CHAP. I, The Definition, Description, and Construction, of the Numericall Table of Proportion.

I. A Table of Proportion is an Instrument framed by Logarithms, and in­vented for the more ea­sie resolving of Arith­meticall and Geometricall Operations.

In Naturall or Vulgar Arithmetick, the Propositions are resolved by using the Numbers themselves, as if 4 were given to be multiplyed by 2, we say, two times four makes 8, the Product. In Artificiall Arithmetick, if rhe same Question were propounded, insteed of 4 and 2, we take their Logarithms; so if the Logarithm of 4 (being 0,602060) be added to the Logarithm of 2 (being 0,301030) their summe is 0,903090 which being found in the Table of Logarithms, is the Logarithm of 8, the Product, as before. Howbeit here, in the use of this Instrument we need not Multiply or Divide, Adde ar Substract, which for the most part perplex and discourage the Practitioner, but by the motion of certain Chesse-men (fitted for that purpose) we perform with pleasure and delight, the hardest Pro­positions of Arithmetick and Geometry without charging the minde or memory with any thing, which may seeme bur­thensome or distastfull.

II. This Instrument is twofold, Nu­mericall or Trigonometricall.

III. The Numericall Table of Pro­portion is an Instrument, by help whereof, and of certain moveable Chesse-men, all Questions Arithmeticall and Geometri­call (performed by Multiplication, Di­vision, or the Golden Rule, and not Tri­gonometricall) together with mean Pro­portionals, & the Extraction of the Roots of all Square-numbers under 11 places, and of all Cube-numbers under 16 pla­ces, as well in mixt and broken, as in whole numbers (when the term requi­red exceeds not 100000) are with great [...]ase and exactnesse 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 Trigonometricall.

IV. Of the Numericall Table of Pro­ [...]tion these things offer themselves to [...] considered, viz. The Description and [...]onstruction, or the Ʋse.

V. For the more plain describing of [...]is Instrument, it may be said to consist [...] two parts, viz. The Body of the Ta­ [...]e it self and substantiall part, or the [Page 4]Appendants and Circumstantiall part thereof.

VI. The Bodie of the Table it self is a Scale of unequall parts broken off in­to Fractions, and hereafter (for distin­ction sake) called the Scale of Numbers. This Scale is nothing else but a line of Numbers broken off into 36 fracti­ons or equal parts; Now what a line of Numbers is, hath been heretofore taught by Mr. Gunter in his Book of the Crosse-staffe, and is well enough known to all modern Artists.

VII. A Fraction of the Scale of Numbers is an equall part of the same Scale, consisting of Lines, Spaces, and Divisions: So this Scale is broken of or divided into thirty six of those equal parts or fractions, numbred at their right ends by 1, 2, 3, &c. to 36, of which the part signed at the left end thereo [...] by 100, is the first Fraction, that signe by 100, is the second, &c.

VIII. Each of these Fractions con­sists of three lines and two spaces: so the pricked line which you finde place under each Fraction, is not to be take [Page 5]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 [Page 6]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 Ʋnits: 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 [Page 7]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 [Page 8](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 [Page 9]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 [Page 10]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 [Page 11]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 [Page 12]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; [Page 13] 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 [Page 16]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, w^ch 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 [Page 17]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 [Page 18]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 [Page 20]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) [Page 21]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. [Page 22]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 [Page 23]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 [Page 25]this case I place three pointed Chesse­men 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 [Page 26]fourth may be discovered by this Rule and the 19 Rule of the first Chapter, which being plain, I omit to exempli­fie: 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, &c. In like manner, if two letters were given, and it be desired to finde 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, &c. [Page 27]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 finde out a fourth, which may bear the like distance from the third, that the second beares 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 thorow the last letter so counted, and place it upon the perpendicular, where that last letter begins; this done, observe how many intire 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 [Page 28]adde the intire 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 adde 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 Minimes, as before with the di­gits, advancing also that Chesseman for­ward, according to the aggregate of the Minimes, 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 Minimes of the third letter in the first Alphabet (being C) viz. at a; the se­cond at four digits and four Minimes of the last letter in the second Alphabet (being F) viz. at b; and the third at four digits and four Minimes of the third letter in the fourth Alphabet (being C) viz. at c, and let it be desired to finde a fourth point, which shall bear like distance from c, that b bears [Page 29]from a. Here, by the rule aforegoing, I finde 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 intire letters intercepted: This done, I observe one intire 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 adde four for the number of intire 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 intire digits comprehended betwixt it and the place from whence I tooke it: Lastly, having observed two minimes at a, four minimes at b, and four likewise at c, and added them to­gether, their summe is ten, according [Page 30]to which I yet again advance the fourth Chesseman ten minimes 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, &c. And here note, that the demonstration of this Rule may be produced from the nature and properties of Arithmeticall proportion, which (for brevitie sake) I leave to the further scrutinie of the Practitioner: In some cases also the first and second termes will fall out to be so neer together, that you may easily discover the like distance be­twixt the third and fourth termes upon view, without any farther trou­ble.

IV. What hath been here (by the two [Page 31]last Rules) practised upon the top-rank of Alphabets (with the Transversals, Digits and Minimes thereunto belong­ing) may be likewise performed by the Alphabets repeated through the body of the Table, and their respective Trans­versals, Digits and Minimes placed un­der the Fractions: And that, albeit the termes or points given are propounded up­on severall Transversals; so as the Transversall, 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 minimes 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 mini­mes 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 minimes 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 [Page 32]distance from the third point given, as the second beares from the first.

Here first of all, having placed at each of the terms given a pointed Chessman, I finde (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 termes, 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 Minimes 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 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 [Page 33]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 [Page 34]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 [Page 35]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 [Page 36]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 [Page 37]of a Chesseman upon the Table, when you are constrained to over-shoot the table either on the right or left hand, take the Fraction next to it, either a­bove or below it, viz. if on the right hand, then the Fraction below it, but if on the left hand, then that above it, as in the two last premised examples you finde it practised.

Again, if f be the first term, g the second, and k the third, the fourth term will fall upon the third Fraction at 4 digits and 4 minimes of the third In­terval, viz. at the point q; and in that case you do not onely fall off at the lower end of the side-rank, taking it again at the top, but likewise overshoot the table upon the right hand, and take it again upon the left, and (in that re­spect) take not the fraction whereunto you are directed by the fourth term found in the side-rank, but take the next under it: On the other side, if k were the first term, h the second, and f the third, the fourth term will reside upon the 27 Fraction at 4 digits and 4 minimes of the 30 Interval, viz. at the [Page 38]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 [Page 39]at the terms given, and likewise upon the side rank at the ends of the Fractions, upon which they are re­spectively scituate) I find the third term to fall upon the 33 Fraction; and then observing eight Letters or Inter­vals to be intercepted betwixt the first and second termes, accounting as ma­ny from the second towards the third, I find the Chesseman of the third term to be likely to fall upon the said thir­ty third Fraction in the one and twen­tieth Interval, signed by C, and there­fore draw that Chesseman back to the twenty perpendicular upon the same thirty third Fraction: this done, and I observing one digit at the first term, and four at the second, I double those four, and adde them to the one, all which amounting to nine, I ad­vance the Chesseman of the last term accordingly, setting it in the middle of the one and twentieth In­terval; then finding also at the first term two minimes, and four at the second, I likewise double the [Page 40]four, and adde them to the 2, all which amount to 10, according to which summe I advance the Chesseman of the third term ten minimes farther, and so at last I finde the said third term to fix upon the 33 Fraction at four digits and four minimes of the 21 Interval, which is the term required. And if at any time in working questions of this kinde you happen to descend below, or ascend above the side-rank, or other­wise overshoot the table either on the right hand, or left, you are (in such ca­ses) to use the Rules aforegoing, but still doubling the digits and minimes of the second term, as in the premised example. In like manner may you al­so (if you please) discover a fourth term to those three known, and so (con­sequently) a fift, sixt, seventh, &c. in infinitum.

VII. A point upon any one of the transversals being given, to find halfe the distance betwixt that point, and the beginning or left end of that transversal; follow this direction: Take half the [Page 41]Alphabets, half the letters, half the di­gits, and half the minimes intercepted be­twixt the beginning of that line and the point given, and so shall you have your desire. So if the point c upon the chief transversal were propounded, half the distance betwixt the beginning or left end thereof, and that point will be found at two digits and two minimes, of the letter E in the second Alphabet, viz. at the point S, for, in this case, there being three Alphabets and two letters intercepted betwixt the begin­ning of that traversal, and the letter wherein the point given is scituate, I take one Alphabet and three letters for the three Alphabets, and one letter more for the two odde letters; then for the four digits I take two digits, and for the four minimes, two minimes; all which being accounted from the be­ginning of that transversal will fall at S, the point required: the same may likewise be acted upon any of the repeated Alphabets and transversals in the body of the table.

VIII. Ʋpon any one of the Trans­versals, [Page 42]to discover the third part of the distance betwixt the beginning or left end thereof, and any point thereupon pro­pounded: this is the Rule; Take the third part of the distance in Alphabets, letters, digits, and minimes, and so shall you attain the point or term required. So the point c upon the chief transversal being again propounded, the point t will be third part of the distance inqui­red. For in liew of the three Alphabets I take one; for the third part of the two odde letters, I take four digits; for the four other digits I take one digit and two minimes. And for the four last minines I take one minime and some­what more, by which meanes t will be found at last the point songht for: Thus likewise you may be practised upon the repeated Alphabets and transver­sals.

IX. A Fraction of the Scale of Numbers being given, to finde upon the side rank of Alphabets the half distance betwixt it, and the first Fraction (in­cluding the first Fraction for one) pro­ceed in this mnnnner; first, having pla­ced [Page 43]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 odde; if even, then take half the summe thereof, and place another plain Chesseman at the right end of the Fraction next un­der that half summe: but if the num­ber be odde, neglecting the odde Fra­ction, proceed with the even number, as before, nnd so you shall accomplish your desire.

Example. Let the Fraction sign­ed at the right end thereof by 21 be given, and let the half distance betwixt it and the first Fraction be demanded. Here, the number above it is 20, whereof the half is 10; where­fore I taking a Chesseman, place it at the right end of the Fraction, signed by 11, which is the half distance de­manded. And if the 22 Fraction were propounded, the half distance would still remain the same: How­beit (in that case) the odde Fra­ction signed by 21, would remain over and besides the two moities, [Page 44]which neverthelesse will produce no errour in the use of the table, as shall appear hereafter.

X. A Fraction of the Scale of Num­bers being propounded, to discover upon the side-rank of Alphabets the third part of the distance betwixt it and the first Fraction (including the first Fraction for one) use this Rule; Having placed a Chesseman at the right end of the Fra­ction given, as before, observe whether the number of the Fraction placed next above it may be divided into three even parts; if so, then take the third part thereof, and place another Chesseman at the right end of the Fraction next under that third part: but if that number will not admit such an equall division, then neglecting the odde Fraction or Fracti­ons so remaining, proceed with the num­bers, which do so equally divide them­selves, as before, and so you shall disco­ver the third part you look for.

Example. Let the 22 Fraction be given, and the third part of the distances required: Here, the number next above it is 21, whereof the third part is 7; [Page 45]wherefore finding 7 amongst the num­bers placed at the right ends of the Fractions, I place another Chessman at the right end of the eighth Fraction, which denotes the third part required: Howbeit the 23 Fraction being given, an odde Fraction will remain over and above the number, which so equally divides it self into three parrs, as afore­said, and if the 24 Fraction were pro­pounded, two such odde Fractions would remain; which (neverthelesse) causeth no inconvenience in the pra­ctice of this Instruments, as shall be manifested in the proper place.

CHAP. IV. The Application of the Rule of Proportion.

WE have done with Numeration, Application infues, which teach­eth the use of this Instrument for the easie and ready resolution of divers Propositions in Arithmetick and Geo­metry, as followeth;