both ends together and take halfe thereof for a mean breadth so finde you 18 then is it all one as if your Board were 18 inches and you would know how much in length makes a foot Take 12 and fit it in 18 and take it over in 12 and so much makes a foot Let a board be broad at one end ten inches and a quarter and at the other seven and a halfe now the desire is to know how much in length makes a foot Adde both the numbers together and take halfe which maketh 8 inches and seven eight parts of an inch for the common breadth then bring 8 inches and seven eights of an inch 12 inches into eights and it maketh 71 eights and 96 eights Take then 96 in some Scale and fit that in 71 then let the Scale rest then take it over in 12 and that apply to the same Scale where the 96 was taken and it sheweth 16 and a quarter and so many inches in length make one foot of Board CHAP. XIII To finde how many square feet any whole Board containeth without finding how much in length makes a foot IMagine a Board be 15 foot long and 16 inches broad and it is required to finde how many square foot of Board it containeth Take the length of 15 on some Scale of equall parts and fit that in 12 the inches in a foot alwayes there let the Scale rest then take it over in 16 the breadth and apply it to the same Scale where the length was taken it sheweth 20 and so many square foot is found to be therein contained Let a Board be 17 foot and a quarter long and 16 inches and a halfe broad and the desire is to know how many foot it containeth Take 17 and a quarter the length and fit it in 12 and take it over in 16 and a half and that apply to the same Scale whe 17 and a quarter the length was taken it sheweth 23 and two thirds and so many foot it containeth Or you may bring 17 and a quarter into quarters makes 69 and in like manner 12 into quarters makes 48 and take it over in 16 and a halfe the breadth so finde you 23 and two thirds as before CHAP. XIV To measure Board that is broader at the one end then at the other in the same manner SUppose a Board be broad at the one end 18 inches and at the other end 14 and long 21 foot I demand how many square foot it containeth Adde the breadth at both ends together makes 32 inches whose halfe is 16 inches for a mean breadth then proceed as before take 21 and fit it in 12 and take it over in 16 or fit it in five times 12 and take it over in five times 16 so finde you 28 for the area required Again let a Board be broad at the one end 11 inches and a halfe and at she other 7 and three quarters and 15 foot and three quarters long now the Area is required First adde them both together and take half makes 9 five eight parts for the mean breadth Then take 15 three quarters the length on any Scale and fit in 12 and take it over in 9 five eights and that applyed to the same Scale where the length was taken and it sheweth how many foot it containeth Or bring 12 and 9 five eights into eights make 96 and 77 then fit fifteen three quarters the length in 96 and take it over in 77 and that sheweth on the same Scale where the 15 three quarters was taken twelve two thirds the Area desired CHAP. XV. To measure Timber SUppose a piece of Timber be 18 inches broad and deep 16 inches it is required to finde how much in length doth make a foot Take twelve the inches in a foot on any Scale of equal parts fit that in the breadth eighteen and take that over in twelve alwayes Again set that distance in sixteen the depth and take it over in twelve still and that apply to the same Scale where the twelve was taken shew six and so many inches in length make a foot the thing required Again let a piece of Timber be broad sixteen inches and deep thirteen and a halfe and it is required to finde how much in length make one foot As before fit twelve in sixteen and take it over in 12 still that apply to the same Scale where the twelve was taken sheweth eight inches and so many inches in length make one foot Again let a piece of Timber be fifteen three quarters broad and eleven three quarters deep I demand how much shall make a foot Bring fifteen three quarters and twelve into quarters makes sixty three and forty eight then take twelve on some Scale of equall parts and fit it in sixty three and take it over in 48 and that distance fit in eleven one quarter and take it over in 12 Or as before bring eleven one quarter and twelve both into quarters makes forty eight and forty five then fit it in forty five and take it over in forty eight and that applyed to the same Scale where the first twelve was taken sheweth nine foure fifths and so many inches in length will make one foot If a piece of Timber be seven one quarter broad and five a half deep it is required to finde how much in length shall make a foot Bring seven one quarter and five an a half into quarters makes twenty nine eight hundred twenty two likewise twelve makes 48 then take twelve on any Scale of equall parts and fit it on twenty nine and take it over in forty eight which distance fit again in twenty two and take it over in forty eight and that applyed to the same Scale where the twelve was taken sheweth forty three one third part and so many inches in length make a foot which was required to be done CHAP. XVI To measure Timber that is broader at one end than at the other SUppose a piece being broader at the one end than at the other be given to be measured First take some place neer the bigger end for a meane part then take the breadth and depth thereabout which suppose to be twenty and fifteen then proceed as before so finde you 5 three quarters and so many inches in length make a foot CHAP. XVII How Perpendicular heights may be found without either Instrument or Arithmetick TAke a trencher or any simple boards end of what fashion soever such as you can get draw thereon a line towards one of the sides as the line AB and on the point A raise a perpendicular as AD then in the line AB knock in two pins one at A and the other at B then on the point or pin at A hang a thrid with a plummet then lift up this board with the end A towards the height required till you bring the two pins into one straight line with your eye and the top of the height
of timber whose end shall have more sides than foure may be measured after this manner adde all their sides together and take halfe that number for one side of an unequal squared piece of Timber then let fall a perpendicular from the centre or midst of the figure to the midst of some one side and take that length for the other side of the same piece with which two sides proceed as before is shewed Suppose the figure A to be the end of a piece of Timber of five sides being all equall and each side conteining 12 inches which being added together into one sum will make 60 the half whereof wil be 30 for the breadth of your piece then take the length of the perpendicular falling from the center A to the midst of one of the sides which here is 8 inches for the thicknesse of the same piece with which breadth and thicknesse proceed in all things according to the former part of the tenth Chapter This rule is generall in all kinde of regular Polygons how many sides soever they have Here I might have proceeded to have shewed by what means Pyramidall or picked Timber or Steeples may be measured but considering how little this appertaineth to Carpenters and how sufficiently they be handled by Master Diggs in his Geometricall works I forbeare here to write of them CHAP. XVII How to finde the length of a Foot of Board at any breadth given THe breadth of a Board being given with the number of 12 the side of a square foot of Board you may by the sixt Chapter finde how much in length will make a foot at any given breadth by finding a third proportionall number which shall be to 12 as 12 is to the given breadth As suppose a Board to be 16 inches broad and I would know how much in length will make a foot thereof CHAP. XVIII The breadth and thickness of a piece of Timber given to find how much in length shall make a foot of square Timber at that breadth and thicknesse SUppose a piece of Timber to be 18 inches broad and 14 inches thick First make any angle as DFB and place 18 inches from F to A this is the supposed breadth of your piece then place 12 the side of a Cubicall foot of Timber from F to E and draw the line AE So likewise place 12 from F to G from which point G draw the line GH parallel to AE till it cutteth FD in H so shall HF be the length of asquare foot of flat measure at the former breadth given thus far according to the last Chapter Now to proceed place 14 the thicknesse of your piece from F to C and draw the line CG and lastly from the point H draw the line HI parallel to CG till it cutteth FB in I so shall IF be the length of a foot required which being applyed to your Scale will reach almost unto 7 inches it wanteth but one seventh part of an inch and such is the length of a foot of Timber whose breadth is 18 inches and thicknesse 14 inches CHAP. XIX Of the Table for Board and Square Timber and also for round Timber COncerning the use of these Tables I would have you to understand that I have supposed the Inch to be divided into 10 equall parts and each part divided into 10 equall parts and so the whole inch will contain 100 equall parts A Table for Board measure Inches Feet Inches 10 part of In 10 part of a 10 part 1 12 00 0 0 2 06 00 0 0 3 04 00 0 0 4 03 00 0 0 5 02 04 8 0 6 02 00 0 0 7 01 08 5 7 8 01 06 0 0 9 01 04 0 0 10 01 02 4 0 11 01 01 0 9 12 01 00 0 0 13 Â 11 0 7 14 Â 10 2 8 15 Â 09 6 0 16 Â 9 0 0 17 Â 8 4 7 18 Â 8 0 0 19 Â 7 5 7 20 Â 7 2 0 21 Â 6 8 5 22 Â 6 5 4 23 Â 6 2 6 24 Â 6 0 0 25 Â 5 7 6 16 Â 5 5 3 27 Â 5 3 3 28 Â 5 1 4 29 Â 4 9 6 30 Â 4 8 0 A Table of square Timber measure Inches Feet Inches 10 part of In 10 part of a 10 part 1 144 00 0 0 2 36 00 0 0 3 16 00 0 0 4 9 00 0 0 5 5 09 1 2 6 4 00 0 0 7 2 11 2 6 8 2 03 0 0 9 1 09 3 3 10 1 05 2 8 11 1 02 2 8 12 1 00 0 0 13 Â 10 2 2 14 Â 08 8 1 15 Â 07 6 8 16 Â 6 7 5 17 Â 5 9 7 18 Â 5 3 3 19 Â 4 7 8 20 Â 4 3 2 21 Â 3 9 1 22 Â 3 5 7 23 Â 3 2 6 24 Â 3 0 0 25 Â 2 7 6 26 Â 2 5 5 27 Â 2 3 7 28 Â 2 2 0 29 Â 2 0 5 30 Â 1 9 2 A Table of round Timber measure Inches Feet Inches 10 part of In 10 part of a 10 part 1 113 01 7 1 2 28 03 4 2 3 12 06 8 5 4 7 00 8 5 5 4 06 3 0 6 3 01 7 1 7 2 03 7 0 8 1 09 2 3 9 1 04 7 6 10 1 01 5 7 11 Â 11 2 2 12 Â 09 4 2 13 Â 08 0 3 14 Â 06 9 2 15 Â 06 0 3 16 Â 5 3 0 17 Â 4 6 9 18 Â 4 1 9 19 Â 3 7 6 20 Â 3 3 9 21 Â 3 1 1 22 Â 2 8 0 23 Â 2 5 6 24 Â 2 3 5 25 Â 2 1 7 26 Â 2 0 0 27 Â 1 8 6 28 Â 1 7 3 29 Â 1 6 1 30 Â 1 5 1 The first columne towards the left hand doth contein any number of inches from one to 30. In each of these Tables is set down the length of a foot in feet inches the tenth part of an inch and so to the tenth part of one tenth part of an inch that is to the hundreth part of an inch Of Board Measure An example upon each Table will give more light than many words and therefore first of Board suppose a Board to be 7 inches broad then find 7 in the first columne towards the left hand and over against it under the title of Board Measure you shall find one foot 8 inches 5 tenths of an inch and 7 tenths of one tenth part of an inch and such is the length of a foot of Board at that breadth And so if a Board be 14 inches broad look 14 in the column towards the left hand and against it under the title of Board Measure you shall find 10 inches two tenths of an inch and eight parts of one tenth part of an inch for the length of a foot at that breadth and the like is to be observed for Timber Of square Timber Suppose a piece of
was required to be done CHAP. VI. To lay down sodainly 2 3 or more lines in proportion required IT is required to lay down foure lines in proportion one to another as these foure numbers following The numbers given 60 A 50 B 32 C 23 D Open by chance your Scale and there let it rest then take it over in 60 and in 50 and lay them both down also take it over in 32 and 23 and lay them down and so have you four lines A B C D in proportion according to the four numbers given CHAP. VII In a Map or Plot the length of any line being known thereby to find the length of all or any of the rest AS in the Plot ABCDEF let the length of the line AB be known to be 47 parts on some Scale now it is required to finde the length of the line CD Take the known line AB and fit that in 47 and let the Scale rest then take CD and bring it along the equall parts till it be equally fitted on each side which is 73 parts so is CD 73 of the same as AB is 47 the like of all the rest CHAP. VIII Vnto two lines given to find a third in proportion THe two lines given are A and B and it is required to finde a third in proportion Take the two lines given and apply them to any Scale of equall parts and see how many parts they contein and let A contain 24 parts and B 36 of the same parts then take 36 the length of the line B on some Scale of equall parts and fit that on 24 the line A then let the Scale rest then take it over in the line B viz. 36 and that distance lay downe for the line C which shall be 54 a third line in proportion required The Reason For as 36 is 24 one time half so is 54 once 36 and a half and so consequently 54 is the third proportionall required CHAP. IX Vnto 3 Lines given to find a fourth in proportion that is to perform the Rule of Three in Lines AS let A B C be the 3 lines unto the which it is required to finde a fourth in proportion that is as the first is to the second so is the third to the fourth Take the three lines one after another with your Compasses and apply them to any Scale of equall parts to know their length and suppose you find them as the numbers which stand by them CHAP. X. To divide a line given into two such parts bearing proportion one to the other as two numbers given AS let it be required to divide the given line AB into two such parts bearing proportion one to the other as 28 to 21 viz. that AC may be to CB as 28 to 21. Adde your two given numbers together viz. 28 and 21 make 49 then take with your Compasses the given line AB and fit it in 49 there let the Scale rest then take it over in 28 which set from A to C so is AC to BC as 28 to 21 which was required to be done CHAP. XI To measure flat Measure A Board being 16 inches broad now it is required to finde how much in length makes one foot Take on any Scale of equall parts 12 the number of inches in one foot and fit that in the breadth of the board which is 16 there let the Scale rest then take it over in 12 alwayes and that apply to the same Scale of equall parts where the 12 was taken and it sheweth 9 and so many inches in length make a foot of board required for if a board have 16 inches in length and 9 in breadth these two numbers multiplyed together make 144 inches the number of square inches conteined in a foot of square board or glass c. Let a board be seven inches three quarters broad now it is required to finde how much in length makes one foot Take as before 12 of some Scale of equall parts and fit it on seven three quarters the breadth thereof and then take it over in 12 as before but to fit it in seven three quarters would open the Scale too wide therefore take four times seven three quarters which is 31 fit 12 in that and take it over in four times 12 which is 48 and that distance applyed to the same Scale where the 12 was taken sheweth 18 three fifths and so many in length shall make one foot If a board be two inches broad how much in length shal make a foot Multiply two the breadth of the Board and 12 the inches in the foot by 10 makes 20 and 120 then take of some small Scale 120 which may be done upon some Scale placed upon your Rule and fit that on 20 on the Scale and take it over in 12 and fit it in 2 3 or 4 times 20 and take it over in so many times 12 and that apply to the same Scale where the 120 was taken and it sheweth 72 inches and so many in length is a foot of Board the Board being two inches broad Let a Board be three inches and three quarters broad now you desire to know how much in length maketh a foot Bring three inches and a quarter into quarters and it maketh 13 quarters then multiply 12 the inches in a foot into quarters and it maketh 48 take then 48 parts of some small Scale and fit that in 13 then let the Scale rest and take it over in 12 and apply that to the same Scale where the 48 was taken sheweth 44 and one third part and so many inches in length is required to make a foot But having taken your 48 on some small Scale and are to fit it on 13 now if it open your Scale too wide you may fit it over in two or three times 13 and take it over in so many times 12 as fit 48 in four times 13 that is in 52 and take it over in four times 12 that is in 48 and it sheweth 44 and one third as before being applyed to the same Scale where the 48 was taken Again let a Board be 5 inches and three eight parts of an inch broad and it is required to finde how many inches in length make a foot Bring five and three eights into eights makes 43 and 12 into eights make 96 then take 96 and fit it in 43 or in twice 43 there let the Scale rest and take it over in 12 and also apply it to the same Scale where 96 was taken and it sheweth 26 and three quarters and so much in length makes a foot of Board the breadth being 5 inches and three eight parts of an inch which is the thing desired CHAP. XII To measure Board that is broader at one end then at the other SUppose a Board be broad at one end 20 inches and at the other 16 now it is required to finde how much in length makes one foot throughout the whole Board Adde the breadth at
THE ARTIFICERS PLAIN SCALE OR The Carpenters new Rule In two Parts The first shewing how to measure all Superficies and Solids as Timber Stone Board Glasse c. Geometrically without the help of Arithmetick it being a new way not heretofore practised The second shewing how to measure Board and Timber Instrumentally upon the Scale it selfe without Arithmetick or Geometry but what is common to every man ALSO How to take Heights and Distances severall wayes and to draw the Plot of a Town or City By Thomas Stirrup Philomat London Printed by R. W. Leybourn for Thomas Pirrepont at the Sun in Pauls Church yard 1651. To THE READER Gentle Reader ALthough many excellent both in Arithmetick Geometry upon infallible grounds have put forth divers most certain and sufficient rules for the measuring of board timber yet very few of our common Artificers have been furthered thereby because they have not the art of Arithmetick upon which most of their rules depend The consideration of which with the aptnesse which I see in some of them for the raising of a Perpendicular and the drawing of a Parallel Line upon which most of this Book depends this I say hath been the cause which hath moved me to give them some rules Geometrical whereby they may measure both board timber without the help of Arithmetick Therefore to thy view Gentle Reader that wanteth the art of Arithmetick doe I prefer this short and plain Treatise wherein in the beginning is declared the infallible grounds upon which the whol work doth depend then doth follow the applying of those rules to the present purpose with the declaration of three tables one for Board and one for square Timber and the third for round Timber very fit for all such as stand in need thereof and yet want both Instruments and Arithmetick whereby to use the same In the second Part of this Book is shewed a second way whereby you may measure Board and Timber by Rule and Compasse only without drawing of lines also how to take Heights and Distances several ways without Instrument all which are grounded upon infallible principles Geometricall Thus desiring thee to accept of this little Booke as a taste of my good will towards thee which I wish even so to further thee as I know it sufficient for the true measuring both of Board and Timber Farewell THE CONTENTS THe meaning of certain terms of Geometry used in this Book Page 1 How to raise a perpendicular on any part of a right line given Page 5 How to let fall a perpendicular from a point assigned to a line given Page 7 To a line given to draw a parallel line at any distance required Page 8 To perform the former proposition at a distance required and by a point limited Page 10 Having two lines given to find a third proportionall line to them Page 12 Having three lines given to finde a fourth proportionall line to them Page 14 The making of a Rule or Scale for the measuring of Board and Timber Page 17 How any Board may be measured Geometrically Page 20 How Timber may be measured Geometrically Page 26 Of Round Timber Page 33 How Triangled Timber or Timber which hath but three sides may be measured Page 42 How Timber whose end is a Rhombus is to be measured Page 45 How Timber whose end is a Rhomboiades is measured Page 47 How to measure Timber whose end is a Trapesiam Page 49 How to measure Timber whose sides are many as 5 6 7 8 9 10 or more so they be all equall Page 51 How to finde the length of a foot of Board at any breadth given Page 54 The breadth and thicknesse of a piece of Timber given to finde how much in length shall make a foot of square Timber at that breadth and thicknesse Page 56 How to finde a mean proportionall line between two lines given Page 66 The second Part. OF the Scale and the graduations or divisions thereof and how they are to be used Page 71 To divide a line given into any number of equall parts Page 73 To take any part or parts of a line Page 74 A line conteining any part or parts of a line thereby to finde the whole line Page 75 A line being given conteining any number of equall parts to cut off from it so many as shall be required Page 77 To lay down sodainly two three or more lines in proportion required Page 78 In a Map or Plot the length of any line being known thereby to find the length of all or any of the rest Page 80 Unto two lines given to finde a third in proportion Page 82 Unto three Lines given to finde a fourth in proportion that is to perform the Rule of Three in Lines Page 84 To divide a line given into two such parts bearing proportion one to the other as two numbers given Page 86 To measure flat Measure Page 87 To measure Board that is broader at one end than at the other Page 91 To finde how many square feet any whole Board conteineth without finding how much in length makes a foot Page 93 To measure Board that is broader at the one end then at the other in the same manner Page 95 To measure timber Page 97 To measure timber that is broader at one end than at the other Page 100 How Perpendicular heights may be found without either Instrument or Arithmetick Page 101 How to take the altitude or height of a building by a bowl of water Page 105 How to take the altitude of a Building by a line and plummet the Sun shining Page 107 How to finde the altitude of a Building by two sticks joyned in a right angle Page 109 To finde a Distance by the two sticks joyned square Page 112 How to describe a Town or City according to Chorographicall proportion by the help of a plain glasse Page 116 An Advertisement To the READER FOrasmuch as throughout this whole Book there is mention made of Rules and Scales the making whereof is different from those which are vulgarly made and sold if any therefore be desirous to have any particular Rule mentioned in this book or one Rule to performe all the work in generall he may have them exactly made by Master Anthony Thompson in Hosier lane neer Smithfield THE ARTIFICERS Plain Scale CHAP. I. The meaning of certain terms of Geometry used in this Book BEcause all Carpenters or other Artificers in their Trade or Calling doe in a manuer and according to their fashion use some kind of Geometry although themselves be ignorant thereof therefore I did consider that they might bee sooner brought to measure Board and Timber by that art of Geometry seeing they have their Rule and Compasses by them then by Arithmetick being but few of them can write and therefore uncapable of that art and of them few which can write not one in ten that hath Arithmetick which is the only cause as I suppose that
most of them are so ignorant in this art which doth so much concern them notwithstanding all those excellent Rules which have been formerly delivered by the learned But now to our intended purpose SEeing I shall have occasion in this Work to use some terms of Geometry by which I may with more ease deliver and you with more plainnesse perceive my minde in these things I have therefore set down the meaning as plainly as I can of some Geometricall terms which most serve for our present purpose 1 An Angle is nothing else but a corner made by the meeting of two lines for I speak not of solid angles 2 A right Angle which we call a square angle is that whose two lines comprehending or making the angle stand perpendicular or plumb the one to the other 3 A Perpendicular line is that which stands plumb upright upon another leaning neither the one way nor the other 4 A Superficies is that which hath only Length and Breadth and no Thicknesse at all 5 A Solid or a Body is that which hath Length Breadth and Thicknesse 6 Parallels are those lines that differ every where alike or are not neerer together in one place then another 7 A Figure is any kinde of Superficies or Solid that is bounded about as Triangles Squares Circles Globes Cones Prismes and the rest 8 The Base of a Figure is any side thereof upon which it may be supposed to stand or if you take any side of a Figure for the Ground or Bottome or lower part thereof that same is the Base 9 The height of a Figure is the length of a Perpendicular or plumb line falling from the top thereof to the Base or bottome thereof CHAP. II. How to raise a Perpendicular on any part of a right line given LEt AB be a right line given and let C be a point therein whereon I would raise a perpendicular open the Compasses to any convenient distance and setting one foot in the point C with the other mark on either side thereof the equall distances CE and CF then opening your Compasses to any convenient wider distance with one foot in the points E and F strike two arch lines crossing each other as in D from whence draw the line DC which is a perpendicular to AB or as we call it a square line to the line AB Or you may from the given point C prick out any five equal distances and opening your Compasses to 4 of them with one foot in C strike an arch or piece of a Circle towards N then opening your Compasses to all 5 divisions with one foot in 3 cross the same arch line in N from whence draw the line NC which is a perpendicular to the line AC as before for if 3 lines be joyned together so they be in such proportion as 3 4 and 5 they will make a right angle CHAP. III. How to let fall a Perpendicular from a point assigned to a line given LEt the point given be D in the former Chapter and let the line whereon it should fall be AB open the Compasses to any convenient distance setting one foot in the point D make an arch or piece of a Circle with the other foot till it cut the line AB twice that is at E and F then finde the middle between those two Intersections and from that middle draw a line to the point D which is the point given and that line shal be perpendicular or plumb from the point D to the line AB as was required CHAP. IV. To a line given to draw a parallel line at any distance required SUppose the line given to be AB unto which I must draw a parallel Open your Compasses to the distance required and setting one foot of your Compasses in the end A strike an arch on that side the given line whereon the parallel is to be drawn as the arch C then doe the like in the end B as the arch line D then draw the line CD so as it may but touch or be a touch line to these two arches C and D and this line so drawn shall be parallel to the line AB as was required CHAP. V. To perform the former proposition at a distance required and by a point limited ADmit AB in the former Chapter to be a right line given whereunto it is required to have a parallel line drawn at the distance and by the point C. Place therefore one foot of your Compasses in C from whence take the shortest distance to the line AB as CA at which distance with one foot in the end B with the other strike the arch line D by the extream part of which arch line D and the point C draw the line CD which is parallel to the given line AB which was required CHAP. VI. Having two lines given to finde a third proportionall line to them THe two lines given are A and B and it is required to finde a third line which shall bee in such proportion to A as A is to B. Make any angle whatsoever as the angle HEC. Here note that an angle is always represented by three letters whereof the middle letter represents the angle intended Then place the line A from the angle E to D and the line B from E to F and draw the line DF. Place also the line A from E to H and lastly by the 4 Chapter from the point H draw the line HC parallel to FD. So shall EC be a third proportionall line to the two given lines as was required CHAP. VII Having three lines given to finde a fourth proportionall line to them THE three lines given are A B and C and let it be required to finde a fourth line which shall have such proportion to A as B hath to C make any angle as DGK now seeing the line C hath the same proportion to the linne B as the line A to the line sought for therefore place the line C from G to H and the line B from G to F then draw the line FH now place the line A from G to I by which point I draw the line EI parallel to FH till it cutteth DG in E so have you EG the fourth proportionall line required which is 24. For as the line 12 is to the line 16 so is the line 18 to the line 24 which is the length of the line we sought for These two last Chapters would I have you diligently to consider and throughly to learne because it is the ground-work of that which I intend to deliver in this Booke which being well understood will bring much pleasure and profit to the unlearned Artificer for whose sake this was written CHAP. VIII The making of a Rule or Scale for the measuring of Board and Timber This line thus divided is called a Scale which is no other thing but a right line divided into any number of equall parts be they greater or lesser wider or narrower so they be equall
to the 9th Chapter as if it had been a board This Chapter would I have you well to consider because I do not intend to repeat what I have heare delivered but only describe unto you the end of some pieces according to their formes and so give you some Rules for to measure them by this Chapter CHAP. XI Of round Timber HEre first I would have you to understand what the Circumference the Center and Diameter of a Circle is the Circumference is the line incompassing the Circle the Center is the point in the middest thereof the Diameter is a right line passing by the Center through the whole Circle and divideth the same into two equal parts either halfe of which Diameter is called the Semidiameter Now having found the Circumference of a round piece of Timbe by girding it about with some line I thinke it is heare needfull to give you a Rule for the finding of the Diameter of the same piece Now the Circumference and Diameter being found you may finde the solid content after this manner First take one halfe of the Circumference for the breadth of your piece and one halfe of the Diameter for the thicknesse thereof according to which breadth and thicknesse you may proceed in all things by the former part of the tenth Chapter as if it were an unequall squared piece of Timber as in the figure A take 22 inches the Circumference of the piece for the breadth thereof Or take a quarter of 44 that is 11 for the one side and the whole 14 for the other And take 7 which is the halfe of 14 the Diameter for the thicknesse thereof and so with this breadth and thicknesse proceed in all things according to the former part of the tenth Chapter Of the half-round or quarter or any other portion or part of a Circle FOr this halfe Circle take halfe the arch line CDB which is 11 for the breadth of your piece And one halfe the Diameter which is 7 for the thicknesse thereof and proceeding with this breadth and thicknesse by the tenth Chapter you shall finde the content Now having a piece of Timber whose end shall be like unto this portion of a Circle noted with these letters ABCD before we can give the content thereof it will be needfull to to finde out the Center which for to doe work as followeth A Segment of a Circle being given to finde out the Center and consequently the Diameter and so if need be the whole Circle The center being found draw the lines EA and EC and cast up the whole figure ABCE as before is shew'd and then by the next Chapter finde the content of the Triangle ACE and take it from the content of the whole figure ABCE and that which is lift shall be the content of the figure ABCD as was required By this Rule observed with discretion may all manner of Segments or parts of a Circle whether greater or lesser then a Semicircle be easily measured without further instruction Hithereto have wee shewed the measuring of such Timber as is most in use that is to say of equal squared and also of unequall squared Timber so likewise have we shewed how round Timber and its parts may be measured by the former Rules so So now will I shew how some pieces of extraordinary formes may be brought to be measured by the former Rules CHAP. XII How triangled Timber or Timber which hath but three sides may be measured TRiangles are made of straight lines oâ— crooked or of both together but I speake only of Right lined Triangles which is nothing else but a figure made of three right lines as the figure ABC Triangles are divers both in respect of their sides and angles and may be measured divers wayes but let this one way serve for all take half of the base and suppose it to be one side of a squared piece of Timber take the whole hieght or perpendicular for the other side of the same piece and so measure it by the former part of the tenth Chapter in all respects as there is shewed Let the Triangle ABC be the end of a piece of Timber to be measured which hath but only three sides CHAP. XIII How Timber whose end is a Rhombus or Diamond form is to be measured A Rhombus or Diamond is a figure of foure equall sides but no right Angles such as is the figure ABCD for the measuring whereof observe this example Let the said figure ABCD be the end of a piece of Timber to be measured now taking the length of the side or base AB which is 14 inches for one of the sides of a squared piece of timber and the length of the perpendicular DE which wil be found to be 12 and something better then the eighth part of one more for the other side of the same piece with which two sides as if it were an unequall squared piece of timber proceed in all things according to the former part of the tenth Chapter CHAP. XV. How Timber whose end is a Rhomboides or Diamond-like is measured A Rhomboides or Diamond-like is a figure whose opposite sides and opposite Angles are only equall and it hath no right Angles Such as is the figure FGHI and may be measured after this manner take the length of the side HI or FG which is 16 inches for one side of a squared piece of timber and take the perpendicular FL which is 10 inches for the other side of the same piece so you may measure it by the former part of the tenth Chapter as if it were an unequall squared piece of 16 inches broad and 10 inches thick All other four sided figures besides the true Square and the unequall Square in the tenth Chapter and the Rhombus in the last Chapter and the Rhomboides in this are called Trapezias or Tables CHAP. XV. How to measure Timber whose end is a Trapeziam A Trapeziam is any irregular four sided figure of what fashion soever as the figure ABCD is a Trapeziam and may be cast into two Triangles by drawing the Diagonall line AC and so each Triangle measured as is before shewed which being done adde the contents of them both together and you shall have the content of the whole Trapeziam ABCD. Or you may more readily measure it thus Take one half the Diagonall line AC which in this example will be 8 inches for one side of your piece and take the two perpendiculars BF and DE and joyn them both together in one sum so shall you have in this example 10 inches for the other side of your piece with which two sides as if it were an unequall squared piece of Timber proceed as before in the former part of the tenth Chapter CHAP. XVI How to measure Timber whose sides are many as 5 6 7 8 9 10 or more so they be all equall MAny sided figures are those which have more sides than foure and are generally called Pollygons A piece
required and directly where the thrid falleth there mark it with a prick of your Compasse as at E and draw the line AE now measure the distance between your standing and the base of your altitude which here wee will suppose to be 36 foot as from F to G and take 36 from your Scale set it down from A to D from which point D raise a Perpendicular to cut the plumbe-line AE in E so shall DE be the height required which being applyed unto your Scale will reach unto 32 and so many foot is the altitude GH Here note that the altitude thus found is from the levell of the eye upwards and therefore the height frō the eye downwards is to be added thereto to make it compleat CHAP. XVIII How to take the altitude or height of a building by a bowl of water PLace on the ground a Bowle of water which done erect your body straight up and goe back in a right line from the building till you espie in the center or middle of the water the very top of the altitude which done observe the place of your standing and measure the height of your eye from the ground together with the distance from your standing to the water and the distance from the water to the base or foot of the altitude which being all exactly taken will help you to the altitude required by the Rule of proportion Which will be found to be 66 foot and 8 inches CHAP. XIX How to take the altitude of a Building by a line and plummet the Sun shining If CD the shadow of the line and plummet 4 foot 5 11 give EC 7 foot in altitude what altitude doth 14 foot give which is the shadow of the altitude required Multiply and divide according to the Rule and you shall finde in your Quotient 22 foot which is the true altitude of the building required CHAP. XX. How to find the altitude of a Building by two sticks of one length joyned in a right angle CAuse two sticks to be joyned in a right angle as is in the figure MN and OP having at O a hole made wherein to hang a thrid plummet The two sticks being thus prepared come to the building whose altitude you require which building let be AB then apply the end of your crosse staffe noted with D to your eye hold it up and down till the third and plummet hang just upon the perpendicular then goe backward or forward till your eye at D looking over E espy the top of the building at A which found marke well the place of your standing which is at F and measure the distance from your eye to the ground which is DF and set that same distance off from F to C then measure the distance from C to B for that is the true height of the building AB CHAP. XXI To finde a Distance by the two sticks joyned square THis experiment is grounded upon the fourth proposition of the 6 Booke of Euclid Let the distance which you desire to know be AB set up a staffe at A of four foot long or more or lesse at your pleasure as the staffe AC at the end of the staffe C place a thrid as CD Then hanging the angle of the square on the top of the staffe at C move it up or down till you see the farthest part of your longitude the square so remaining and the staffe not removed draw the string that is fastened at C close by the side of the square till it touch the ground at D then measure how many times the distance DA is conteined in the Staffe for so many times is the Staffe conteined in the longitude Example The Staff supposed four foot high placed at A and the Square being CHAP. XXII How to describe a Town or City according to Chorographicall proportion by the helpe of a plain glasse TO performe this conclusion you must resort to some high place in the Town or Countrey you would describe from whence you may behold all the Castles Ports Harbours Bays Gates Forts and such other notable places as you intend to describe which place being chosen provide a plain glasse which in the midst of the Platforme hang parallel to the Horizon in the doing of which you must be very carefull so that moving up and down the platforme you may in the Center of the Glasse see all those notable places The foundation being laid let us now proceed to the worke and first of all on your platforme you must draw a Meridian line which must passe just under the Glasse so that if a perpendicular line were let fall from the Center of the Glasse to the platforme it might cut the Meridian line at right Angles and by having this line drawn you may draw the line of East and West at right Angles to the Meridian and in like manner the two and thirty points of the Compasse with Circles and Parallels as is usuall in the projecting of Sea-charts so that thereby you may know how all the chief places in the Town are situate and how they bear from you This done move Circularly about the Glasse observing always when you espie any marke in the Center of your Glasse to set up a staffe writing thereupon the name of the place whether it be Village Port Road or such like you shall in the end situate as it were the whole Countrey in due proportion upon your platform so that measuring the distance of every staffe set up from the Center of your platforme and the distance likewise of every staffe from other you may by the Rule of Proportion finde out the distance of every Town Village Fort Haven and the like from your platform and also the distance between any two places there described This Experiment is marveilous pleasant to practise and most exactly serving for the description of a plaid Champion Countrey which when you have thus traced out upon the platform you may by the help of Scale and Compasses project in paper or parchment with a Scale of Leagues Miles Furlongs Paces or other measures as liketh you best FINIS