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
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
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