Ludus Mathematicus: OR, THE MATHEMATICAL GAME: Explaining the description, construction,, and use of the numerical 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 versed in the Mathematics) and finding also by experience, that it may prove useful for others (and, Bonum quò communas eò melius) I have permitted it to launch into the Ocean of censure: Howbeit, I present it chief to such as (in some competent manner) have already acquainetd themselves with the modern use of Arithmetic, 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 kind of Artificial Arithmetic, it may (at first) seem somewhat more difficult. But unto such as are not at all versed in Arithetick, I may object Plato's Inscription placed over the door of his Academy concerning Geometry (including also Arithmetic) 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 clearly do, (especially towards the beginning of the Scale) when the term or terms out of which the Question is to be produced, are rational 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 mixed 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 (nevertheless) will occasion no inconvenience in the practice of this Instrument, the broken part of a number of such an extent being not considerable in Questions of ordinary practice, as is well known to all Artists: This advertisement I have thought fit to premise, lest it might seem 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 numerical 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 find a fourth in a direct proportion ibid. 2. To three numbers given, to find a fourth in an inversed proportion. p. 52 3. One number being given to be multiplied by another, to find 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 proportional unto them, and to three, a fourth, and to four, a fift, etc. 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. interval r. internal. 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 numerical Table of Proportion. I. A Table of Proportion is an Instrument framed by Logarithms, and invented for the more easy resolving of Arithmetical and Geometrical Operations. In Natural or Vulgar Arithmetic, the Propositions are resolved by using the Numbers themselves, as if 4 were given to be multiplied by 2, we say, two times four makes 8, the Product. In Artificial Arithmetic, if the same Question were propounded, instead 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 sum 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, Add are Subtract, 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 Arithmetic and Geometry without charging the mind or memory with any thing, which may seem burdensome or distasteful. II. This Instrument is twofold, numerical or trigonometrical. III. The numerical Table of Proportion is an Instrument, by help whereof, and of certain movable Chesse-men, all Questions Arithmetical and Geometrical (performed by Multiplication, Division, or the Golden Rule, and not trigonometrical) together with mean Proportionals, & the Extraction of the Roots of all Square-numbers under 11 places, and of all Cubenumbers under 16 places, as well in mixed and broken, as in whole numbers (when the term required exceeds not 100000) are with great ●ase and exactness Resolved. For we ●ntend not here to meddle with any questions, that are performed by the Doctrine of Triangles, referring them ●o be handled in the use of the Table of Proportion trigonometrical. iv Of the numerical Table of Pro●●●tion these things offer themselves to 〈◊〉 considered, viz. The Description and construction, or the Use. V For the more plain describing of ●●is Instrument, it may be said to consist 〈◊〉 two parts, viz. The Body of the Ta●●e itself and substantial part, or the Appendants and Circumstantial part thereof. VI The Body of the Table itself is a Scale of unequal 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 cross-staff, and is well enough known to all modern Artists. VII. A Fraction of the Scale of Numbers is an equal 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, numbered at their right ends by 1, 2, 3, etc. to 36, of which the part signed at the left end thereo● by 100, is the first Fraction, that sign by 100, is the second, etc. VIII. Each of these Fractions consists of three lines and two spaces: so the pricked line which you find 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 join 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 join 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 join 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 entire 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 join with the right end of that above it, signed by 35 E. and so of the rest, in ascending upwards, until you mount to the beginning of the Scale. X. The entire Scale of Numbers is first divided into a thousand unequal 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, etc. XI. Each of these Hundreds are again subdivided into ten other unequal 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 real 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 guess 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, etc. 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) guess 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 equal parts, containing in length the hundred part of the whole intended Scale of Numbers (which Scale of equal parts must be supposed to be divided into 1000 equal 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 equal parts to the said 43; this done, if you apply that extent of the compasses towards the right hand from the beginning of your intended Scale of Numbers, the movable 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 equal to this whereof we treat; this Scale being entirely taken together, as one continued Scale, according to the ninth Rule aforegoing) it contains 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 movable 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 equal 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, until 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 substantial part, or Body of the Table itself; in the next place follows the circumstantial part, or Appendants thereof to be handled. XIII. The Appendants of the Table are either external, and placed without it; or internal, 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 Appendent placed at the top above it is the whole length of the Table divided into 36 equal parts, numbered by 1, 2, 3, etc. 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 join interchangeably 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 sixth 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 join interchangeably 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 find 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 Appendent 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 ready 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 cross one another at right angles, and are either Perpendiculars or Transversals. XXII. The Perpendiculars are pricked lines drawn downwards through the Body 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 transversal: And each of those Transversals placed under the Fractions respectively, is termed the transversal of the Fraction, under which it is so placed; and therefore the right end of each of them is to be conceived to join with the left of the next under it; as also the left end of each of them to join 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 equal parts, called Digits, and each of those six parts are again supposed to be subdivided into six other equal parts, termed Minims'. CHAP. II. Numeration upon the Scale of Numbers. I. THus far the description and construction of this Instrument; the use follows, which consists in Numeration and Application. II. Numeration upon the Table teacheth how to find 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 find 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 find the point upon the Table, where that number is represented; proceed thus: First, find 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 number amongst them, which being less, cometh nearest unto them; this done, upon that Fraction make search for the Hundred, which gins 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 number 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 from 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 find exactly the three first figures thereof at the left ends of the Fractions, as before, I take 317, which being less, comes nearest 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 find 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 find it represented upon the same 19 Fraction just above the letter y, for (in this case) there being a cipher 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 sixth hundred of that Fraction. VI If a number consisting of three places, or (besides the three places) having cyphers 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 cyphers 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 cyphers 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 cyphers annexed unto it: So if 1, or 10, or 100 or 1000, or 10000, or 100000, or 10000426, etc. 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, etc. IX. When the number propounded is mixed, reduce the broken part thereof to a decimal Fraction, and then find 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 decimal, viz. 75; the entire 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 decimal 6708, that entire 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 encumber the Practitioner) to Decimals. Upon these Tabular Scales you shall find the compound Fractions described in the upper Scales thereof, and in the lower their respective Decimals; the first of them (being broken into ten equal 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 only) reduceth Avoirdupoiz Great weight: The third Avoirdupoiz Little weight, and all other measures or weights, which divide themselves into halves, quarters, etc. And the fourth is made for the reduction of Time, Dozen, and Inches: so upon the first Tabular Scale the decimal of 8 s. 3 d. 3 q. is .4156, and the decimal of nine penny weight and seven grains is .4646. Also upon the second, the decimal 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 several 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 decimal 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 decimal, and then find it upon the Scale of Numbers, as if it were a whole number: So 1/4 or 25 is found at the fixed 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 find 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 terms assigned thereupon. II. Three letters in either rank of Alphabets being propounded, to find a fourth, which shall bear like distance from the third, that the second bears from the first; proceed thus, Count the entire Alphabets and Letters, which are intercepted betwixt the letters of the first and second terms, then from the letter of the third term account as many entire 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 terms 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 sixth 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, etc. 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 of the first Chapter, which being plain, I omit to exemplify: And in all these Cases and the like, it mattereth not whether you account the Alphabets and letters from the first term to the second, or to the third; as in the last example, if I account an Alphabet from the first term to the third towards the right hand, and then the like from the second term the same way, the fourth term will then also fall at E in the second Alphabet, as before: The like experiment you shall also find in the side-rank, etc. In like manner, if two letters were given, and it be desired to find a third, which may bear like distance from the second that the second bears from the first; In this case also count as many letters from the second towards the third, as you find intercepted betwixt the first and the second, and so shall you likewise have your desire. For example, if E in the first Alphabet, and A in the third were given, the third letter will fall to be C in the fourth Alphabet, etc. The like experiment may be also acted upon the side rank, as may plainly appear without further instruction. III. When three points are assigned upon the Chief transversal, to find out a fourth, which may bear the like distance from the third, that the second bears from the first, proceed thus; Having placed (as before) a Chesseman at each of the points given, and (by the Rule aforegoing) found the letter under which the fourth term is likely to fall, and there also placed another Chesseman, as before; draw back that last Chesseman quite thorough the last letter so counted, and place it upon the perpendicular, where that last letter gins; this done, observe how many entire digits are comprehended betwixt the first given point, and the next perpendicular towards the second point, as also how many such digits are contained betwixt the second given point, and the next perpendicular towards the first given point, and to these add the entire digits that are found betwixt the third given point, and the next perpendicular towards the same hand, according to which the digits of the second point were taken off. All this performed, if you add all these three numbers of digits together, and according to that aggregate advance the last Chesseman forward again, and proceed in like manner with the Minims, as before with the digits, advancing also that Chesseman forward, according to the aggregate of the Minims, over and above the digits so found, you will at last fall upon the fourth point required. Example, Admit the first point or term to be given at four digits, and four Minims of the third letter in the first Alphabet (being C) viz. at a; the second at four digits and four Minims of the last letter in the second Alphabet (being F) viz. at b; and the third at four digits and four Minims of the third letter in the fourth Alphabet (being C) viz. at c, and let it be desired to find a fourth point, which shall bear like distance from c, that b bears from a. Here, by the rule aforegoing, I find F in the fift Alphabet to be the letter where the Chesseman of the term inquired will rest, and therefore (according to this Rule) draw it back and set it upon the 29 perpendicular,, viz. at the beginning of the last of the entire letters intercepted: This done, I observe one entire digit betwizt the point a, and the nezt perpendicular towards b, signed by 4, and four digits betwixt b, and the next perpendicular towards a, signed by 12, these being added together make five, unto which I also add four for the number of entire digits contained betwixt c, and the perpendicular signed by 21, all rhese added together make nine, according to which I advance the Chesman of the fourth point or term (from the perpendicular 29, where I last placed it) to the letter e, to the end there may be nine entire digits comprehended betwixt it and the place from whence I took it: Lastly, having observed two minims at a, four minims at b, and four likewise at c, and added them together, their sum is ten, according to which I yet again advance the fourth Chesseman ten minims forward, and so at last the point or term required will be found to reside at the point d; so likewise, if d were the first term, c the second, and b the third, the fourth term would fall at a; also if b were the first, c the second, and d the third, the fourth term would then also fall at a, by going beyond the line, according to the 16 Rule of the first Chapter before-cited. Again, if c were the first, b the second, and a the third, the fourth term would be found at d, etc. And here note, that the demonstration of this Rule may be produced from the nature and properties of Arithmetical proportion, which (for brevity sake) I leave to the further scrutiny of the Practitioner: In some cases also the first and second terms will fall out to be so near together, that you may easily discover the like distance betwixt the third and fourth terms upon view, without any farther trouble. IU. What hath been here (by the two last Rules) practised upon the top-rank of Alphabets (with the Transversals, Digits and Minims thereunto belonging) may be likewise performed by the Alphabets repeated through the body of the Table, and their respective Transversals, Digits and Minims placed under the Fractions: And that, albeit the terms or points given are propounded upon several Transversals; so as the transversal, upon which the fourth term will fall, be also assigned. Example. Let the first term be given upon the transversal of the fift Fraction at four digits and four minims of the third Interval, signed by C, viz. at the point f, and the second term upon the transversal of the 19 Fraction at four digits and four minims of the twelfth Interval, signed by F, viz. at the point g. And the third upon the transversal of the 10 Fraction at four digits and four minims of the 21 Interval, signed by C, viz. at the point h, and let the demand be to find a 4th. point upon the transversal of the 24 Fraction, which may bear such like distance from the third point given, as the second bears from the first. Here first of all, having placed at each of the terms given a pointed Chessman, I find (as in the first example of the last Rule) eight Intervals, (or rather one Alphabet and two letters) to be intercepted betwixt the first and second terms, and therefore accounting as many (towards the same hand) from the third term, I find the fourth term to be likely to fall upon the Transversal of the said 24 Fraction in the 30 Interval, signed by F; where having placed another pointed Chesseman, I bring it back to the beginning of the last accounted letter, and then proceeding with the digits and Minims of the terms propounded, and advancing that fourth Chesseman 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 Minims 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, etc. V Having three points given upon three several transversals, to discover the transversal upon which the fourth term will fall, and also the point of that transversal, where that fourth term will bear like distance from the third point, that the second bears 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 situate 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 situate. 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 minims 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 minims 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 minims 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 join 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 minims 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 minims of the said third Interval, viz. at the point m. In like manner, if the three terms 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 Minims, 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 join 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 Minims 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 minims 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, than the Fraction below it, but if on the left hand, then that above it, as in the two last premised examples you find 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 minims of the third Interval, viz. at the point q; and in that case you do not only 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 minims of the 30 Interval, viz. at the point r. And (in that case also) you do not only 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 aforegoing examples. VI After the same manner may you also discover a third term to two terms propounded, save only, that (in regard the second term doth in a sort in that case represent the two middle terms) you are to double the digits and minims of the second term, and then add them to the digits and minims of the first term, to the end, you may understand by that sum 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 ends of the Fractions, upon which they are respectively situate) 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 terms, 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 add 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 minims, and four at the second, I likewise double the four, and add them to the 2, all which amount to 10, according to which sum I advance the Chesseman of the third term ten minims farther, and so at last I find the said third term to fix upon the 33 Fraction at four digits and four minims of the 21 Interval, which is the term required. And if at any time in working questions of this kind 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 minims 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, sixth, seventh, etc. in infinitum. VII. A point upon any one of the transversals being given, to find half 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 minims 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 minims, 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 situate, I take one Alphabet and three letters for the three Alphabets, and one letter more for the two odd letters; then for the four digits I take two digits, and for the four minims, two minims; 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. Upon 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 minims, 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 lieu of the three Alphabets I take one; for the third part of the two odd letters, I take four digits; for the four other digits I take one digit and two minims. And for the four last minines I take one minime and somewhat more, by which means 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 find 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 b● even or odd; if even, then take half the sum thereof, and place another plain Chesseman at the right end of the Fraction next under that half sum: but if the number be odd, neglecting the odd 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 odd Fraction signed by 21, would remain over and besides the two moities, which nevertheless will produce no error 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 equal division, then neglecting the odd 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 odd Fraction will remain over and above the number, which so equally divides itself into three parrs, as aforesaid, and if the 24 Fraction were propounded, two such odd Fractions would remain; which (nevertheless) causeth no inconvenience in the practice of this Instruments, as shall be manifested in the proper place. CHAP. IU. The Application of the Rule of Proportion. WE have done with Numeration, Application infues, which teacheth the use of this Instrument for the easy and ready resolution of divers Propositions in Arithmetic and Geometry, as followeth; Prop. 1. To three numbers given, to find 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 Arithmetic and Geometry: 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 terms propounded, you shall there find 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 were given for the first term, 18452 for the second, and 32292 for the third (in working upwards upon the 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 proportional 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 minims 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 minims of the Fraction next above it in the same Interval, viz. the point p, upon the 13 Fraction, which represents 22715, the fourth proportional 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 minims of the third Interval, viz. at the point q, which (by the fourth Rule of the second Chapter) represents 11422, the fourth proportional 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 minims of the 30 Interval, viz. at the point r, which represents these figures 52169: Howbeit, because common sense tells me, that the fourth term to the other three last given terms 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 minims of the 17 Interval, which point yields me these figures ●0975, whereof I take the two fi●●●, (viz. 1●) 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 signify 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 minims of the three terms propounded, this general Rule, or Corollary may be inferred. In all questions that may be performed by the Golden Rule, the digits and minims to be taken off from the first term are always so taken off from that side of the first term which inclines towards the second term, and then the digits and minims of the other two terms are always taken off upon the contrary side to those of the first term; as is manifest by all the examples aforegoing, which Rule being always duly observed, you may with greater confidence proceed to resolve any question propounded. And because this Corollary is always to be kept in memory, I have expressed it in this Distich, Aurata in Regula bis laeva aut dextra petatur, Dum contragreditur Terminus ips● prior. Thus Englished: For th' Rule of Three each hand may be pursued two times, Whiles that the foremost term against them always climbs. Prop. 2. To three numbers given, to find a four●● in an inversed proportion. This Rule of Three Inverse is th● 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 penny white loaf weighs 8 ounces, and 9 peny-weight; how much aught a penny white loaf to weigh, when wheat is at 23 shillings, six pence the quarter? Here the terms 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 proportional, you shall find it to fall upon the sixth Fraction at three digits and three minims 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 penny white loaf ought to weigh (according to the abovesaid rate) when wheat is sold for 23 s. 6 d. the quarter. Prop. 3. One number being given to be multiplied by another given number, to find the product. In multiplication there are four terms Geometrically proportional, whereof the first is always an unity, or 1, the multiplicator and multiplicand are the two means, 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 always 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 itself) but only 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 always 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 terms 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 minims 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 terms given are, 1— 4.3315— 3.1582— Whose fourth term I discover to fall upon the sixth 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 penny w. and 15 grains, the term required. In Multiplication observe these Rules. In performing Multiplication you always operate upon the Table towards the right hand, and upon the side-rank of Alphabets always downwards: for an unity or 1 being always the first term, you always 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 always 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 always 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 minims which are to be taken off from the points of the terms given, are always so to be taken off upon the left hand, and never upon the right. Here, by the terms given are intended only the Multiplicand and Multiplicator; for, the first term (viz. 1) hath no digits or minims 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 minims attending them, which are always 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 find the Quotient. As in Multiplication, so in Division there are four terms Geomettically proportional; whereof the Divisor is always 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 always one of the terms you need not thereat place a pointed Chesseman to denote it; but only 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 always 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 Quotient required. Example. 39893 being given to be divided by 287, the three terms 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 sixth 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 cyphers, 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 find the fourth term to reside upon the 12 Fraction at three digits and five minims 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 minims observe this Rule. When the digits and minims are taken off from the right hand of the Divisor, take the digits and minims 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 minims in Multiplication and Division, let this Hexameter be remembered. Multiplicá laeuè, sed divide dextrifinistre. Multiply by th' right hand, with both the hands divide. Prop. 5. Two numbers being given, to find a third Geometrically proportional unto them, and to three a fourth, and to four a fift, etc. This Proposition may be resolved by the directions given in the fixed Rule of the last Chapter; for, having two terms 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 sixth 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 fixed Rule of the last Chapter) will be found upon the 33 Fraction at two digits and two minims 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 will be 32, etc. And so you may (by this means) erect a rank of numbers Geometrically proportional, which in Arithmetic is called Geometrical Progression. Prop. 6. To extract the Square root of any number under 10000000000. FIrst, prepare the Square-number given for extraction (as in Vulgar Arithmetic) 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 minims 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 minims 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 odd 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 odd 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 situate, 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 sixth Fraction at one digit of the 31 Interval, and the half distance thereof at 3 minims 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 odd 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 minims of the 34 Interval; All this thus acted, I find the Chesseman last placed to meet wtih the 21 Fraction (being the true Fraction of the root, as aforesaid) at three minims 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 odd 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 find it practised in the last Example. Note also, that when you have more figures discovered upon the Table for the root, than the number given requires, those that exceed are a decimal Fraction belonging to the root; likewise when a mixed 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 Arithmetic) 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 situate; this done, by the eighth Rule of the last Chapter, find 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 odd Fraction or Fractions upon the side rank, than 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 Cu●●-●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, etc. (by the eighth Rule of the last Chapter) at three digits and four minims of the sixth 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 minims 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 look 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 odd Fractions upon the side rank, proceed (as in the last Rule) to find the third part of the distance, etc. 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 than 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 odd 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 situate, 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 minims of the third interval, and the third part of the distance, etc. (by the eight Rule of the last chapter) at four digits four minims and somewhat more of the first interval, 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 odd 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 find the Chesseman last placed to meet with the 22 Fraction (being the true Fraction of the root, as aforesaid) at four digits and four minims and somewhat 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 find these figures represented 55686, whereof (common sense 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 show a further use of this Instrument for the resolving of divers other Propositions in Arithmetic and Geometry; as, Betwixt two numbers given, to discover one, two, or more mean proportionals; To three numbers given, to find a fourth in a duplicated or triplicated proportion; To work Rules of plural Proportion; The double Golden Rules direct and Inverse; The Rules of Fellowship, Alligation, False position, etc. But I have deemed these (at present) sufficient to satisfy the curiosity of the Practitioner, who in obtaining the knowledge of these, (if he esteem them worthy his pains) may be thereby so perfectly acquainted with the nature of the Table, that he may afterwards be able to resolve not only the Propositions but all others, which may be performed by Arithmetic either Vulgar or Artifitial; And (perhaps upon further scrutiny) some others also, which cannot be resolved without Symbolical Arithmetic, usually called Algebra: All which I will hereafter eudeavour also to explain, (as vacancy from other more pertinent affairs will permit) together with the Fabric and Use of a trigonometrical Table of Proportion for the resolution of Plain and Spherical Triangles, if I shall find the pains herein already taken may obtain grateful reception: This Tractate being (indeed) only intended as an Eschantillon, or glimpse of that which may be performed upon this and the other abovesaid Table applicable to Trigonometry. Praestat pauca avide discere, quàm multa eumtaedio devorare. Erasm. in Coll. rel. FINIS.