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-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
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-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-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
For the performance thereof observe these ensuing directions By the Instructions delivered in the second Chapter aforegoing find the numbers given upon the Scale of Numbers setting at each of them a pointed Chesseman as also three other plain Chessemen upon the side rank of Alphabets at the left ends of their respective Fractions This done if by the fourth and fifth Rules of the last Chapter you will discover a fourth term to the three termes propounded you shall there finde the number you look for Example If 12980 represented upon the fift Fraction at the point f be the first term given 32192 represented upon the nineteenth Fraction at the point g the second and 18452 represented upon the tenth Fraction at the point h the third the fourth term by the fourth and fifth Rules of the last Chapter will tall upon the four and twentieth Fraction at the the point k by the last Rule of the second Chapter gives you the number 45907 the fourth proportional required so if 45907 vvere given for the first term 18452 for the second and 32292 for the third in working upwards upon the side-side-rank and towards the left hand upon the table the fourth term will be found to rest upon the fift Fraction at the point f representing 12980 as before In like manner if g viz. 32292 were the first term given h viz. 18452 the second and k viz. 45907 the third the fourth term would fall upon the 15 Fraction at the point l but because in that case you go beyond the table towards the right hand you are to take instead thereof according to the direction given in the third example of the fift Rule of the last Chapter the point m upon the 16 Fraction which represents 26231 the fourth proportionall required so likewise if h representing 18452 be the first term g representing 32292 the second and f representing 12980 the third the fourth term will reside upon the 14 Fraction at four digits and four minimes of the 30 Interval viz. at the point n. Howbeit in this case also you are not take that point but because you overshot the Table upon the left hand you are instead thereof to take the digits and minimes of the Fraction next above it in the same Interval viz. the point p upon the 13 Fraction which represents 22715 the fourth proportionall required according to the fourth example of the said fift Rule of the last Chapter Again if f viz. 12980 be the first term g viz. 32292 the second and k viz. 45707 the third In this case the fourth term will according to the fift example of the fift Rule of the last Chapter at last reside upon the third Fraction at four digits and four minimes of the third Interval viz. at the point q which by the fourth Rule of the second Chapter represents 11422 the fourth proportionall sought for On the other side if k viz. 45907 were the first term h viz. 18452 the second and f viz. 12980 the third the fourth term will according to the last example of the said fift Rule of the last Chapter at last fall upon the 27 Fraction at four digits and four minimes of the 30 Interval viz. at the point r which represents these figures 52169 Howbeit because common sence tells me that the fourth term to the other three last given termes cannot be so great nor yet so little as 521.69 therefore I conclude the term required to be in this case 5216.9 or 5217 ferè If a Chest of Sugar that weighs 7 C. 2 qu. and 17 lb. cost 36 l. 14 s. 10 d. what is the price of 2 C 1 q and 4 lib. thereof according to the same rate Here after the reduction of the broken parts of the number given into Decimals the first term is 7.6518 the second 36.7417 and the third 2.2857 with which three terms working upon the Table according to the precepts before premised I find the fourth term to be fixed upon the second Fraction at two digits and two minimes of the 17 Interval which point yields me these figures â—0975 whereof I take the two fiâ—â—â— viz. 1â— for 10 l. and the other three for a decimal Fraction of a pound Sterling which after Reduction amounts to 19 s. 6 d. And therefore I conclude that 2 C. 1 qu. and 4 lb. of that Sugar is worth 10 l. 19 s. 6 d. which was the term required for when I have those five figures given me upon the Table for the fourth term common reason tells me they cannot signifie 109.75 for that were too great nor 1.0975 for that were too little and therefore in this case I take 10.975 viz. 10 l. 19 s. 6 d. being the fourth term sought for Now from this example and the rest before premised for the ready working of the digits and minimes of the three termes propounded this generall Rule or Corollary may be inferred In all questions that may be performed by the Golden Rule the digits and minimes to be taken off from the first term are alwayes so taken off from that side of the first term which inclines towards the second term and then the digits and minimes of the other two termes are alwayes taken off upon the contrary side to those of the first term as is manifest by all the examples aforegoing which Rule being alwayes duly observed you may with greater confidence proceed to resolve any question propounded And because this Corollary is alwayes to be kept in memory I have expressed it in this Distick Aurata in Regula bis laeva aut dextra petatur Dum contragreditur Terminus ipsâ— prior Thus Englished For th' Rule of Three each hand may be pursued two times Whiles that the foremost term againsâ— them alwayes climes Prop. 2. To three numbers given to finde a fourâ—â— in an inversed proportion This Rule of Three Inverse is thâ— same with that of the Rule of Three direct if instead of the first term you take the third term given to be the first in the question by transposing the last into the place of the first Example If when the price of wheat is 40 shillings the quarter a peny white loaf weighs 8 ounces and 9 peny-weight how much ought a peny white loaf to weigh when wheat is at 23 shillings six pence the quarter Here the termes given are viz. 40 the first 8.45 after Reduction the second and 23.5 the third which as they are propounded in the question stand in this form 40 8.45 23.5 But being inverted stand thus 23.5 8.45 40 Unto which three having by the directions aforegoing made search upon the Table for a fourth proportionall you shall find it to fall upon the sixt Fraction at three digits and three minimes of the 25 Interval which point affords you these Figures 14.383 which after Reduction amount to 14 ounc 7 peny-weight 16 grains being the term required for so much a peny white loaf ought to weigh according to the abovesaid rate when wheat is
taken off from the right hand of the Divisor take the digits and minimes placed on the left hand of the Dividend and when on the left hand of the Divisor take them from the right hand of the Dividend For the ready discovery and taking off the digits and minimes in Multiplication and Division let this Hexameter be remembred Multiplicá laevè sed divide dextrifinistre Multiply by th' right hand with both the hands divide Prop. 5. Two numbers being given to find a third Geometrically proportionall unto them and to three a fourth and to four a fift c. This Proposition may be resolved by the directions given in the fixt Rule of the last Chapter for having two termes given and placing Chessemen upon them as also at the right ends of their respective Fractions as in the aforegoing Propositions if you by the said sixt Rule find a third term to the two other given terms that third is the term you look for Example If 2 and 4 be the two terms given a third proportional unto them by the fixt Rule of the last Chapter will be found upon the 33 Fraction at two digits and two minimes of the 19 Interval which point represents 8 the third term required In like manner you may proceed to find a fourth term to those three which will be 16 and a fift to those 4 terms found which wil be 32 c. And so you may by this means erect a rank of numbers Geometrically proportionall which in Arithmetick is called Geometricall Progression Prop. 6. To extract the Square root of any number under 10000000000. FIrst prepare the Square-number given for extraction as in Vulgar Arithmetick by subscribing a point under each other figure beginning with the last first so these numbers following being given for extraction and prepared as aforesaid will stand thus 3 2 8 1 5 3 2 2 5     1 4 5 2 7 5 3 2 2 5      And so many points as are in that manner subscribed of so many figures will the root consist viz. in these examples of five figures Place this between page 64 and 65 Example Let 328153225 be the square number given the root thereof required This number admits five points to be subscribed under it as appears before and is found upon the 19 Fraction at five digits of the 21 Interval also the half distance thereof by the seventh Rule of the last Chapter is likewise discovered at two digits and three minimes of the 11 Interval as also the half distance in the side rank by the tenth Rule of the said last Chapter upon the tenth Fraction wherefore if I place another pointed Chesseman upon the said 10 Fraction at two digits and three minimes of the 11 Interval being the same with the half distance upon the Fraction of the number given that point will discover these figures 18115 being the root required But when the first point towards the right hand happens to fall under the second figure of the number given and there be also an odde Fraction upon the side rank proceed as in the last Rule to find the half distances upon the Fraction of the number given and also upon the side rank Howbeit to discover the true Fraction upon which the root in such case is to be found account three Alphabets downwards from the half distance upon the side-rank in regard the first figure of the number given hath no point under it there place another plain Chesseman Also in regard of the odde Fraction upon the side rank account in in like manner three Alphabets towards the right hand from the half distance upon the Fraction where the number given is scituate and there likewise place another pointed Chesseman All this performed in the angle of position where the last placed Chesseman meets with the true Fraction of the root before found upon the side rank you shall discover the root required Example Let the Square root of the number subscribed be desired 1 4 5 2 7 5 3 2 2 5      This number is found upon the sixt Fraction at one digit of the 31 Interval and the half distance thereof at 3 minimes of the 16 Interval as also the half distance in the side rank upon the 3 Fraction but because I find no point under the first figure I account upon the side rank 3 Alphabets downwards from that 3 Fraction and thereupon set another plain Chesseman at the left end of the 21 Fraction Also in regard in this case the fift Fraction in the side rank is an odde Fraction I likewise account three Alphabets towards the right hand from the half distance upon the Fraction where the number given resides and thereupon place another pointed Chesseman at three minimes of the 34 Interval All this thus acted I finde the Chesseman last placed to meet wtih the 21 Fraction being the true Fraction of the root as aforesaid at three minimes of the said 34 Interval where having placed another pointed Chesseman I discover these figures 38115 being the root sought for And here let me give you this Rule once for all That whensoever there is no point under the first figure of the number given you are to account upon the side rank three Alphabets downwards from the half distance there found and when there is an odde Fraction upon the side rank you are likewise to account three Alphabets upon the Fraction of the number given towards the right hand from the half distance found upon that Fraction as you finde it practised in the last Example Note also that when you have more figures discovered upon the Table for the root then the number given requires those that exceed are a decimal Fraction belonging to the root likewise when a mixt number is given you are to subscribe the points only under the significant figures thereof And therefore if these two numbers viz. 4 3 6 2 3   and 1 7 6 2 8     were given the Square root of the first would be 208.86 and of the other 41.985 All which observations and the like after some practice upon the Table common reason will dictate unto you Prop. 7. To extract the Cube-root of any number given under 1000000000000000. Prepare the Cube-number given to be extracted as in Vulgar Arithmetick by subscribing a point under every third figure and beginning with the last first So the number hereafter following prepared for extraction will stand thus 2 2 1 9 8 9 4 0 6 6 1 2 5         And so many points as are in this manner subscribed of so many figures will the root consist according to the aforesaid observation of the Square-root Place a pointed Chesseman at the number given and likewise another upon the side rank at the left end of the Fraction upon which the number given is scituate this done by the eighth Rule of the
last Chapter finde the third part of the distance betwixt the point of the number given and the left end of the Fraction whereon it is placed as also upon the side rank by the eleventh Rule of the same Chapter the third part of the distance betwixt that Fraction and the first Fraction All this performed when the first figure of the number given towards the left hand hath a point placed under it and you find no odde Fraction or Fractions upon the side rank then the point where the third part of the distance of the Fraction of the number given meets with the Fraction of the third part of the distance in the side rank will discover unto you the Cuâ—â—-â—oot required Example Let 2219894066125 be the Cube number to be extracted this number admits five points to be subscribed under it as appears above and is found upon the 13 Fraction at five digits of the 17 Interval also the third part of the distance c. by the eighth Rule of the last Chapter at three digits and four minimes of the sixt Interval and likewise the third part of the distance in the side rank by the 11 Rule of the last Chapter upon the 5 Fraction wherefore if I place another pointed Chesseman upon the said 5 Fraction at 3 digits and 4 minimes of the 6 Interval being the same with the third part of the distance upon the Fraction of the number given that point represents these figures 13045. being the root you looke for But when the first point towards the right hand happens to fall under the second or third figures of the number given and there be also one or two odde Fractions upon the side rank proceed as in the last Rule to find the third part of the distance c. upon the Fraction of the number given and also upon the side rank Howbeit to discover the true Fraction upon which the root in such case is to be found for every figure which the said first point hath towards the left hand being never more then two account two Alphabets downwards from the third part of the distance found upon the side rank and there place another plain Chesseman also for every odde Fraction which will never likewise exceed two account in like manner two Alphabets towards the right hand from the third part of the distance upon the Fraction where the number given is scituate and there likewise place another pointed Chesseman All this performed in the angle of position where the last placed Chesseman meets with the true Fraction of the root before found upon the side rank you shall discover the root required Example Let the Cube-root of the number under-written be desired 6 4 1 9 2 1 9 2 0 6 4        This number is found upon the 30 fraction at two digits and five minimes of the third intervall and the third part of the distance c. by the eight Rule of the last chapter at four digits four minimes and somewhat more of the first intervall as also the third part of the distance in the side rank by the 11 Rule of the last Chapter upon the 10 Fraction but because I find the first point of the number given to have a figure before it towards the left hand I account two Alphabets downwards from the third part of the distance found upon the side rank viz. From the 10 Fraction to the 22 Fraction and there place another plain Chesseman which 22 Fraction is the Fraction whereupon the root is to be found and therefore I place there another plain Chesseman Again for the two odde Fractions viz. the 28 and 29 I account four Alphabets towards the right hand from the third part of the distance upon the Fraction of the number given and there likewise place another pointed Chesseman All this performed I finde the Chesseman last placed to meet with the 22 Fraction being the true Fraction of the root as aforesaid at four digits and four minimes and somwhat more of the 25 Interval where having placed another pointed Chesseman I discover these figures 4004 being the root required So 172.68 being propounded to be extracted the Cube-root thereof will be found upon the 27 Fraction at three digits of the 31 Interval where you shall finde these figures represented 55686 whereof common sence tells me 5 are the Integers and the rest of the figures are a decimal Fraction of the root so as in that example the true root sought for after separation of the Integral part from the broken part thereof is 5.5686 Here I might proceed to shew a further use of this Instrument for the resolving of divers other Propositions in Arithmetick and Geometry as Betwixt two numbers given to discover one two or more mean proportionals To three numbers given to finde a fourth in a duplicated or triplicated proportion To work Rules of plurall Proportion The double Golden Rules direct and Inverse The Rules of Fellowship Alligation False position c. But I have deemed these at present sufficient to satisfie the curiosity of the Practitioner who in obtaining the knowledge of these if he esteem them worthy his paines may be thereby so perfectly acquainted with the nature of the Table that he may afterwards be able to resolve not only the Propositions above-mentioned but all others which may be performed by Arithmetick either Vulgar or Artifitial And perhaps upon further scrutiny some others also which cannot be resolved without Symbolicall Arithmetick usually called Algebra All which I will hereafter eudeavour also to explain as vacancie froÌ„ other more pertinent affairs will permit together with the Fabrick and Use of a Trigonometricall Table of Proportion for the resolution of Plain and Sphericall Triangles if I shall finde the paines herein already taken may obtain gratefull reception This Tractate being indeed onely intended as an Eschantillon or glimpse of that which may be performed upon this and the other above-said Table applicable to Trigonometry Praestat pauca avide discere quà m multa eumtaedio devorare Erasm in Coll. rel FINIS
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
is represented at the hundred signed by the same three figures So 114 or 11400 or 11400000 or 114005321 are all-represented at the first hundred upon the third Fraction and 322 or 322000 or 32200273 are found at the sixth hundred of the 19 Fraction VII If a number consisting of two places or besides the two places having ciphers in the third fourth or fifth places thereof were propounded it is represented at the hundred which hath those two figures and a cipher annexed unto them So if 13 or 13000 or 130000 or 1300000 or 13000734 were given they are all represented at the first hundred of the fifth Fraction VIII If a number of one figure or place or besides that one place having ciphers in the second third fourth or fift places thereof were given it would be represented at the hundred which is signed by that one figure and two ciphers annexed unto it So if 1 or 10 or 100 or 1000 or 10000 or 100000 or 10000426 c. were assigned they would be all represented at the beginning of the Scale signed by 100 so likewise if 3 or 30 or 300 or 3000 or 30000 or 300000 or 30000342 were propounded they would be found at the fourth hundred of the 19 Fraction c. IX When the number propounded is mixt reduce the broken part thereof to a Decimall Fraction and then finde the whole upon the Scale as if it were a whole number So 5 3 4 being given and the broken part thereof viz. 3 4 reduced to a decimall viz. 75 the intire number given after such reduction will be found 5.75 which is represented at the 13 hundred of the 28 Fraction In like manner 12 l. 13 s. 5 d. being propounded and 13 s. 5 d. the broken part thereof reduced to the Decimall 6708 that intire number will stand thus 12.6708 which is represented at the seventh tenth of the fift hundred of the fourth Fraction The great use and benefit of reducing ordinary broken numbers to Decimals is now so commonly known to most Artists that I conceive it not necessary here to insist long thereupon Only I will here insert certain Tabular Scales which may serve for the ready reduction of compound Fractions viz. of Money Weight Measure and Time which usually incumber the Practitioner to Decimals Upon these Tabular Scales you shall finde the compound Fractions described in the upper Scales thereof and in the lower their respective Decimals the first of them being broken into ten equall parts or Fractions reduceth the Fractions of Money Troy-weight the Integers thereof being a pound sterling for Money and an ounce Troy for Troy-weight The second broken off into two Fractions onely reduceth Avoirdupoiz Great weight The third Avoirdupoiz Little weight and all other measures or weights which divide themselves into halves quarters c. And the fourth is made for the reduction of Time Dozens and Inches so upon the first Tabular Scale the Decimall of 8 s. 3 d. 3 q. is .4156 and the Decimall of nine peny weight and seven grains is .4646 Also upon the second the Decimall of 3 quarters of C. 8 lb. and 7 ounces is .825 The like reduction may also be made upon the other two Tabular Scales according to their severall and respective divisions Howbeit if you please yet to have a more compendious way for the reduction of the Fractions of Money and Troy-weight you may do it by the first of the double Scales drawn at the left end of the Table of Proportion by which pence and farthings for money and grains and half grains for Troy-weight may be readily reduced there being no great difficulty in reducing shillings and peny-weights to Decimals as is well known to all such as are competently acquainted with the nature of Fractions The other little Scales there also placed being for Avoirdupoiz weight and Time give you the Decimall of one quarter which is to be added to the Decimals of one quarter viz. 25 or of two quarters viz. 50 or of three quarters viz. 75 as the question may be propounded X. When the term propounded is a Fraction or broken number convert it to a Decimall and then finde it upon the Scale of Numbers as if it were a whole number So 1 4 or 25 is found at the fixt hundred of the 15 Fraction and 8 s. 3 d. 3 q. or 4156 at the sixth tenth of the seventh hundred of the 23 Fraction XI When a point upon the Scale of Numbers is assigned to finde out the number represented by that point invert the rules aforegoing so shall you discover the number or figures you look for So if the point q were given upon the third Fraction the number or figures represented by it will be found 11422 Also if the point g were assigned upon the 19 Fraction the number or figures represented by it are 32292 as appears by the two examples of the fourth Rule aforegoing The like also may be said of all the other examples above in this Chapter produced CHAP. III. Numeration upon the Alphabets and Transversals I. Numeration upon the Alphabets and Transversals teacheth how to discover distances betwixt points or termes assigned thereupon II. Three letters in either rank of Alphabets being propounded to finde a fourth which shall bear like distance from the third that the second bears from the first proceed thus Count the intire Alphabets and Letters which are intercepted betwixt the letters of the first and second termes then from the letter of the third term account as many intire Alphabets and Letters the same way that done the letter placed next beyond the last letter so accounted is the letter required Example In the top-rank of Alphabets let E in the first Alphabet A in the third and B in the fourth be given In this case I place three pointed Chesse-men in the Chief Transversal viz. one under E another under A and a third under B This done and I finding one Alphabet and one letter betwixt the two first termes E and A and accounting the like from B in the fourth Alphabet towards the right hand at last I fall upon D in the fift Alphabet which is the letter required where I also place another pointed Chesseman So if C in the third Alphabet D in the fift and B in the sixt be propounded C in the second Alphabet will be the fourth term you look for according to the 16 rule of the first Chapter Again if F in the fift Alphabet C in the third and D in the first be given In this case working towards the left hand the fourth term will be A in the fift Alphabet likewise if B in the third Alphabet be the first term D in the first be the second and C in the fourth be the third term the fourth term will be E in the second Alphabet c. After the same manner in the side-rank of Alphabets three letters being given a fourth may be discovered by this Rule and the 19 Rule