hundred just under the letter X. In like manner also may you proceed untill you have described all the divisions of the Scale of Numbers as you see here drawn upon this Instrument This may suffice to have spoken of the substantiall part or Body of the Table it selfe in the next place followes the circumstantiall part or Appendants thereof to be handled XIII The Appendants of the Table are either externall and placed without it or internall and placed within it XIV Those placed without it are either so placed at the top above it or on each side thereof viz. at the ends of the Fractions XV. The Appendant placed at the top above it is the whole length of the Table divided into 36 equall parts numbered by 1 2 3 c. to 36 and signed by six Alphabets each of them consisting of six letters viz. A B C D E and F. And all these Alphabets taken together are bereafter for distinction sake called the Top-rank of Alphabets XVI The two ends of this Top-rank ought to be conceived to joyn interchangably to each other in like manner as if the Alphabets and Letters were placed in a Circle For Example If B in the fourth Alphabet were propounded and I were to account from that letter four Alphabets and three letters towards the right hand The letter A in the fift Alphabet makes one Alphabet and A in the sixt Alphabet is the second Alphabet but now because in proceeding to account another Alphabet I shall go beyond the right end of the line for the third Alphabet I take A in the first Alphabet and for the fourth I take A in the second Alphabet and so have I all the four Alphabets demanded And then I account three letters from the last A taken which leads me to the letter D in the said second Alphabet being the letter required In like manner if I were to proceed towards the left hand and C in the second Alphabet were the term given from whence I am to account three Alphabets
and five letters D in the first Alphabet is the first Letter in that account D in the last Alphabet is the second and D in the fift Alphabet is the third from which if I account five letters the same way viz. towards the left hand at last I shall fall upon E in the fourth Alphabet which is the letter required XVII The Appendants placed on each side of the Table are so placed on the right hand or on the left XVIII That placed on the right hand is another like rank of Alphabets which is hereafter called the side-rank of Alphabets XIX The two ends also of this side-rank ought to be conceived to joyne interchangably to each other as those of the top-rank For Example If D in the third Alphabet were propounded and it be demanded from thence to account downwards five Alphabets and four letters descending downwards I finde C in the fourth Alphabet to be the first C in the fifth the second C in the sixth the third and then C in the first Alphabet is the fourth and C in the second Alphabet is the fifth from whence if I account four letters at last I fall upon A in the third Alphabet which is the letter required so likewise if E in the second Alphabet be given and it be required to account upwards four Alphabets and three letters first F in the first Alphabet is the first F in the last Alphabet is the second F in the fifth Alphabet is the third and F in the fourth is the fourth from whence I account three Letters upwards which guides me to the letter C in the said fourth Alphabet being the letter desired XX. The Appendant placed on the left hand is nothing else but a rank of Numbers expressing the three figures of the first Hundred of every Fraction respectively and serveth for the more readie finding out of numbers upon the Scale as shall be more clearly taught hereafter XXI The Interval appendants placed within the Table are either Alphabets or Parallels The Alphabets are nothing else but the top-rank of Alphabets ten times repeated in the body of the Table The Parallels are certain pricked lines which crosse one another at right angles and are either Perpendiculars or Transversals XXII The Perpendiculars are pricked lines drawn downwards through the Bodie of the Table from every division of the top-rank of Alphabets XXIII The spaces comprehended betwixt every two perpendiculars are called Intervals XXIV The Transversals are also pricked lines drawn under the top-rank and likewise under every Fraction respectively whereof that placed under the top-rank is called the Chief Transversall And each of those Transversals placed under the Fractions respectively is termed the Transversall of the Fraction under which it is so placed and therefore the right end of each of them is to be conceived to joyn with the left of the next under it as also the left end of each of them to joyn with the right end of that next above it In like manner as the Fractions are said to do in the ninth Rule aforegoing XXV The parts of the Transversals comprehended in the Intervals betwixt every two of the Perpendiculars are by points divided into six equall parts called Digits and each of those six parts are again supposed to be sub-divided into six other equall parts termed Minimes CHAP. II. Numeration upon the Scale of Numbers I. THus far the description and construction of this Instrument the use followes which consists in Numeration and Application II. Numeration upon the Table teacheth how to finde out numbers and discover distances thereupon and it is performed either upon the Scale of Numbers or upon the Alphabets and Transversals III. Numeration upon the Scale of Numbers is to finde thereupon any number propounded or any point thereof being assigned to discover the figures or number represented at that point IV. If a number consisting of five places or more be given to finde the point upon the Table where that number is represented proceed thus First finde amongst the numbers placed at the left ends of the Fractions the three first figures of the number given or if you cannot find the three figures exactly take that nūber amongst them which being less cometh neerest unto them this done upon that Fraction make search for the Hundred which begins with those three first figures of the number propounded and for the fourth figure count so many Tenths of that hundred for the fifth figure so many units of the tenth last taken all this performed that place is the point at which the nūber propounded is represented Example let 11422 be the number given to be found upon the Scale of Numbers here 114 the three first figures thereof are found at the left end of the third Fraction which leads me to the first hundred of that Fraction signed by the same figures then for 2 the fourth figure of the number given I count 2 tenths frō the beginning of that hundred which brought me to the second tenth of that hundred for 2 the last figure of the number given I count 2 units of the tenth last taken which leads me to the point of the Scale of Numbers placed just above the letter q which point is the place where the number propounded is represented upon the same Scale so likewise if the number given did consist of more places than five it would be represented at the same point as 11422004500 or 1142212974 are also there represented But if the number given were 32292 because I cannot finde exactly the three first figures thereof at the left ends of the Fractions as before I take 317 which being lesse comes neerest unto them and guides me to the 19 Fraction upon which finding the three first figures of the number given at the sixth hundred thereof I take those three figures to be there represented and proceeding as before I finde the last number given to be represented upon that 19 Fraction at the point placed just above the letter g. Again if the number propounded were 32205 you shall finde it represented upon the same 19 Fraction just above the letter y for in this case there being a cypher in the place of tenths no tenth is to be taken in the discovery of that or the like number upon the Scale V. If a number consisting of four places or over and besides the four places having a cipher in the fifth place be propounded it may be discovered upon the Scale in like manner as the first four figures are found out by the last Rule So if 1142 or 114200000 were given they would be both represented upon the third Fraction at the second tenth of the first hundred as before and if 32290000 or 322905321 were given they would be found upon the 19 Fraction at the ninth tenth of the sixt hundred of that Fraction VI. If a number consisting of three places or besides the three places having ciphers in the fourth and fifth places thereof were propounded it
sold for 23 s. 6 d. the quarter Prop. 3. One number being given to be multiplied by another given number to finde the product In multiplication there are four terms Geometrically proportionall whereof the first is alwayes an unity or 1 the multiplicator and multiplicand are the two meanes and the product is the fourth term demanded for as 1 is to the multiplicand so is the multiplicator to the product or as 1 is to the multiplicator so is the multiplicand to the product Now an unity or 1 being alwayes represented at the beginning of the Scale of numbers as appears by the eighth Rule of the second Chapter you need not there place a pointed Chesseman to denote it being notorious of it self but onely where the multiplicand or multiplicator are found upon the said Scale when therefore any such proposition as that above is made placing one pointed Chesseman upon the Multiplicator and another upon the Multiplicand as also two plain Chessemen upon the side rank at the right ends of their respective Fractious and taking the beginning of the line to be alwayes the first term in the question by the directions given in the first proposition of this Chapter find out a fourth term to those three terms propounded which done that fourth term is the product you look for Example 287 being given to be multiplied by 139 the three termes given are 1 139 287 Unto which if a fourth be sought for by the instructions delivered in the first Proposition of this Chapter it will be found upon the 22 Fraction at four digits and three minimes of the 23 Interval which point gives you these figures 39893 the product required What is a wedge of gold worth that weigheth 4 ounces 6 peny-weight and 15 grains at 3 l. 3 s. 2 d. the ounce Here the weight of the wedge after Reduction is 4.3315 and the rate of an ounce is 3.1582 and therefore the termes given are 1 4.3315 3.1582 Whose fourth term I discover to fall upon the sixt Fraction at two digits and one minime of the 33 Interval which gives me these figures 1368 whereof I take the two first for pounds Sterling and the other two for the decimal Fraction of a pound Sterling which after Reduction amounts to 13 s. 7 d. and somewhat more for common-reason dictates to me that it cannot be 136 l. nor so little as 1 l. and therefore I conclude the product to be 13 l. 13 s. 7 d. as before being the value of the 4 ounc 6 peny w. and 15 grains the term required In Multiplication observe these Rules In performing Multiplication you alwayes operate upon the Table towards the right hand and upon the side-rank of Alphabets alwayes downwards for an unity or 1 being alwayes the first term you alwayes begin the account of the Alphabets and Letters comprehended betwixt 1 and the term placed next to it from the left side of the Table which will alwayes tend towards the right hand and then by consequence in laying down the like distance betwixt the other term and the product you are to proceed the same way viz. towards the right hand for the like reason it is that you are alwayes to work downwards upon the side-rank because there also you are to begin your account from the first Fraction being that whereupon 1 the first term is represented All which plainly appears by the premised examples The digits and minimes which are to be taken off from the points of the termes given are alwayes so to be taken off upon the left hand and never upon the right Here by the termes given are intended onely the Multiplicand and Multiplicator for the first term viz. 1 hath no digits or minimes attending it being represented upon the first perpendicular at the beginning of the Scale of Numbers but the Multiplicand and Multiplicator may have digits and minimes attending them which are alwayes to be taken off upon the left hand according to the direction of this Rule and as is manifest by the examples aforegoing Prop. 4. One number being given to be divided by another given number to finde the Quotient As in Multiplication so in Division there are four termes Geomettically proportionall whereof the Divisor is alwayes the first an unity or 1 and the Dividend the two mean terms and the Quotient is the fourth term required for as the Divisor is to 1 so is the Dividend to the Quotient or as the Divisor is to the Dividend so is 1 to the Quotient and here as in multiplication an unity or 1 being alwayes one of the terms you need not thereat place a pointed Chesseman to denote it but onely where the Divisor and Dividend are found upon the Scale as also two plain Chessemen upon the side rank at the right ends of their respective Fractions and then taking the Divisor to be alwayes the first term in the question by the directions given in the first Proposition of this Chapter finde out a fourth term to those three terms propounded which done that fourth term is the Quotient required Example 39893 being given to be divided by 287 the three termes given are 287 1 39893 Or 287 39893 1 Unto which if a fourth term be found out by the instructions given in the said first Proposition of this Chapter it will be found at the second Hundred of the sixt Fraction which gives this number 139 for the quotient required so likewise if 3989348 were given to be divided by 287 the first three figures of the quotient would be found 139 as before but in that case you are to annex unto them two ciphers to make the quotient consist of five places for that in this question the divisor may be written under the dividend five times as appears by the posture of the numbers hereunto annexed 3 9 8 9 3 4 8 2 8 7 And therefore in that case the quotient required will be 13900 which case with divers others as they happen the Artist after he perfectly understands by practice the nature of this Instrument will be well able by discretion to order as occasion shall serve If a Pipe of wine containing 126 Gallons cost 25 l. 14 s. 5 d. what is the price of a Gallon thereof according to the same rate Here the terms in the question after Reduction are 126 25.721 1 For in this case the question is if 126 Gallons give 25.721 how much will one Gallon yield wherefore proceeding according to the directions aforegoing I finde the fourth term to reside upon the 12 Fraction at three digits and five minimes of the fift Interval where I find these Figures represented viz. 20377 which in reason I conceive to be a decimal Fraction of a pound Sterling and after reduction thereof discover it to represent 4 s. 1 d. and so much is the value of every Gallon in the Pipe and the Quotient required In Division for taking off the digits and minimes observe this Rule When the digits and minimes are