The LINE of PROPORTION or NUMBERS Commonly called Gunters Line Made Easie By the which may be Measured all manner of Superficies and Solids as Board Glass Pavement Timber Stone c. ALSO How to perform the same by a Line of Equal Parts drawn from the Centre of a Two-Foot-Rule Whereunto is added The Use of the Line of Proportion Improved Whereby all manner of Superficies and Solids may both exactly and speedily be measured without the help of Pen or Compasses by Inspection looking only upon the Ruler By WILLIAM LEYBOURN London Printed by J. S. for G. Sawbridge at his House on Clerkenwell-green 1667. LICENSED Nov. 9. 1666. Roger L' Estrange To the Right Honourable Sir WILLIAM BOLTON Knight Lord Major of the City of London And the Right Worshipful the Aldermen of the same City As also to JOHN AUSTEN and THOMAS NEVILE Esquires Commoners Appoined by a Committee of Common-Council to direct the Admeasurement of the Ruines by the late Fire there WILLIAM LEYBOURN One of those Employed by Order in the Survey of those Ruines Humbly presents with the best of his Services this Manual Necessary for all Builders and those that shall Employ them TO THE READER THE Line of Proportion or Numbers commonly called by Artificers Gunter's Line hath been discoursed of by several persons and variously applied to divers uses for when Mr. Gunter had brought it from the Tables to a Line and written some Uses thereof Mr. Wingate added divers Lines of several lengths thereby to Extract the Square or Cube Roots without doubling or trebling the distaence of the Compasses After him Mr. Milbourn a Yorkshire Gentleman disposed it in a Serpentine or Spiral Line thereby enlarging the divisions of the Line Again Mr. Seth Partrîdge contrived two Rulers to slide one by the side of the other having upon them two Lines of one length which exactly and readily performeth all Operations wrought thereby very exactly and speedily without the help of Compasses Now whatsoever all the forementioned Contrivances will perform I have here shewed in this Manual and so ordered the Line that it will perform the work without Compasses by Inspection looking only upon the Ruler And thereby may be measured let the Line be of what length soever not only Board Glass Timber and Stone but also all manner of Hangings Pavements VVainscots Plaistering Tyling Brick-work c. To all which Uses I have particularly applied it as will appear by several Instances in all the forementioned particulars and the rather because this Treatise may be beneficial and useful as well to Gentlemen and others who at this time may have more than ordinary occasion to make use thereof in the Re-building of the Renowned City of London as to Artificers themselves for whose sakes chiefly it was intended Vale. ADVERTISEMENT IF any Gentleman studious in the Mathematicks have or shall have occasion for Instruments thereunto belonging or Books to shew the use of them they may be furnished with all sorts useful both for Sea or Land either in Silver Brass or VVood by Walter Hayes at the Cross-Daggers in Moor-fields next door to the Popes-head Tavern where they may have all sorts of Maps Globes Sea-plats and Mathematical Paper Carpenters Rules Post and Pocket-Dials for any Latitude Steel Letters Figures Signs Planets or Aspects at reasonable Rates How to Measure Board and Timber BY THE Carpenters PLAIN RULE ALL manner of Superficial and Solid Measures may be measured the most absolute and artificial ways that are yet known by the Precepts and Examples in this Book delivered But although every Capacity may not attain to the knowledg and understanding thereof I thought good here to insert the Use of that Rule which is commonly made and sold and which every Artificer continually carries about him It s Description I. Of the FORE-SIDE It consisteth of two flat sides one of which towards either edge thereof is divided into 24 equal parts called Inches and numbered by 1 2 3 4 and so forth to 24 at the end thereof Every one of the parts or Inches is again divided into two equal parts by Lines about half the length of the other representing half Inches and every of these half Inches is divided into two other equal parts called quarters of Inches and each of those again into two other equal parts called half quarters of Inches So that each Inch is divided into eight equal parts representing Inches Halves Quarters and Half-quarters Both the edges on the one side of the Rule are thus divided and numbered only where 24 stands at one end of the Line on one edge there 1 stands on the other edge so that which end of the Rule soever you measure with you may count your number of Inches and parts right without turning of the Rule II. Of the BACK-SIDE On the other side of the Rule you have two other Lines or Scales drawn neer to the edges of the same side one is called the Line of Board-Measure the other the Line of Timber-Measure At the beginning of either of these Lines you have a little Table of Figures the one for Board the other for Timber or Stone The line or Scale of Board-Measure begins at 6 towards your left hand and so goes on to 36 ending just 4 Inches short of the other end of the Rule but sometimes this Line is continued up to 100 but nor often and then it goes nearer to the end of the Rule namely to within an Inch and an half of the end thereof At the beginning of this Line there is a small Table from 1 to 6 Inches which shews in Figures the quantity of the length of a Foot of any Board from one Inch broad to 6 Inches broad and then the divisions supply the greater breadths On the other edge on the same side you have the Line or Scale of Timber-measure This Scale begins at 8 and an half and so goes on by divisions to 36 towards the other end of the Ruler namely 36 ending within almost an Inch and half of the Rules end To this Scale also there belongeth a Table which standeth at the beginning of the Line and goes from 1 Inch to 8 Inches and gives the quantity of the length of a Foot of any Timber under 8 Inches square in Figures as the other did for Board from 1 to 6 And these are called the Tables of Under-measure The Table for UNDER-BOARD-MEASURE 1 2 3 4 5 6 12 6 4 3 2 2 0 0 0 0 4 0 The Table for UNDER-TIMBER-MEASURE 1 2 3 4 5 6 7 8 144 36 16 9 5 4 2 2 0 0 0 0 9 0 11 3 Thus much for the Description of the Lines upon the Carpenters plain Rule Now for Their Use I. Of the Fore-side or Side of Inches This side is only to measure the length and breadth of any thing to be measured in Inches and parts the manner of doing whereof is natural to every man for taking the Rule in the left hand apply it to the
from 2. Inches square to 8 Inches by Inches Halves and Quarters The Tables follow The Table for Board-measure Inches qu. feet in 10 par  1 48 0 0  2 24 0 0  3 16 0 0 I. 0 12 0 0  1 9 7 2  2 8 0 0  3 6 10 2 II. 0 6 0 0  1 5 4 0  2 4 9 6  3 4 4 4 III. 0 4 0 0 Inches qu. feet in 10 par III. 0 4 0 0  1 3 8 3  2 3 5 1  3 3 2 4 IV. 0 3 0 0  1 2 9 9  2 2 8 0  3 2 6 3 V. 0 2 4 8  1 2 3 4  2 2 2 2  3 2 1 0 The Table for Timber-measure II. 0 36 0 0  1 28 4 3  2 23 0 4  3 19 0 3 III. 0 16 0 0  1 13 7 6  2 11 9 1  3 10 1 8 IV. 0 9 0 0  1 7 11 6  2 7 1 3  3 6 4 6 V. 0 5 9 1  1 5 2 7  2 4 9 1  3 4 4 2 VI. 0 4 0 0  1 3 4 2  2 3 4 9  3 3 1 9 VII 0 2 11 2  1 2 8 1  2 2 6 7  3 2 4 7 Place here the FIGURE Of the RULE THE Line of Proportion or Numbers Commonly called Gunters Line Made Easie WHat this Line is and how to make it is best known to those who make Mathematical Instruments but the uses of it are so general that all sorts of men of what faculty soever may apply it to his particular use though it more immediately and particularly concerns such Artificers whose employment consists in Menstruation as Carpenters Joyners Masons Bricklayers Painters Glasiers and such like for that all kind of mensurations either SUPERFICIAL as Board Glass Pavement Tyleing c. Or SOLID as Timber Stone Pillars Piramids c. are by this Line most easily speedily and exactly performed For whatsoever thing concerning measure that may be performed by Arithmetick this Line will do exactly and much sooner as by the working of the several Rules in Arithmetick by this Line shall be plainly made appear CHAP. I. NUMERATION upon the Line BEfore I shew you how to number upon the Line it will be necessary to let you understand how the Line is divided and numbered as also what those divisions and numbers set to them upon the Ruler do sigdifie Know therefore that the Line of numbers begins at the Figure One and so proceeds successively from I to 2.3.4.5.6.7.8.9 and then on farther by 1.2.3.4.5.6.7.8.9.10 at the end of the line The first 1. which standeth at the beginning of the line representeth the One tenth part of any Unite or Intiger as One tenth part of a Foot One tenth part of a Yard Ell Perch Mile c. Or it may signifie One tenth of a Year Moneth Hour c. Or the one tenth of a Pound Shilling or Penny c. Or the one tenth part of any thing either in Number Weight Measure Tine or the like The Figure 2. signifies two tenth parts of any thing The figure 3. three tenth parts The figure 4. four tenth parts c. till you come to the second 1. which standeth in the middle of the line which 1. signifieth One whole Unite or Intiger as One whole Foot Yard Perch c. Now the other intermediate divisions those which stand between the figures 1 and 2 which are in number ten do represent each of them one hundred part of one Unite or Intiger so the first division beyond the figure 1 represents 11 hundred parts of the Intiget the second division 12 hundred parts of the Intiger and so on the figure 2 representing 20 hundred parts of the Intiger and the next division beyond 2 is 21 hundred parts and so on till you come to the figure 1 in the middle of the line which representeth one whole Intiger The figure 2 signifieth two whole Intigers the figure 3 three whole Intigers and so on till you come to 10 at the end of the line which signifieth ten whole Intigers and the intermediate divisions which stand between 1 and 2 in the middle of the line are every of them tenth parts of the Intiger So the Rule contains 10 whole Intigers every of which is divided into 10 parts But if upon the line you would count numbers of more places then two which are all numbers above 10 then the 1 which is at the beginning of the line must be accounted one Intiger and the 1 in the middle of the line ten Intigers and the 10 at the end will be 100 Intigers But yet farther if upon the Line you would expresse numbers of more places than three which are all numbers above 100 Then the 1 at the beginning of the line is to be accounted ten Intigers the 1 in the middle a hundred Intigers and the 10 at the end of the line 1000 Intigers And if you proceed yet farther then the 1 at the beginning must be accounted for a hundred Intigers that in the middle a thousand and the 10 at the end of the line for 10000 ten thousand Intigers In this manner you might proceed farther by counting the first 1 for 1000 10000 c. Intigers but to four places is sufficient which by a Rule of a competent length as of two Foot any question concerning measuring may be by one exactly enough performed The Divisions and Numbers on the line being thus explained it resteth now to shew you how to find that point upon the line which shall represent any number proposed and that I shall shew you in these Propositions following which may fitly be called NUMERATION PROP. I. A whole number consisting of two three or four places being given to finde the point upon the Line which representeth the same NOTE Let your number given be of how many places soever for the first figure of your number you must take the same figure upon the Line For the second figure in your number take the number thereof on the grand or larger intermediate divisions on the Line For the third figure in your number take the number thereof on the smaller intermediate divisions on the line And for your fourth figure you must find its place by estimation Example I. Let it be required to finde the place of 15 upon the Line For your first figure 1 count the 1 in the middle of the Line then for the 5 which is your second figure count five of the grand or larger intermediate divisions upon the line and that point is the very place upon the line representing 15. Note that every fifth of the grand intermediate divisions is drawn forth with a longer line then the rest for case in counting Again To finde the place upon the Line representing 37. For your first figure 3 count the figure 3 upon the Line then for the 7
thing to be measured so have you the length breadth or thickness of the thing desired But II. Of the Backside and I. Of the Line of Board-measure PROB. 1. The breadth of any Board being given to finde how much thereof in length will make a Foot square Look for the number of Inches that your Board or Glass is broad in the Line of Board-measure and the number of Inches and parts of an Inch which stand against that on the other side of your Rule is the quantity of Inches that will make a Foot square of that Board or Glass or what other thing soever it be to be measured Example 1. There is a Board or Plank that is 9 Inches broad how much of that in length will make a Foot square Look for 9 Inches upon the Line of Board-measure which you shall finde at the Figure 9 upon the same Line and just against that on the other side of your Rule you shall finde 16 Inches which shews that every 16 Inches of that piece in length will make a Foot square Example 2. A Pain of Glass is 22 Inches broad how much thereof in length will make a Foot square Look for 22 Inches in the Line of Board-measure and right against it on the other side of your Rule you shall finde 6 Inches and almost an half and so much in length of that breadth will make a Foot square Example 3. If any plain Superficies be 30 Inches broad how much thereof in length will make a Foot square Seek for 30 Inches in the Line of Board-measure and right against it on the other side of the Rule you shall finde 4 Inches and â…˜ that is 4 Inches and 4 fifth parts of an Inch. Example 4. If a Board be 9 Inches and an half broad how much thereof in length will make a Poot square Seek 9 Inches and an half in the Line of Board-measure and against that on the other side of the Rule you shall finde 15 Inches and about one sixth part of an Inch to make a Foot square ¶ NOTE All these Examples might be performed otherwise by the Line for if you take the Rule in your left hand and apply the end thereof noted with 36 to the end of the Superficies the other edge of the Superficies will shew how many Inches Halves and Quarters will make a Foot square This needs no Example PROB. 2. The length and breadth of a Superficies being given to finde how many Square Feet are therein contained By any of the ways before taught finde how much of the breadth given will make a Foot square then run that length from one of the ends of the Superficies as often as you can and so many square feet is there in that Superficies Example A Board is 9 Inches â—road and 15 Poot long how many square Feet are therein contained By the first Example you finde that 〈◊〉 9 Inches broad 16 Inches in length to make a Foot wherefore take 16 Inches of your Rule and run that length along the Board from one end thereof and you shall finde that â—ength to be contained in the Board of 15 Foot long 11 times and 4 Inches over which is ¼ of a Foot so that the Board of 15 Foot long and 9 Inches broad contains 11 Foot and one quarter The like of any other II. Of the Line of Timber-measure PROBL. 1. The Square of any piece of Timber at the end thereof being given to finde how much of that piece in length shall make a Foot solid The Use of the Line of Timber-measure is in all respects the same at that of Board-measure for knowing the square of your piece of Timber at the end thereof you have no more to do than to look for the quantity of the Square thereof in the Line of Timber-measure and right against it on the other side of the Rule you have the quantity of Inches that will make a Foot solid of that piece Example 1. A piece of Timber is 10 Inches square how much thereof in length will make a Foot solid Look for 10 Inches in the Line of Timber-measure and right against it on the other side of the Rule you shall finde 17 Inches and somewhat above a quarter of an Inch and so much of that piece in length will make a Foot solid Example 2. If the Square of a piece of Timber be 21 Inches how much thereof in length will make a Foot solid Seek 21 Inches in the Line of Timber-measure and against it you shall finde on the other side of the Rule almost 4 Inches and so much in length will make a solid Foot of Timber Note 1. If Timber be broader at one end then at the other the usual way is to add both ends together and take half thereof for the true square but if the difference be very much this way is erroneous though for the most part practised Note 2. Also for Round Timber the usual way is to girt it about the middle with a String and take a fourth part thereof for the Square this also is erroneous Therefore for such as desire curiosity and exactness let them repair to the Rules in this Book delivered for that purpose where they may receive ample satisfaction Concerning the Tables at the beginning of the Lines of Board and Timber-Measure The Table of Board-Measure gives the length of a Foot square of any Board under 6 Inches broad therefore by the Table there set you may finde that  Foot In. parts    If a Board be 1 Inches broad 12 0 0 will make a Foot square 2 6 0 0 3 4 0 0 4 3 0 0 5 2 4 5 6 2 0 0 By this small Table you may see that a Board of 4 Inches broad will require 3 Foot thereof in length to make a Foot square Also a Board of 5 Inches broad will require 2 Foot 4 Inches and 4 fifth parts of an Inch. The Table of Timber-measure gives the length of a Foot folid of any piece of Timber or Stone whose square is under 8 Inches Wherefore by the Table at the beginning of the Line of Timber-measure you may finde that If a piece of Timber be 1 Inch quare 256 0 0 will make a Foot solid 2 36 0 0 3 16 0 0 4 9 0 0 5 5 9 0 6 4 0 0 7 2 11 0 8 2 3 0 By this Table which is the same in effect with that which standeth at the end of the Line of Timber-measure you may see that a piece of Timber that is 4 Inches square requires 9 Foot in length to make a solid Foot Also a pitce of 5 Inches square require 5 Foot 9 Inches and 1 16 parts of an Inch to make a solid Foot And so of the rest But because these Tables go only to whole Inches I have here added two Tables one for Board the other for Timber the Table for Board from one quarter of an Inch to 6 Inches in breadth and the Table for Timber
count 7 of the intermediate divisions and that point is the place upon the Rule representing 37. Example 2. Let it be required to finde the place of 134 upon the line For your first figure 1 count 1 upon the line for your second figure 3 count three of the grand divisions and for the third figure 4 count 4 of the smallest intermediate divisions and that very point is the place upon the Line representing 134. Again To finde the place representing 308. For your first figure 3 count the 3 upon the Line for your second figure o which is a Cipher count none of the grand divisions but for your last figure 8 count 8 of the intermediate divisions and that point shall be the place upon the line representing 308. Example 3. Let it be required to finde the place of 1350. For your first figure 1 take 1 on the middle of the Line For your second figure 3 take the figure 3 upon the line upwards for the 5 count five of the grand intermediate divisions and that is the place of 1350. Again To finde the place of 1626. For your first figure 1 count the 1 on the middle of the Line for your second figure 6 count the figure 6 upon the line upwards then for your third figure 2 count two of the grand divisions and for your last figure 6 estimate six tenth parts of the next grand division which is something more then half the distance because 6 is more then half 10 and that is the point upon the line representing 1626. Note By these Examples last mentioned you may perceive that the figures 1.2.3.4.5.6.7.8.9 do sometimes signify themselves alone sometimes 10.20.30 c. sometimes 100.200.300 c. as the work performed thereby shall require The first figure of every number is alwaies that which is here set down and the rest of the figures are to be supplyed according as the nature of the Question shall require And by this variation and change of the powers of these numbers from 1 to 10 or 100 or 1000 any proportion either Arithmetial or Geometrical may be wrought One whereof I will insert for your better exercise of numbring on the Rule by the often practice whereof you will find the work facile and delightful which shall be this following PROP. 2. Having two numbers given to finde as many more as you please which shall be in continual proportion one to the other as the two numbers given were FOr the working of this proposition this is THE RULE Place one Foot of the Compasses in the first given number on the line and extend the other Foot to the second given number then may you turn the Compasses from that second number to a third from that third to a fourth from that fourth to a fifth a sixth a seventh c. towhat number of places you please Example 1. Let the two given numbers be 2 and 4. Place one Foot of your Compasses in 2 and extend the other Foot to 4 then that Foot which now standeth in 2 being turned about will reach from 4 to 8 and from 8 to 16 from 16 to 32 from 32 to 64 from 64 to 128. But when your Compasses stand in 64 if you turn them about yet farther they will fall beyond the end of the line wherefore you must place one Foot in some other 64 nearer the beginning of the line and then the other Foot will reach to 128 and from 128 to 256 and from 256 to 512 and from 512 to 1024 but here it will go off of your line again whereof as before you must choose another 512 nearer the beginning of the line and there placing you Compasses they will reach to 1024. from 1024 to 2048 from 2048 to 4096 c. Example 2. But if the given numbers were 10 and 9 decreasing then place one Foot in 10 at the end of line and extend the other downwards to 9 the same extent will reach still backwards to 8.1 or 8 1 10 and from 8.1 to 7.29 and still backwards from 7.29 to 6.56 Likewise if the two first numbers had been as 1 to 9 the third proportional would have been 81 the fourth 729 and the fifth 656 with the same extent of the Compasses Again Let the two numbers be 10 and 12 place one Foot in 10 and extend the other to 12 that extent will reach from 12 to 14.4 and from thence to 17.28 But if the numbers were 1 and 12 then the third proportional would be 144 and the fourth 1728 and all with the same extent of the Compasses CHAP. II. MULTIPLICATION by the Line IN Multiplication the Proportion is this As 1 upon the Line Is to one of the numbers to be multiplyed So is the other of the numbers to be multiplyed To the Product of them Which is the number sought Example 1. Let it be required to multiply 5 by 7. The proportion is As 1 ∶ to 5 ∷ so is 7 ∶ to 35. Therefore Set one Foot of your Compasses in 1 and extend the other Foot to 5 with that extent of the Compasses place one Foot in 7 and the other Foot will fall upon 35 which is the Product Example 2. Let it be required to multiply 32 by 9. The Proportion is As 1 ∶ to 9 ∷ so 32 ∶ to 288. Set one Foot in one and extend the other Foot to 9 that same extent will reach from 32 to 288 which is the product or sum of 32 being multiplyed by 9. Otherwise Set one Foot in 1 and extend the other to 32 the same extent will reach from 9 to 288 as before Example 3. Let it be required to multiply 8 75 100 by 5 45 100. The Analogy or Proportion is As 1 ∶ to 8.75 ∷ so 6.45 ∶ to 56.44 fere Set one Foot in 1 and extend the other to 8.75 the same extent applyed forward upon the line will reach from 6.45 to 56.44 fere Or if you set one Foot in 1 and extend the other to 6.45 The same extent will reach from 8.75 to 56.44 almost namely to 43 ¾ as before CHAP. III. DIVISION by the Line IN Division three things are to be minded viz. The Dividend or number to be divided The Divisor the number by which the Dividend is divided The Quotient which is the number sought And so often as the Divisor is contained in the Dividend so often doth the Quotient contain Unity For the working of Division this is the Analogy As the Divisor is to Unity or 1. So is the Dividend to the Quotient Example 1. Let it be required to divide 35 by 7. The Proportion is As 7 ∶ to 1 ∷ so 35 ∶ to 5. Set one Foot of the Compasses in 7 and extend the other Foot downwards to one that same extent will reach from 35 downwards to 5 which is the Quotient and so many times is 7 contained in 35. Otherwise Extend the Compasses upwards from 7 to 35 that same extent will reach upwards from 1 to 5
28 that extent doubled will reach from 154 to 616 for first it will reach from 154 to 308 and from thence to 616 and that is the Area or Content of a Circle whose Diameter is 28. Example 2. If a piece of Land that is 20 Pole square be worth 30 pounds what is a piece of Land of the same goodness worth that is 35 Pole square Extend the Compasses from 20 to 35 that extent doubled will reach from 30 to 91.8 that is 91 pound 8 10 of a pound which is 16 shillings and so much is such a piece of Land worth II. Of the Proportion of SUPERFICIES to LINES In this Case Extend the Compasses unto the half of the distance between the two numbers of the same denomination that same extent shall reach from the third number to the fourth required Example 1. Let there be two Circles given the Area or Content of the one being 154 and its Diameter 14 The Area of the other Circle is 616 what is the length of its Diameter Upon your Line divide the distance between 154 and 616 into two equal parts then with that distance set one Foot in 14 and the other shall fall upon 28 which is the length of the Diameter of the other Circle whose Area is 616. Example 2. There is a piece of Land containing 20 Pole square worth 30 pound there is another piece worth 91 pound 16 s. how many Pole square ought that piece to contain Take with your Compasses halfe the distance between 30 l. and 91 l. 16 s. Then set one foot in 20 pole and the other foot will reach to 35 Pole and so many Pole square must the Land be that is worth 91 l. 16 s. CHAP. VII OF TRIPLICATE PROPOTION by the Line TRiplicate proportion is such a proportion as is between Lines and Solids or between Solids and Lines 1. Of the Proportion between LINES and SOLIDS In this Case Extend the Compasses from the first number to the second of the same denomination that extent being tripled shall reach from the third number to the fâ—rth Example There is a Bullet whose Diameter is 4 Inches weying 9 l. What shall another bullet of the same mettall weigh whose Diameter shall be 8 inches Extend the Compasses from 4 to 8 the two Diameters the same extent being tripled will reach from 9 to 72. which is the weight of a bullet whose diameter is 8 inches II. Of the proportion of SOLIDS to LINES In this Case Extend the Compasses unto the third part of the distance between the two numbers of like denomination that same extent shall reach from the third to the fourth number required Example The weight of a Cube being 72 pound the Side whereof was 8 inches and the weight of another Cube of the same matter weighting 9 pound what must the side be Upon your line divide the distance between 9 and 72 into three equal parts then set one foot of that distance in 8 and the other foot shall rest in 4 the length of the side of the Cube required CHAP. VIII The Extraction of the SQARE-ROOT By the Line TO Extract the Square-Root is to find a mean proportional Number between 1 and the number given and therefore is to be found by dividing the space between them into two equal parts Example Let it be required to find the Square-Root of 36. Extend the Compasses from 1 to 36 the middle way upon the Line between these two numbers is 6 which is the Square-Root of 36. In like manner may you find the Square-Root of 81 to be 9 of 144 to be 12 of 256 to be 16 and of other numbers as in this Table Root Square Root Square 1 1 11 121 2 4 12 144 3 9 13 169 4 16 14 196 5 25 15 225 6 36 16 256 7 49 17 289 8 64 18 324 9 81 19 361 10 100 20 400 If you suppose the number to have pricks over every second Figure as is usual in the Arithmetical Operation then if the last prick towards the left hand fall over the last Figure which will always be when the number of Figures are odde then it will be best to place Unity at the 1 in the middle of the line so that the Root and the Square may both fall forward towards 10 at the end of the Line But if the number of Figures be even it will then be best to place Unity at 10 at the end of the Line so the Root and the Square both will fall backwards towards the middle of the Line CHAP. IX The Extraction of the CVBE-ROOT By the Line TO Extract the Cube-Root is to find the first of two mean proportionals between 1 and the number whose Cube-Root you require and is therefore to be found upon the Line by dividing the space between them into three equal parts Example Let it be required to find the Cube-Root of 216. Extend the Compasses from 1 to 216 one third part of that distance shall reach from one to 6 which is the Cube-Root of 216. In like manner may you find the Cube-Root of 729 to be 9 of 1728 to be 12 of 110592 to be 48 of 493039 to be 79 c. as in this Table Root Cube Root Cube 1 1 11 1331 2 8 12 1728 3 27 13 2197 4 64 14 2744 5 125 15 3375 6 216 16 4096 7 343 17 4913 8 512 18 5832 9 729 19 6859 10 1000 20 8000 CHAP. X. The Use of the Line applied to SUPERFICIAL-MEASURE Such as Board Glass Wainscot Pavement Hangings Paintings c. of what kind soever THe Measures by which Board Glass Timber Stone and such like are measured is by the Foot a Foot containing 12 Inches and each Inch into eight parts called halves quarters and half-quarters but this kind of division not being consentaneous or agreeable to the divisions upon your Line of Proportion where between 1 and 2 is divided not into 8 but into 10 parts the like between 2 and 3 into 10 parts and so between 3 and 4,4 and 5 c. Therefore I hold it requisite both for ease and exactness to have every Inch on your two-foot Rule divided not into 8 but into 10 equal parts which hereafter throughout this Book we will call Inch-Measure Again Whereas your Foot is divided into 12 equal parts called Inches I would have your Foot divided into 10 equal parts and each of those parts sub-divided into 10 other equal parts so will you whole Foot contain 100 equal parts which will be agreeable to the divisions of your Line and facilitate the work as by the Examples in this kind given will be made to appear and this we shall hereafter call Foot-Measure But if any person be so wedded to Inches halves and quarters that he will not be beaten out of his opinion but persist therein and yet is desirous to have knowledge in the use of this Line I say such person may have added to the side of his Inches halves and
third parts of the content of the Circle at the Bung. Wherefore Extend the Compasses from 1 to 5236 the same extent will reach from 1024 the square of 32 the diameter at the Bung to 536.166 inches the content of two third parts of the circle at the Bung. II. For one third part of the Circle at the Head As 1 to this general number 2618 So is 324 the square of the diameter at head 18 to 84.823 Inches which is one third part of the content of the circle at the head Wherefore Extend the Compasses from 1 to 2618 the same extent will reach from 324 the square of 18 the diameter at the Head to 84.823 Inches the content of one third part of the diameter at the head III. For the number of Square Inches in the Vessel Add these two numbers 536.166 and 84.823 They make 620.989  40 Which multiplied by 40 the length of the Vessel produceth 24839.560 And so many Square Inches are contained in such a Vessel whose diameter at the Head is 18 Inches at the Bung 32 Inches and is 40 Inches long IV. For the Content in Wine or Ale Gallons Divide this number 24839.56 by 231 for Wine 282 for Ale and the Quotients shall tell you the number of Gallons and parts of a Gallon Wine  gall parts 231 24839.56 107.52  231   1739   1617   1225   1155   706   693   13  Ale  gall parts 282 24839.56 88.08  2256   2279   2256   2356   2256   100  By this work you may perceive that this Vessel containeth 107 Gallons 53 parts of Wine measure 88 Gallons 08 parts of Ale measure How to multiply and divide by the Line is taught in the Second and Third Chapters of this Book and therefore were needless here to repeat it again But I chose rather to do it Arithmetically for the better illustration and for the satisfaction of such as have a delight in Numbers How to Measure Board Glass Timber Stone c. By a Line of Equal Parts drawn from the Centre of a Two-foot Joynt-Rule ALL Proportions that may be wrought upon a straight Ruler by the Line of Proportion or Numbers the same may be wrought by a Line of Equal Parts drawn from the Centre of an opening Joynt And whereas this Line of Equal Parts is numbered from the Centre of the Rule towards the end thereof by 1 2 3 4 c. to 10 that these Figures as in the other Line do sometimes signifie themselves only sometimes 1 2 3 c. do signifie 10 20 30 c. sometimes 100 200 300 c. according to the quality of the Question propounded By this Line you may also Multiply divide work the Rule of Proportion and perform divers things which the Line of Numbers performeth and some others which that will not but I shall here only shew you how Board Glass Timber Stone c. may be thereby measured which I shall do in these following Propositions And I. For SUPERFICIAL MEASURE as Board Glass c. I. In INCH-MEASURE PROP. I. A Plank being 27 Inches broad and 263 Inches long how many Square Inches are contained therein As I is to 27 So is 263 to 7101. Take in your Compasses the distance from the Centre to 27 the breadth upon your Line of Equal parts with this distance set one foot in 10 at the end of the Line and open the Rule till the other foot fall in 10 on the other Leg of the Rule The Rule thus standing take with your Compasses the distance between 263 on one Leg of the Rule to 263 on the other Leg this distance will reach from the Centre of the Rule to 7101 and so many square Inches are in that piece PROP. 2. If a Board or Plank or piece of Pavement or of Glass be 20 Inches broad how much thereof in length shall make a Foot square As 20 is to 144 So 1 to 7.2 Take 144 out of your Line of Equal parts from the Centre and setting one foot in 20 open the other Leg till the other Compass point fall in 20 also The Rule thus standing take the distance between 10 and 10 and that distance will reach from the centre of the Rule to 7 Inches 2 10 parts of an Inch and so much in length will make a Foot Square II. In FOOT-MEASURE PROP. 3. A Room is 52 Foot broad and 110.5 Foot long how many Square Foot are there in that Room As 52 is to 1 So is 110.5 to 5746. Take in your Compasses 52 the breadth with this distance open the Ruler in 10 and 10 it so resting take the distance between 110.5 and 110.5 on either side that distance applied to the Centre of the Rule will reach to 5746 and so many Square Foot is in that Room PROP. 4. A Plank being 2 Foot 25 parts broad how much in length thereof shall make a Foot Square As 2.25 the breadth is to 1 or 10 So is 10 to 44 the length of a Foot Take in your Compasses the distance from the Centre of your Rule to 1 then set one foot in 2.25 and open the other Leg till the other Compass point fall in 2.25 on the other side the Rule thus standing take the distance between 10 and 10 that distance applied from the Centre of the Rule will reach to 44 parts of a Foot and so much in length will make a Foot III. In YARD-MEASURE PROP. 5. A Room is hung with Tapestry containing 130 Yards 25 parts in compass and in depth 5 Yards 20 parts how many Yards of Tapestry is in that Room As 1 to 5.20 So 130.25 to 677.4 Take 5.20 in your Compasses and that distance put over in 10 and 10 the Rule thus standing take the distance between 130.25 and 130.25 on each Leg of the Rule that distance will reach from the Centre of the Rule to 677 Yards 4 tenths of a Yard II. For SOLID-MEASURE as Timber Stone c. By the Line of Equal parts I. In INCH-MEASURE PROP. I. A piece of Timber being 30 Inches broad 21 Inches 6 parts deep and 183 Inches long how many Foot is contained in that piece of Timber 1. As 1 is to 30 So is 21.6 to 648. Take the distance from the Centre to 30 then set one foot in 10 and open the Rule till the other Compass point fall in 10 on the other Leg of the Rule Then take the distance between 21.6 and 21.6 that distance will reach from the Centre of the Rule to 648 the content of the base or end of the piece of Timber in Inches Then 2. As 1728 the number of Inches in a Foot solid is to 648 the content of the base So is 183 Inches the length to 68 Foot 62 parts the content in Feet Take in your Compasses the distance from the Centre to 1728 with this distance set one foot in 648 and
of one and the same Radius being set upon a plain Ruler of any length the larger the better having the beginning of one Line at the end of the other the divisions of each Line being set so close together that if you finde any number upon one of the Lines you may easily see what number stands against it in the other Line This is all the Variation and what this easie contrivance will effect will appear by the uses following The Lines are the same with the Line of Proportion or Numbers mentioned and treated of in the former part of this Book and therefore how to number upon them is shewed in the first Chapter of this Book and therefore needs not here again be repeated Also Multiplication Division the Golden Rule Duplicated and Triplicated Proportion the Extraction of Roots c. delivered in the second third fourth fifth Chapters c. as also in Measuring of Superficies and Solids and the mensuration of other Figures treated of through the whole Book these Lines thus disposed will effect with Compasses But some of those Uses which they will effect in measuring without the help of Compasses I will here shew CAUTION What measure soever you measure by let the Integer or grand Measure be divided into 10 or 100 parts it matters not of what length your Lines of Proportion be for to them all measures are alike Thus If you measure any thing by the Foot let your Foot be divided into 100 parts If by the Yard divide your Yard into 100 parts If by the Ell divide that into 100 parts So likewise if by the Perch Square c. or by what Measure soever let the grand Measure as I said before be divided into 100 parts CHAP. I. OF SUPERFICIAL MEASURE BY Superficial Measure is meant all kind of flat Measure such as is Board Glass Pavement Hangings Plaistering Tyling Land-measure c. And these several things are measured by distinct Measures as some by the Foot others by the Yard others again by the Ell some by the Rod and some by the Square of all which I shall give Examples and I. Of FOOT-MEASURE Example 1. If a Board be 2 Foot 64 parts broad how much in length of that Board will make a Foot Square Look upon one of your Lines it matters not which for 1 Foot 64 parts and right against it on the other Line you shall finde 61 and so many parts of a Foot will make a Foot square of that Board Example 2 A Plank is 3 Foot 50 parts broad how much thereof in length will make a Foot Finde 3 Foot 50 parts upon one Line and right against it on the other Line you shall finde 28 parts and 4 7 or something more that half a part Example 3. If a Board be 75 parts of a Foot broad how much thereof in length shall make a Foot square Look upon one of your Lines for 75 and right against it you shall finde 1 Foot and 33 parts and so much in length makes a square Foot Note If the breadth of any thing given be more than one Foot then the length of a Foot square must be less than a Foot as in the two first Examples it was But if the breadth given be less than a Foot as in this last Example then the length of a Foot square must be more than a Foot Example 4. A Pain of Glass is 35 parts broad how much in length makes a Foot Finde 35 in one Line against it you shall finde 2 Foot 854 7 parts and so much in length makes a square Foot Example 5. A Pain of Glass is 3 Foot broad how much in length makes a Foot Finde 3 Foot in one Line against it in the other you shall finde 33 â…“ parts and so much in length makes a Foot square Example 6. If a piece of Glass be 1 Foot 98 parts broad how much in length will make a Foot Look 1 Foot 98 parts in one Line and against it in the other you will finde 5 Foot and half a part and so much in length makes a Foot II. Of YARD-MEASURE Example 1. A Gallery is Wainscoted 2 Yards 56 parts deep how much of that length will make a Yard square Seek 2 Yards 56 parts in one Line and against it in the other you shall finde 39 parts and somewhat more and so many parts of a Yard will make a Yard square Example 2. A Room is Wainscoted 1 Yard 13 parts high how much in length thereof will make a Yard square Look one Yard 13 parts in one Line against it in the other you will finde 88 parts and above half a part and so much in length makes a Yard square Example 3. If the Frieze about a Room be 62 parts of a Yard broad how much in length thereof will make a Yard square Finde 62 parts in one of your Lines and against it in the other you shall finde 1 Yard 61 parts and somewhat more and so much in length makes a Foot square Example 4. There is a Gallery paved with Marble being 5 Yards 70 parts broad how much of that in length will make a Yard square Seek 5 Yards 70 parts in one Line and against it in the other you shall finde 17 parts and an half and so much in length of that Pavement will make a Yard square Example 5. A Parlour being 7 Yards 29 parts broad hath a Cieling of Fret-work plaistered how much of that breadth will make a Yard Square Finde 7 Yards 29 parts in one of your Lines and right against it in the other Line you shall finde 13 parts and 7 10 which is above half a part So that 13 parts and a little more than half a part will make a Yard square of that Cieling Example 6. A Plaisterer hath Rendered the inside of a Wall containing 2 Yards 36 parts in height how much of that will make a Yard square Finde 2 Yards 36 parts in one of your Lines and right against it on the other you shall finde 42 parts 3 10 of a part that is something more than one third part of a part and so much in length makes a Yard Square III. Of MEASURE by the ELL Example 1. There is a Room hung with Tapestry which is 4 Ells 25 parts high how much Tapestry in length will make an Ell square Note Here by Ells we understand Flemish Ells for by that Measure are Hangings sold which Ell contains three quarters of our Yard that is 75 parts of our Yard So that if an Upholster have his Flemish Ell divided into 100 parts he may measure his Hangings as in the Examples following is shewed Here because the Hangings are 4 Ells 25 parts deep Look for 4 Ells 25 parts in one of your Lines right against which in the other you shall finde 23 parts and a half and so many parts of his Ell will make a Flemish Ell square Example 2. Tâ—â— Embroidery of a pair of Vallens about a Bed is 28
parts of a Flemish Ell deep how much of that Embroidery in length will make a Flemish Ell square Look for 28 parts in one of your Lines and against it in the other Line you shall finde 3 Ells and 57 parts of an Ell and so much in length will make an Ell square Example 3. A Gallery being 3 Ells 98 parts deep is hung with Arras how much of that depth will make an Ell square Seek 3 Ells 98 parts in one Line against which in the other you shall finde 25 yarts and 1 10 of a part and so much in length will make an Ell square IV. Of MEASURE by the ROD. Example 1. There is a Brick-wall which is 75 parts of a Rod high how much in length of that Wall will make a Rod square Note That all Wall-work is by the Bricklayers measured by the Rod which contains 16 Foot and a half in length Wherefore let his Rod being 16 foot and an half in length be divided into 100 equal parts and then let him work as followeth The Wall being 75 parts of a Rod high Look for 75 parts in one Line and in the other Line right against 75 you shall finde 1 Rod 33 parts of a Rod and so much of that Wall in length is contained in a Square Rod. Example 2. A Carpenter hath Rail'd and Paled in a Garden with Pales 52 parts of a Rod high how much of that Paling shall make a Rod square Seek 52 parts in one Line against it in the other Line you shall finde 1 Rod 92 parts and so much in length will make a Rod square of that Paling Example 3. A Bricklayer hath made a Shewer to carry Water the Bottom Sides and Arch together contains 1 Rod 64 parts how much of that Drain or Shewer makes a Square Rod Finde 1 Rod 64 parts in one of your Lines and right against that number you shall finde in the other Line almost 61 parts and so many parts of a Rod in length will make a Rod square And here note That though I have here put these two last Examples that Paling is not measured by the Square Rod but let the height thereof be what it will it is measured by the Rod in length In like manner is Hedging Ditching and many other things that are measured by the Rod. Example 4. If a piece of Land be 2 Rod 31 parts broad how much in length thereof shall make a Rod square Seek 2 Rods 31 parts upon one of your Lines and over against it upon the other Line you shall finde 42 parts and about â…“ of a part and so much in length makes a square Rod. Example 5. A piece of Land being 80 parts of a Rod broad how much thereof in length shall make a Rod square Look for 80 parts in one Line and in the other Line opposite thereunto you shall finde 1 Rod 22 parts and so much in length makes a Rod square V. Of MEASVRING by the SQUARE There are two things principally which are measured by the Square and they are Tyling of Houses and Flooring of Rooms and in this reckoning they account a Square to be 10 Foot every way So that for this kind of Measure divide a Line or Rod of Ten Foot long into 100 parts and it is fit for the purpose Exampler 1. A Barn the breadth of the Tyling whereof on both sides is 1 Square 30 parts how much in length of that Tyling will make a Square Square Finde 1 Square 30 parts upon one of your Lines and right against it on the other Line you shall finde 77 parts almost and so much in length of that Tyling will make a Square Square Example 2. The Tyling of a house is 76 parts of a Square broad how much in length thereof will make a Square Square Seek 76 parts in one Line and against it in the other you shall finde 1 Square 31 parts and an half almost and so much in length will make a Square Square that is 10 Foot every way in all 100 Foot CHAP. II. OF SOLID MEASURE BY Solid Measure is meant such Measure as hath Length Breadth and Thickness such as Timber Stone or the like But before Timber or Stone can be measured you must finde the content of the square of the base thereof which is taught by the Problem at the end of the Tenth Chapter but that being performed by Compasses I will here shew how it may be by these Lines thus disposed performed without and that shall be my first Proposition or Example Example 1. Let a piece of Timber or a Stone be 80 parts of a Foot deep and 3 Foot 75 parts broad how much in length of that piece will make a Foot square Here by any of the former Rules of Superficial Measure finde at 80 parts broad how much in length will make a Foot which you shall finde to be 1 Foot 25 parts For If you finde 80 parts the depth of the piece in one Line against it in the other you shall finde 1 Foot 25 parts Take 1 Foot 25 parts of your Foot Rule and measure it along the breadth of the piece which is 3 Foot 75 parts and see how often it is contained therein which you shall finde to be three times wherefore you may conclude that the square of the base of that piece of Timber whose depth is 80 parts and whose breadth is 3 Foot 75 parts is just 3 Foot Now the square of the base of the piece being thus obtained you may finde the length of a Foot square thereof in this manner Example 2. Let the Square of the base of a piece of Timber or Stone be 3 Foot how much in length of that piece will make a Foot solid Look for 3 Foot in one of your Lines and in the other right against it you shall finde 33 parts and â…“ part of a part and so much in length will make a Foot solid Example 3. Let a piece of Stone or Timber be 2 Foot 50 parts broad and 50 parts deep how much of that Stone in length shall make a solid Foot By any of the ways before prescribed you shall finde that the depth of your Stone being 50 parts it will require 2 Foot in length thereof to make a Foot square Wherefore measure how often you can finde 2 Foot in the breadth of your Solid which you may finde only once and 50 parts more which is one quarter of two Foot Wherefore the Square of this Solid contains 1 Foot 25 parts Wherefore Look in one of your Lines for 1 Foot 25 parts and right against it you shall finde 80 parts and so much in length will make a Foot solid Example 4. The Square of the Base of any regular Solid being given together with the length of the same Solid to finde how many solid Feet are contained in the same Let the forementioned Solid serve for this Example also whose length was 35 Foot We found that the Square of the Base was 1 Foot 25 parts and that 80 parts in length would make one solid Foot Wherefore take 80 parts of your Rule and run it along the piece so often as you can which you shall finde to be 43 and 60 parts which is just three Quarters so that in this piece of Timber there is 43 Foot and three Quarters I might add many more Examples of this kinde and some to other purposes but these are sufficient for the purpose intended and so I shall conclude this Treatise leaving the farther practise thereof to your self For Usus optimus Magister CHAP. III. OF CIRCULAR MEASURE By having either the Circumference or Diameter of any Circle given thereby to finde the Side of a Square Equal to the same Circle or the Side of a Square that may be inscribed within the same Circle IN the Thirteenth Chapter of this Book you have six Examples by having the Circumference or Diameter of any Circle given thereby to finde the Side of a Square equal the Superficial Content c. But I have seen upon some Two-Foot-Rules Lines to effect this thing by only opening the Compasses to the distance given upon one Line and applying the same to some of the other Scales One of those Scales is noted at the end thereof with C signifying the Circumference of any Circle the other with D signifying the Diameter the other with S E signifying Square Equal to the Circie the other with S. W signifying Square within Example So that if you should have given you the Diameter of a Circle being 15 Inches Out of the Line noted with D take 15 Inches apply that distance to the Line noted with C it will reach to 47 Inches and 13 100 parts of an Inch and so much is the Circumference of that Circle Again The Diameter being 15 Inches as before take that distance out of the Line D and it will reach upon the Line S E to 13 Inches 29 100 parts and that shall be the Side of a Square equal to the Circle whose Diameter is 15 Inches Again The Diameter being 15 Inches if you take that distance out of the line noted with D it will reach upon the Line S W to 10 Inches 60 1 parts of an Inch and that is the length of the Side of the greatest Square that can be drawn within that Circle whose Diameter is 15 Inches The like may be done if the Circumference were given by taking the Circumference thereof out of the Line noted with C and applying it to the other Scales This I thought convenient to adde here because sometimes these Lines are put upon Two-foot Rules FINIS