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
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
same will reach from 1 to 2.67 So that 2 inches 67 100 parts will make a Foot solid of that piece of Timber or Stone This may be done another way by this Analogy or Proportion 1. As. 12 to 30 the breadth in inches So 21.6 the depth in inches to a fourth number which here will be about 54. 2. As this fourth number 54 is to 144 So is 1 to 2.67 the length of a Foot solid Wherefore Extend the Compasses from 12 to 30 the breadth that extent will reach from 21.6 the depth to a certain place upon the Line about 54 where keep the point of the Compass fast and open the other to 144 then will this extent of the Compasses reach from 1 to 2 inches 67 parts the length of a Foot solid as before II. In Foot-Measure only Example 1. Let a Stone or a piece of Timber be 2 Foot 50 parts broad 1 Foot 80 parts deep and 15 Foot 25 parts long how many solid or cubical Feet doth such a piece contain The proportion is 1. As 1 is to 2.50 Foot the breadth So is 1.80 Foot the depth to 4.50 Foot the base in such measure 2. As 1 unto 4.50 the base So 15.25 the length to 68.62 the content in Feet Extend the Compasses from 1 to 2.50 the breadth the same will reach from 1.80 the depth to 4.50 the base Again Extend the Compasses from 1 to 4.50 the base that extent will reach from 15.25 the length to 68.62 the content in Feet Example 2. In the forementioned piece of squared Stone or Timber being 2 Foot 50 parts broad and 1 Foot 80 parts deep Let it be required to finde how much thereof in length will make a Foot The proportion is 1. As 1 is to 2.50 the breadth So is 1.80 the depth to 4.50 the content of the base in Foot-measure 2. As 4.50 the base is to 1 So is 1 Foot to 222 parts the length of a Foot solid Wherefore Extend the Compasses from 1 to 2.50 the breadth the same extent will reach from 1.80 the depth to 4.50 the content of the base Again Extend the Compasses from 4.50 the base to 1 the same will reach from 10 to 222 parts the length of a cubical or solid Foot of that Stone or piece of Timber III. In FOOT-MEASURE and INCH-MEASURE together Example Let a squared Stone or piece of Timber be 30 Inches broad 27.6 Inches deep and 15 Foot 25 parts long how many cubical or solid Foot of Stone or Timber is shere in that piece The proportion is 1. As 1 is to 30 inches the breadth So is 21.6 inches the depth to 640 the content of the base in inches 2. As 144 the inches in a Foot superficial is to 648 the content of the base in inches So is 15.25 the length of the piece in Foot-measure to 68 Foot 62 part Wherefore Extend the Compasses from 1 to 30 the breadth the same will reach from 21.6 the depth to 648 the content of the base Again Extend the Compasses from 144 to 648 the content of the base the same extent will reach from 15.25 the length of the piece to 68.62 the solid content of the Stone or Timber in Feet and 100 parts of a Foot By having the same things given in the same piece of Stone or Timber or in any other the work may be varied several ways The Analogies or Proportions I will only give you leaving the practise thereof to your self Breadth of the piece 30 inches Depth of the piece 21.6 inches Length of the piece 15.25 foot The Proportions 1. As 144 to 30 the breadth So 21.6 the depth to a fourth Number From which fourth Number if you extend your Compasses to 1 and place one foot in 15.25 the length of the piece the other Foot shall fall upon 68.62 the content of the Stone Or 2. As 12 unto 30 the breadth So 21.6 the depth to some fourth Number From this fourth Number Extend the Compasses to 12 that distance will reach from 15.25 the length of the piece to 68.62 the content of the piece CHAP. XVI How to measure Stone or Timber by the Line by having the Square of the Base and the Length of the Piece given both in Foot and Inch Measure HOw to finde the length of a Side of a Geometrical Square that shall be equal to any Parallelogram or Long Square is taught at the latter end of the Tenth Chapter of this Book by which Rule it may at any time be found That being done there I shall only here begin with Examples Example 1. There is a Squared piece of Timber whose length is 183 Inches and the Side of the Square equal to the base or end thereof is 25 Inches 45 parts how many Foot doth that piece contain 1. As 41.57 to 25.45 the side of the Square So is 183 the length in Inches to a fourth Number 2. And that fourth Number to 68.62 the content in Feet Extend the Compasses from 41.57 to 25.45 the side of the Square the same will reach from 183 the length to some other part of the Line from whence if you again extend the same distance the point will rest upon 68 Foot 62 100 parts of a Foot and so many Foot is in the piece Example 2. Let the side of a Square equal to the Base of a piece of Stone or Timber be 2 Foot 12 parts and the length of the same piece 15 Foot 25 parts how many solid Foot is there in that piece 1. As 1. to 2 Foot 12 parts the side of the Square So 15 Foot 25 parts the length to a fourth Number 2. And that fourth Number to 68.62 the content in Feet Extend the Compasses from 1 to 2.12 the side of the Square that will reach from 15.25 the length to some other number on the Line from whence the Compasses being extended the movable point will fall upon 68.62 the content as before Example 3. The side of a Square equal to the Base of a Stone being 25 Inches 45 parts and the length of that Stone 15 Foot 25 parts how many Foot doth it contain 1. As 12 to 25.45 the Square in Inches So is 15.25 Foot the length to a fourth Number 2. And that fourth Number to 68.62 the content Extend the Compasses from 12 to 25.45 the side of the Square the same will reach from 15.25 to some other point upon the Line from whence the Compasses being extended the movable point will fall upon 68 Foot 62 parts the content of the Stone Example 4. There is a piece of Timber whose side of the Square of the Base is 25 Inches 45 parts how much in length of that piece will make a Foot solid 1. As 25.45 the side of the Square is to 41.57 So is 1 Foot to a fourth Number 2. And that fourth Number to 6 Inches 67 parts Wherefore Extend the Compasses from 25.45 the side to 41.57 the same will reach from 1 to some other point from
whence the Compasses being extended will reach to 6.67 the length of a Foot solid of that piece of Timber Example 5. The length of the side of a Square equal to the Base of a piece of Timber being 2 Foot 12 parts to finde how much in length of that piece will make a Foot solid in Foot-measure As 2.120 the side of the Square is to 1.000 So is 1.000 to a fourth Number And that fourth Number to 0.471 parts of a Foot to make a Foot square Extend the Compasses from 2.120 the side of the Square to 1000 the same extent will reach from 1000 downwards to some other point upon the Line and from thence downwards to 222 parts of a Foot and so much in length will make a Foot solid CHAP. XVII Concerning Timber that is bigger at one end than at the other either Round or Square and how to measure it I. For SQUARED-TIMBER IN large Timber-trees when they are squared there is a great disproportion between the Squares of both ends wherefore some do use to take the Square of the middle of the piece for the mean or true Square but this is not exact though much used but the best way is this Finde by the Problem at the end of the tenth Chapter of this Book the length of the side of a Spuare equal to both the ends of the piece add these two sides together and take the half thereof for the true Square and with that Square you may by the Rules of the last Chapter measure it as if it were perfectly Square II. For ROUND-TIMBER The ordinary way used for the measuring of Round Timber is to girt it about the middle with a Line and to take one fourth part thereof for the side of a Square equal thereto but this is false though most men use it Custom having made it bear the face of Truth for it is more in measure than in reality it should be But the exact way of measuring of Round Timber especially if it be growing is this About the middle thereof in some smooth place girt the same about with a string Then have you this proportion As 1000 is to the number of Inches about So is 2821 to the length of the side of a Square equal thereunto So if a Tree being girt about as abovesaid should contain in circumference 47 Inches 13 parts If you extend the Compasses from 1000 to 47 Inches 13 parts the same extent will reach from 2821 to 13 Inches 29 parts which is equal to the side of a Square equal to that Tree which being obtained the Tree may be measured divers ways according to the Examples in the last Chapter CHAP. XVIII Concerning the measuring of Regular Solids as Cylinders Globes Cones and such like I. Of the CYLINDER A Cylinder is a round Figure of equal circumference in all parts thereof as a standing Pillar a Rowling-stone for Garden-walks c. To measure such a Figure there are several ways both by having the circumference given when it is standing or by having the diameter at the end thereof when it is lying or by having the side of a Square equal to the base thereof I. By having the Diameter given Example 1. The Diameter being 15 Inches how much in length makes a Foot As 15 the diameter to 46.90 So is 1 to a fourth And that fourth to 9.78 the length of a Foot Extend the Compasses from 15 the diameter to 46.90 that extent will reach from 1 to another point upon the Line and from thence to 9 Inches 78 parts the length of a Foot solid Example 2. The diameter being 1 Foot 25 parts how much in length makes a Foot in Foot-measure As 1.25 the diameter in Feet unto 1.128 So 1 to a fourth number And that to 8.14 the length of a Foot solid in Foot-measure Extend the Compasses from 1.25 the diameter to 1.128 the same will reach from 1 to some other number and from thence to 1 Foot 128 parts of a Foot Example 3. Having the diameter 15 Inches and the length 105 Inches how many solid Inches doth the Cylinder Contain As 1.128 to 15 Inches the diameter So is 105 Inches the length to a fourth number And that to 18555.34 Inches the content Extend the Compasses from 1.128 to 15 the length the fame extent will reach from 105 the length to some other number and from thence to 18555.34 Inches the content of the Cylinder in Inches Example 4. Having the diameter 1 Foot 25 parts and the length 8 Foot 75 parts to finde the content in Feet As 1.128 to 1.25 the diameter So 8.75 the length to a fourth And that fourth to 10.74 Foot the content Extend the Compasses from 1.128 to 1.25 the diameter that extent will reach from 8.75 the length to some other number and from that to 10 Foot 74 parts the content Example 5. Having the diameter 15 Inches and the length 105 Inches how many Foot doth it contain As 46.90 to 15 Inches the diameter So is 105 Inches the length to a fourth And that fourth to 10 Foot 74 parts the content Extend the Compasses from 46.90 to 15 the diameter that extent will reach from 105 the length to another number and from that to 10 Foot 74 parts the content Example 6. The diameter being 15 Inches and the length 8 Foot 75 parts how many Foot doth it contain As 13.54 to 15 Inches the diameter So 8.75 Foot the length to a fourth And that fourth to 10.74 the length in Feet Extend the Compesses from 3.54 to 15 the length that extent will reach from 8.75 the length to another number and from thence to 10.74 Foot the content in Feet II. By having the Circumference given Example 1. The Circumference of a Cylinder is 47 Inches 13 parts how much thereof in length shall make a Foot solid As 47.13 Inches the circumference to 147.36 So 1 to a fourth number And that to 9.78 Inches the length of a Foot Extend the Compasses from 47.13 the circumference to 147.36 that extent will reach from 1 to a fourth number and from thence to 9 Inches 78 parts the length of a Foot solid Example 2. Having the circumference of a Cylinder 3 Foot 927 parts to finde the length of a Foot solid thereof in Foot-measure As 3.927 Foot to 3.545 So 1 to a fourth number And that to 815 parts of a Foot the length Extend the Compasses from 3.927 the circumference to 3.545 that extent will reach from 1 to some other number and from thence to 815 parts of a Foot for the length of a solid Foot of that Cylinder Example 3. The circumference of a Cylinder being 47 Inches 13 parts and the length thereof 105 Inches how many Inches is there in such a Cylinder As 3.545 to 47.13 the circumference So 105 Inches the length to a fourth number And that to 18555 the content in Inches Extend the Compasses from 3.545 to 47.13 the circumference that extent will reach from 105 the
length to another number and from thence to 18555 the number of solid Inches in the Cylinder Example 4. The circumference being 47 Inches 13 parts and the length 105 Inches as before how many solid Foot in that Cylinder As 147.36 to 47.13 Inches the circumfer So 105 Inches the length to a fourth number And that to 10 Foot 74 parts the content Extend the Compasses from 147.36 to 47.13 the circumference that extent will reach from 105 the length to another number and from that to 10 Foot 74 parts of a Foot the solid content Example 5. Let the length of the Cylinder be 8 Foot 75 parts and the circumference 3 Foot 927 parts how many Foot doth it contain As 3.545 to 3.927 Foot the circumference So 8.75 Foot the length to a fourth number And that to 10 Foot 74 parts the content Extend the Compasses from 3.545 to 3.927 the same extent will reach from 8.75 the length to 10.74 the content in Feet Example 6. Let the circumference given be 47 Inches 13 parts and the length 8 Foot 75 parts how many solid Foot doth the Cylinder contain As 42.54 to 47.13 Inches the circumference So is 8.75 Foot the length to a fourth And that fourth to 10.74 Foot the content Extend the Compasses from 42.54 to 47.13 the circumference that extent will reach from 8.75 the length to another number and from thence to 10 Foot 74 parts the content of the Cylinder in solid Feet III. By having the Side of a Square equal to the Base or End of a Cylinder Example Let the Side of a Square equal to the Base or End of the Cylinder be 13 Inches 29 parts and the length thereof 105 Inches how many square Feet are contained in that Cylinder As 41.57 to 13.29 Inches the side of the Square So is 105 the length in Inches to a fourth number And that to 10 Foot 47 parts the content of the Cylinder in Feet and parts Extend the Compasses from 41.57 to 13.29 Inches the side of a Square equal to the base of the Cylinder that extent will reach from 105 Inches the length to another number and from thence to 10 Foot 47 parts the content of the Cylinder in Feet II. Of the CONE A Cone is a round Figure having for the base thereof a Circle the side whereof riseth from the circumference of the Circle round about the same equally till it meet in a point just over the centre of the Circle and is in the form of a Spire-Steeple And it is thus measured Example 1. Let there be a Cone the diameter of whose Base is 10 Inches and whose height is 12 Inches I would know how many solid or cubical Inches are contained therein The diameter being 10 the content of the Circle or Base will be found to the 78 Inches 54 parts as by the fifth Example in Chap. 13. of this Book The Area of the Base being thus found the proportion is As 3 to 78.54 Inches the content of the base So is 14 Inches the height to 314 Inches 16 parts of an Inch for the content of the Cone in Inches Extend the Compasses from 3 to 78.54 the base that extent will reach from 12 the height to 314 Inches 16 parts the content of the Cone in solid Inches Example 2. Let the diameter of the Base be 12 Inches as before and the length of the side he 13 Inches kow many solid Inches is there in this Cone 1. Extend the Compasses from 1 to 5 Inches half the diameter of the Base that extent will reach from 5 to 25. 2. Extend the Compasses from 1 to 13 the length of the side that extent will reach from 13 to 169. 3. From this 169 take the 25 before found and there remains 144. 4. Upon your Line take half the distance between 1 and 144 and you shall finde it to be 12 which 12 is the height of the Cone So the height being had you may finde the content as in the last Example III. Of SPHERICAL BODIES A Spherical Body is such a Body whose Superficies in all the parts of it are equally distant from the centre of the body as Globes Bullets c. Example 1. The Circumference of a Globe or Bullet being 28 Inches 28 parts to finde the length of the diameter As 22 to 7 So is 28.28 the circumference to 9 Inches the diameter Extend the Compasses from 22 to 7 the same extent will reach from 28.28 the circumference to 9 Inches the length of the diameter of that Bullet Example 2. The diameter of a Spherical Body being given is 9 Inches and its circumference is 28 Inches 28 parts how many square Inches is there in the superficies of that Spherical Body As 1 is to 9 Inches the diameter So is 28.28 Inches the circumference to 254.5 Inches the superficial content Extend the Compasses from 1 to 9 the diameter the same extent will reach from 28.28 the circumference to 254 Inches 5 parts the superficial Inches in this Spherical Body Example 3. The diameter of a Spherical Body being 9 Inches how many solid Inches are therein contained 1. As 1 is to 9. the diameter Se is 9 to a fourth number And that fourth number to 729 the cube of the diameter 2. As 9 the diameter to 729 its cube So is 11 to 891 Inches the solid content of the Spherical Body Extend the Compasses from 1 to 9 that extent will reach to 81 and from 81 to 729 the cube of the diameter Then Extend the Compasses from 9 the diameter to 729 its Cube that extent will reach from 11 to 891 Inches the solid content of the Spherical Body I might here add the manner how to measure other kind of Bodies both Regular and Irregular as Ellipses Parabolas c. Also of Prismes Scalene Cones Spheroiades c. But these being out of the reach of ordinary Artificers for whose sakes this Treatise was chiefly composed I shall here conclude this Treatise of the Use of the Line of Proportion with a short Supplement of Gauging of Vessels CHAP. XIX Concerning the Gauging of Vessels By the Line BEfore you can measure your Vessel to finde the content thereof in Gallons or parts you must finde the content thereof in solid Inches and to effect this you must finde the content of two third parts of a Circle agreeable to the diameter at the Bung and one third part of another Circle agreeable to that of the diameter at the Head these two added together and multiplied by the length of the Vessel that product will be the content of that Vessel in Inches Example Let there be a Vessel whose diameter at Head 18 diameter at Bung 32 length is 40 And let the content thereof first in Inches and then in Gallons be required I. For the two third parts of the Circle at the Bung. As 1 to this universal number 5236 So 1024 the square of the diameter at the Bung 32 to 536.166 Inches which is two
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
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
it will fall out of the line wherefore you must set one foot in 37.8 in the lower part of the Line and then the other will fall upon 176.74 And thus you must do in other cases whenever your Compass point goes beyond your Line CHAP. XIV II. Of the Triangle A Triangle is a Figure consisting of three sides and three angles the longest side whereof we call the base and a line drawn from the Angle opposite to the base we call the perpendicular To measure Triangles there are several ways I will only shew you one or two to be done by the Line Example 1. There is a Triangle whose base is 14 foot and his perpendicular 6 foot I would know how many square Foot is contained in this Triangle The proportion is As 2 is to 6 the perpendicular So is 14 the base to 42 the Area Or As 1 is to 3 half the base So is 14 to 42 the Area Or As 2 is to 6 the perpendicular So is 7 half the base to 42 the Area Or As 1 is to 6 the perpendicular So is 14 the base to 84 the double Area All these ways produce the same effect but the first is the best Wherefore The base of your Triangle being 14 and the perpendicular 6 Extend the Compasses from 2 to 6 the same extent will reach from 14 to 42 the Area III. Of the Trapezia A Trapezia is any right lined figure consisting of four unequal sides and as many angles For the measuring of it you must first reduce it into two Triangles by drawing a line or diameter from one opposite angle to another the longest way then from the two angles opposite to this line let fall two perpendiculars so is the Trapezia divided into two Triangles The manner how to measure it is this Example There is a Trapezia whose diameter is 12.34 and one perpendicular is 4.20 the other 5.07 I would know the Content or Area thereof The two perpendiculars added together make 9.27 Then the analogy is As 2 is to 9.27 the sum of the perdendiculars So is 12.34 the base to 57.19 the Area There are as many ways to measure Trapezias as in the last Example I gave you for Triangles but this is the best And here note That if you are to measure any irregular piece of what nature soever whether Land Board Glass Pavement or the like your best and exactest way is to reduce them to Trapezias and measure them as before is taught IIII. Of Regular Figures of 5 6 8 10 or 20 equal sides These Figures by Geometricians are called Regular Polygons and the way to measure them is by adding all the sider together Then measure the length from the centre of the figure to the middle of one of the sides by the help of these two you may finde the Area of the Figure Example Let there be a Regular Polygon of 11 equal sides each side being 7 Inches and let the length of the line from the centre to the middle of one of those sides be 12 Inches Add all the sides together they make 77 then As 2 is to 77 the sum of the sides So is 12 the length of the line from the middle of the figure to 462 the content of the figure CHAP. XV. The Use of the Line applied to SOLID-MEASVRE Such as Timber Stone c. TImber and Stone are usually measured by the same Rule or Measure as Board and Glass are namely by Feet and Inches Therefore such a Rule as was mentioned in the beginning of the Tenth Chapter is fit for this business also Before we come to shew the way of measuring of Stone or Timber it will be necessary to premise thus much That the base or end of every piece of Timber or Stone is or must be supposed either exactly square that is every side alike or else one of the sides longer than the other wherefore the first thing to be done is to finde the Area or superficial content of the base or end of any piece of Timber or Stone to be measured which may be done several ways either in Inch-Measure as by the first Example of the first part of the tenth Chapter or in Foot-Measure by the first Example in the second part of the same Chapter or both in Foot-measure and Inch-measure as in the first Example of the third part of the same tenth Chapter and therefore need not be here repeated again Wherefore we will proceed to our intended purpose of Measuring first by Inch-measure only secondly by Foot-measure only and thirdly by both together as we did before in the measuring of Board c. 1. In Inch-Measure only Example 1. There is a piece of Timber 30 Inches broad 21 Inches 6 parts deep and 183 Inches long how many square Inches is there in this solid piece of Timber The proportion is 1. As 1 unto 30 Inches the breadth So is 21.6 Inches the depth to 648 Inches the content of the base of the piece 2. As 1 unto 648 the content of the base So is 183 Inches the length of the piece to 118584 the solid content in Inches Wherefore Extend the Compases from 1 to 30 the breadth the same will reach from 21.6 the depth to 648 the content of the base Again Extend the Compasses from 1 to 648 the content of the base that extent will reach from 183 the length to 118584 Inches the solid content But so many places of Figures cannot well be estimated upon your Line except it be very large but by following Examples you shall have your desire accomplished exactly and easily Example 2. To finde the content of the same piece of Timber in Foot-measure the dimensions being given in Inches and parts The proportion is 1. As 1 to 30 the breadth So 21.6 the depth to 648 the content of the base as before 2. As 1728 the number of solid Inches in a Foot of Timber is to 648 the content of the base So is 183 the length in Inches to 68 Foot and 62 100 parts of a Foot as before Wherefore as before Extend the Compasses from 1 to 30 the breadth the same will reach from 21.6 the depth to 648 the content of the base as before Again Extend the Compasses from 1728 to 648 the base the same extent will reach the same way from 183 the length to 68.62 the content of the piece of Timber in Feet and parts that is 68 Foot and above half a Foot Example 3. Let a squared Stone or piece of Timber be 30 Inches broad and 21 Inches six parts deep how much in length shall make a Foot square of that piece of Timber or Stone You may finde the content of the base as in the last Example to be 648 Inches then the proportion is As 648 the content of the base is to 1728 the inches in a Foot So is 1 to 2 inches 67 parts the length of a Foot solid Therefore Extend the Compasses from 648 the base to 1728 the