The carpenters rule, or, a booke shewing many plain waies, truly to measure ordinarie timber, and other extraordinarie sollids, or timber with a detection of sundrie great errors, generally committed by carpenters and others in measuring of timber; tending much to the buyers great losse. Published especially for the good of the Companie of Carpenters in London, and others also; ... By Richard More carpenter. More, Richard, carpenter. This text is an enriched version of the TCP digital transcription A07688 of text S101895 in the English Short Title Catalog (STC 18075). Textual changes and metadata enrichments aim at making the text more computationally tractable, easier to read, and suitable for network-based collaborative curation by amateur and professional end users from many walks of life. The text has been tokenized and linguistically annotated with MorphAdorner. The annotation includes standard spellings that support the display of a text in a standardized format that preserves archaic forms ('loveth', 'seekest'). Textual changes aim at restoring the text the author or stationer meant to publish. This text has not been fully proofread Approx. 117 KB of XML-encoded text transcribed from 33 1-bit group-IV TIFF page images. EarlyPrint Project Evanston,IL, Notre Dame, IN, St. Louis, MO 2018 A07688 STC 18075 ESTC S101895 99837698 99837698 2036 2240880659

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 Early English books online. (EEBO-TCP ; phase 2, no. A07688) Transcribed from: (Early English Books Online ; image set 2036) Images scanned from microfilm: (Early English books, 1475-1640 ; 1028:10) The carpenters rule, or, a booke shewing many plain waies, truly to measure ordinarie timber, and other extraordinarie sollids, or timber with a detection of sundrie great errors, generally committed by carpenters and others in measuring of timber; tending much to the buyers great losse. Published especially for the good of the Companie of Carpenters in London, and others also; ... By Richard More carpenter. More, Richard, carpenter. [8], 56 p. : ill. Imprinted by Felix Kingston, At London : 1602. Title page stained. Reproduction of the original in the British Library. eng Lumber -- Mensuration -- Early works to 1800. eebo The carpenters rule, or, a booke shewing many plain waies, truly to measure ordinarie timber, and other extraordinarie sollids, or timber: with a dete More, Richard, carpenter 1602 60 19575 14 0 4 0 0 0 9.2 B The rate of 9.2 defects per 10,000 words puts this text in the B category of texts with fewer than 10 defects per 10,000 words. 2010-05 Assigned for keying and markup 2010-05 Keyed and coded from ProQuest page images 2010-07 Sampled and proofread 2010-07 Text and markup reviewed and edited 2011-06 Batch review (QC) and XML conversion

THE CA●●●NTERS RVLE , Or , A BOOKE SHEWING MANY 〈◊〉 waies , truly to measure ordinarie Timber , and other extraordinarie sollids , or Timber : WITH A DETECTION OF SVNDRIE great errors , generally committed by Carpenters and others in measuring of Timber ; tending much to the buyers great losse .

PVBLISH●● ESPECIALLY FOR THE GOOD of the 〈…〉 Carpenters in London , and others also ; and is very 〈◊〉 for Masons , Shipwrights , Ioyners , and 〈…〉 to measure Timber and Board , and 〈…〉 other superficies and sollids .

By R●HARD MORE Carpenter .

AT LONDON Imprinted by FELIX KYNGSTON . 1602.

TO THE RIGHT WORSHIPFVLL SIR HENRY BILLINGSLEY Knight , Alderman of the Citie of London .

RIght worshipfull , when I had often considered the great errors , and by error the great losse comming to this Citie , in measuring of Timber after the common wayes ; and had long expected some mans indeuour for redresse , but saw none : I was in a sort prouoked ( though the vnfittest of a number ) to attempt this thing . And because it is not otherwise likely , but that I in this case , shall haue many opposers : I haue therefore emboldned my selfe , humbly to desire your worships protection : crauing also your fauourable acceptance , of this my poore labour ; that so it may be the better accepted for the common good .

Neither is my desire , only to entreate your worships fauour and protection , but also to testifie my dutifull thankfulnes for your great paines and no small cost , in publishing in our English 〈…〉 Euclide● Elements of Geometrie . A booke , from which 〈…〉 well may , so my selfe needes must confesse , that I haue receiued all that little insight in Geometry , which I haue attained vnto . Yea such is that booke ( as I may well say ) that if mens tongues should be silent , yet their actions would not spare , plainly to declare the worthines thereof , to the praise of your labours . And thus in all duetie , crauing your worships protection of 〈◊〉 small fruite of your owne labours ; crauing also pardon for my boldnesse , I humbly take my leaue : desiring God to blesse and keepe you in all good estate , to the glorie of his name , the benefit of the Common-wealth , and the good of your selfe , in this life and the next .

Your Worships 〈◊〉 all dutie RICHARD MORE .
TO THE WORSHIPFVL , THE MASTER , WARDENS , AND Assistants of the Companie of Carpenters of the Citie of London : And to all other the curteous Readers .

AFter I had long considered , and that not without some griefe , the great losse that commeth to those of my Companie and others , by errors ordinarily committed in measuring of timber ; I did often thinke with my selfe what might be the readiest course to redresse them . One while I thought to haue declared to you the Master and Wardens , what errors I had obserued in our ordinarie measure , and so to haue desired you in all dutie to haue prouided a remedie . But when I perceiued , that custome had caused error to be receiued as a truth , and that therefore men would not forsake them , vnlesse they were plainly conuinced to haue erred : And when I did also see , that most men are very ignorant in true measure , though they seeme and professe to know much therein ; and did also remember that not only Carpenters in this Citie doe in this sort erre , but also Carpenters elsewhere , as also Shipwrights & others , and that throughout the whole land , for the most part : then this course seemed to me to be too priuate , and such as was like to doe but little good . The best course that I could thinke vpon , was , to publish some booke , wherein not onely true measure should be set downe , but also the common errors plainly laid open to the capacitie of the simplest ; that so all men might take knowledge thereof .

My selfe I knew to be very vnfit for this worke , both because of mine inabilitie , as also by reason of some exceptions that diuers might take against me , to their owne hindrance and my discouragement . But when I saw none addressed hereunto then ( remembring the saying of a Heathen , that men are 〈◊〉 for themselues , but for their Countrey ) I did set aside all lets , and both boldly and rudely haue put out this my simple booke , in manner as you see .

The booke I haue diuided into three parts . In the first part I declare the errors most commonly committed , and taken for truth , in measuring of timber . In the second part I shew how ordinarie timber may be measured , both by sundrie plaine waies , as also by waies more artificiall . And in the third part I shew , how extraordinarie timber and sollid formes may be measured .

The errors ordinarily committed in measuring timber , are many . But of them all , especially two , bringeth losse to the buyer . The first is by buying of waynie timber and measuring it as square . The other is by taking halfe the breadth and thicknes of a peece being added together , for the square thereof .

These errors are such , as bring apparant dammage : and therefore they are not to be respected lightly . I would they might not bee said to be a great cause , if not of the ouerthrow , yet of the impairing of some mens estates .

For as for wainie timber , who knowes not , that as it is often measured and hewed , a loade and a halfe will not goe so farre in vse , as a loade of good timber being well hewed and iustly measured .

And as for haluing the summe of the two sides for the square , the dammage hereby is exceeding much ; not onely to those that buy much Timber at the first hand , but also , and that especially ( which is the more pitie ) to the poorer sort that buy at the second hand . For my selfe haue abated of that which workmen themselues haue measured a peece vnto by the foresaid false way , after 3 , 4 , 5 , yea 6 foote of timber in 20 : and yet I haue giuen them no more then their measure .

I cannot stand to amplifie these errors in this place : they are more fully set downe in the booke it selfe . By which ( if they be well weighed and considered vpon ) it will plainly appeare to be true as I haue said , namely , that there commeth great losse by the ordinarie measuring of timber .

Now seeing the case so stands , my desire and request is , euen to euery man , but especially to you the Master , Wardens , and Assistants of our Companie ; that you would put your helping hand to reforme these errors . Many reasons might be alleaged to induce you hereunto , but for some reasons best knowne to my selfe , I omit them all , not doubting of your care and regard hereunto .

I haue heard some men wish , that wee had an act of Parliament procured for redresse of false measure . But truly that seemeth to be the law of our land alreadie , though not the statute law : for the law intendeth that the buyer should haue that which he buyeth . And when we buy a loade of timber , doe we not intend that we haue bought 50. foote of sollid timber , euery foote being full 12. inches euery way , that is , in length , breadth & thicknes ? Why then may we not require so much for a loade , and why is not the seller bound to deliuer it ? And if a man that buyeth twentie bushels of wheate , will not take ninteene ; or if he buy a yard of cloath , he will not take three quarters : Then why should we ( buying a loade of timber ) take 40. foote for it , yea and lesse too ( as oft times it fals out ) whereas wee should haue 50. foote .

I know that many men will obiect many things ; and it is not for me at this time to refute them . Onely whereas some may say , that what is wanting in the measure , is allowed in the price . I answere : 1. That is not alwaies . 2. But where allowance is made , it is not according to the value of the losse comming by the false measure . 3. Againe , it is a preposterous course , to raise or let fall a price by altering of a set measure to an vncertaintie , whereas we vse in all trades to raise or let fall the price according to the goodnes or badnes of the wares , keeping stil the same measure for them both . And surely herein we are but deceiued : For oft times we haue twelue pence bated in the price of a loade , and lose foure or fiue shillings in the measure thereof .

Therefore let vs either haue iust measure , if we buy by measure ; or els let vs buy it by gesse . For it is a shame to pretend to measure truly , and yet doe nothing lesse .

As touching my selfe , I desire that my endeuour may be fauourably accepted : for I write with no other mind then as tendring my Companies good . If any thing that I haue said , be impugned as false , I will be readie whensoeuer you will call me , to proue my affirmations . If any man of knowledge shall except against me , for my rude writing and grosse demonstrations : I pray them to vnderstand , that I writ not for them , but for the simple ; and therfore I demonstrate grossely , not being well able to doe better : Also , that I am not a scholler but a Carpenter , and therefore could not but write rudely . Vnto such also I doe further say , that my desire is , that they would take knowledge , that this hath been one cause of my writing , euen to prouoke them ( seeing our present error and losse , with which peraduenture they were not so well acquainted as my selfe ) to take some paines to reforme the same , in better sort then either I haue or can doe .

While this booke was in printing I came to the sight of a Ruler , sometimes inuented by one Master Bedwel ; which as it is easie , so it is most speedie , and not lesse certaine ( being truly made ) for the measuring of timber and boord ; which I expect and hope will be shortly published for the common good . But I will not trouble you with more words . Onely let me put you in minde of this one thing : That in as much as nothing is more fit for Carpenters to make them readie , not only in measure but also in other things , then Geometrie : that therefore such as are of reasonable capacitie , would spend some part of their spare times to studie the same , in some measure at the least . For your furtherance herein , there are especiall good helps , both by the Lecture at Gresham Colledge euery Thursday in the Terme times ( if you were but a little entered ) as also by Euclides Elements of Geometrie , which the right Worshipfull Sir Henry Billingsley Knight , to the great good of the Common-wealth , though to his own great trauell and charge , hath translated and published in our English tongue . But I am ouer tedious : I pray you accept of this my poore endeuour , as a fruit of my heartie good will to my Companie , and others whom it may concerne . If it take none effect , I shall be sorrie : but if it doe any good , I haue enough for my labour .

RICHARD MORE .
THE FIRST PART OF THE CARPENTERS RVLE , shewing diuers grosse errors , generallie committed by Carpenters and others in measuring of Timber , arising from the Rule , and the vse thereof .
CHAPTER . 1. The error in the Rules , and the reason thereof .

THose that indeuour to amend things amisse , must first of all labour to proue the faults , especially when the parties deceiued , are not yet perswaded , that they are out of the way . And because men little regarde to returne vnto the right , when they doe not perceiue that they are far in the wrong : I haue therefore thought good to spend this first part of my booke , in shewing and declaring , that great error is to too ordinarily committed ( by the most part of Carpenters and others in and about London , and elsewhere ) in measuring of timber .

These errors doe arise , either by meanes of the Rule or Ruler , with which they measure Timber ; or else by the misapplying or vsing thereof . I meane , the ordinarie way of measuring Timber is very erronious . But first of the faults to be obserued in the Ruler it selfe .

The ordinarie Rule which Carpenters , Shipwrights , and others doe vse to measure Timber withall , was inuented and published by Master Leonard Digger . Digges Tectonicon . Now those Rules that agrée with that booke are so true , as that I déeme them ouercurious that except against them ; and they will serue truly to measure Timber withall . But the most which haue that kinde of measure on them which Master Digges describeth , are very false . I meane , those diuisions or strikes which are set on them for measuring of Boord and Timber , are not in their right places .

And as those strikes and diuissons agrée not with the truth , so vpon diuers Rules you shall finde them to disagrée one from another : Yea hardly shall you sée two Rules that do euery where agrée . Neither is this error insensible , and so not to be respected ; but apparantly grosse , and therefore not to be tollerated . For my selfe haue séene some Rules that in some of their diuisions haue erred from the truth and from one another , after the rate of sixe foote in an hundreth of Boord , and after foure foote in a lode of Timber . Neither is this error rare and in some alone , but so generall , as that if a man would examine them , he should be forced to say , that true Rules are very scant .

If I were demaunded the cause of this fault or error in the Rule , I should answer , that in mine opinion it ariseth hence . Many will take vpon them to make Rules , who haue not Master Digges his booke to make them by ; or if they haue it , vnderstand it not , neither the ground from whence the Rule is made , by which they might examine the Rules which they make : which ground is set downe in the eleuenth and twelfth chapters of the second part of this booke . But ordinarily men make them one by another , wherof hath growne the former error .

Thus much for the Rule . Now should follow to shew the errors in measuring Boord and Planke : But because I haue obserued no maine error in measuring of ordinarie planke or Boord ( though great fault in buying of waynie Boord , and measuring it as square ) which I speake of in the next chapter following . I will therefore say nothing thereof now , but passe to errors in measuring of Timber .

CHAP. 2. The great losse that commeth in buying of timber and Boord that is wainie , and measuring it as square .

EVery man that buyeth Boord or Timber , intendeth to haue for a foote as followeth , that is to say , in Boord twelue ynches in length ; and twelue in breadth ; and in timber , twelue ynches in length , twelue in bredth and twelue in thicknes . But who so buyeth Timber or Boord that is waynie , and taketh it as if it were not waynie , hath not his measure for his monie , but farre lesse . This is a great fault , and generall throughout the whole companie , and amongst others also ; yea and apparantly knowne and noted by euery man , and yet not amended nor indeuoured to be amended by any .

This fault in Boord séemeth to come from carelesnesse , and in Timber ( as I take it ) it did first grow from ignorance . For when men knew not now to measure a waynie péece of Timber , nor bow to deduct the waynes , hauing measured it as not waynie ; then they did carelesly ouerpasse this thing as a necessarie euill , or as an inconuenience that could not be auoyded .

But it may be said , that there is allowance in the price . I answere , what is it in a loade of Timber or in a persell of Boord to haue twelue pence abated in the price , and to lose foure or fiue shillings in the measure . This is to get in the hundreth and lose in the Shire .

And besides this losse that commeth by waynie Timber , there is also another great losse thereby not to be forgotten . You know that when the Timber is very waynie , we are compelled to hew away a great part of it , after it is sawed to bring it to some reasonable squarenes fit for vse . If then you shall put this losse to the other , you shall finde , that it is no small damage that commeth by buying of waynie timber . And therefore there should be an extraordinarie care and respect had to the measure , that we lose not by it , séeing so great losse commeth by the stuffe it selfe .

My desire therefore is , that as other faults ; so this may be amended . Not that I would haue Timber hewed dye square , for that were a spoyle of much good Timber : nor that I would haue girt measure vsed , for that ( in this kinde ) is very false , as appeareth afterwards : But I would haue the want or wainynesse deducted or taken out of the whole péece ; which how it may be don , is shewed in the second part of this booke , Chapter 5.

CHAP. 3. That it is very false , to take for the square of a peece of Timber , the halfe of the breadth and thicknes , added together : by which square , should be found the length of a foote in that peece .

IT is a common practise , and that of ould , that who so of the ordinarie and common sort of men , doe measure Timber , they take this course . First they ad or put together the breadth and thicknesse of the péece of Timber to be measured ; then they take the halfe thereof for the square of the same péece ; and then according to that square , they giue the length of a foote by the Rule . As for example , if the péece be ten ynches one way , and fourtéene ynches the other way , they ad together ten and fourtéene , which make twentie foure ; the halfe of twentie foure is twelue , which ( say they ) is the square of this péece of Timber ; which is very false . And yet this course is so generally receiued , and so commonly reputed and taken for good , as that he shall be thought ouercurious of some , that shall but except against it . Howbeit if you will but marke that which followeth , you shall plainely sée that it is an error , and that a great one .

diagram
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Thus then appeareth the error . If you will know the greatnes thereof , you shall vnderstand , that in the former example , and so in all Timber that haue like differences of sides as that péece hath , there is losse alwayes to the buyer , a quarter of that which he buyeth , or one loade of Timber in foure . If the difference of the sides be smaller , then is the losse smaller : but if it be greater , then is the losse greater . And in generall take this for a rule . That there is lost by this kinde of false measuring , a square péece of Timber , as long as the whole péece to be measured , and as broade and thicke as halfe the difference of the two sides of the said péece of Timber to be measured . As for example , let the sides be ten and fourtéene , then their difference is foure ( for fourtéene is more then ten , by foure ) now the halfe of foure is two , which two is the side of the said lost square péece of Timber throughout the péece .

Now séeing this error of taking halfe the sum of the sides for the square , is very manifest ; and séeing also it is an error tending to the great losse of the buyer , yea to the losse of thrée loade in foure , and more also as the differences may be : Therefore I desire that none would follow that corrupt custome , and most false way ; but suffer themselues to be better instructed , in more righter wayes . Which if they refuse to doe , and shall buy much Timber after that rate , their purses are like to pay for it .

But some may say vnto me , this fault is by diuers discerned , and a redresse not altogether vnthought of . I graunt , many doe sée it , and doe indeuour to reforme it thus . If a péece of Timber bee eightéene ynches one way , and sixe ynches the other way , they imagine it to consist of thrée squares of sixe ynches a péece ( as indéede it doth ) the which they measure seuerally , and then adde al their contents together , which they take for the measure of the whole péece .

This is true both in this example , and in all others that will fall out in any number of iust squares . But when the sides of the Timber to be measured , haue not this answereablenes in them , as nine ynches one way and fourtéene the other way ; or seauen and nintéene , or such like : then they are to séeke . For when they haue taken the thicknes out of the breadth as often as they may , then is the remainder vnsquare : In measuring whereof , they faile and erre , as a foresaid ; and yet they professe , not to regard it . For ( say they ) if the difference of the sides be not past thrée or foure ynches , that péece may be measured according to the said erronious way , and no great losse happen thereby . But what the losse is , I haue shewed before , which although it séeme but small , yet in time it ariseth to no small matter .

CHAP. 4. The error in measuring of timber not hewed square but beuell .

THough this error be not of any great moment , because Timber hewed beuell or skew , commeth not often to be measured : yet because sometimes there is such Timber to be sould , I would not haue workemen to be ignorant , that it is an apparant error to measure it , as if it were hewed square .

When the corners of the Timber are hewed vnsquare ; I doe not meane waynie ; for of that I haue spoken before in the second chapter : but when it is hewed beuell or diamond wise ( as sometimes it is by the carelesnesse or vnskilfulnes of the hewer ) then to take the square of such Timber , according to the breadth of the sides thereof , is vntrue ; for the sides are longer then if it were square , and so giues the péece to be bigger then it is .

How great this error is , I stand not to shew at this time : Only note that it doth increase and decrease according as the corners of the Timber doe swarue from square . But how such Timber is to be measured , sée in the third part , Chapter she sixt .

CHAP. 5. The error in the ordinarie measuring of round Timber .

THe former errors doe alwayes bring losse to the buyer , but these that follow doe alwayes bring losse to the seller . And though such round Timber commeth not often to be measured here in London ( the worse for the buyer ) yet would I haue Carpenters know the error which is vsed in measuring the same ; which is this .

They girt the Timber rounde about with a line or thread , and then take one quarter thereof for the square ( as we call it ) of the péece to be measured , and so finde out ( as they thinke ) the length of a foote in that péece of Timber or trée .

This is a grosse error , and yéeldeth not the content of the péece or trée , by one fift part of it and more .

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But how round Timber is to be measured is taught in the sixt chapter of the second part , and in the ninth chapter of the third part of this booke : which when you haue learned and tryed , then you will tell your selfe , that in measuring round Timber after the former way , is very erronious and intolerable .

CHAP. 5. Error in measuring of Timber not fully round , but somewhat flat called Ouall fashion .
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So that to measure Ouals by the former way , is so vncertaine , that you may haue peraduenture more then you should , and peraduenture lesse then you should by a fift part , a fourth part , a third part , and so forth . But because ( ordinarily ) trées or Timber that are Duall fashion , doe not much vary from a round : Therefore the losse is alwayes to the seller , being measured as a foresaid .

Here I had purposed to haue somewhat digressed from my generall purpose , and to haue shewed the great losse that commeth by buying of fagots that are bound flat : But the Statute for fewell in the last Parliament hath preuented me : Where it is enacted , that bands of fagots shall be Round . But yet because I feare al men vnderstand not what is ment by the word Round , or if they doe , yet they will not regarde to redresse it : I pray you therefore take knowledge , that where as men doe commonly buy the flattest fagots for the greatest , they are deceiued ; for indéede they are lesse then the round by farre ; and euermore the flatter they are , the lesser they are . And to speake no more then my selfe haue tryed ; I haue taken foure fagots ( indéede of the flattest ) and haue put the wood of them all into thrée of their owne bands without strayning . Lo then the great losse that comes to folke by this error . That which I say of fagots I intend also to be spoken of billets : For the flatter they are , the lesser they are , being no more in compasse then the statute requires .

I néede not stand to demonstrate that this is an error , if I were perswaded you vnderstood that which is said before of Ouals . But for farther proofe , take a quarte pot of peuter that is round , and make it more flat by crushing in the sides thereof ; and you shall finde that it will not hold a quart of water as before it did , and yet the compasse of the pot is as much as it was before . You may also proue the same by a Boord thus . The fagot band ought to be by the Statute twentie foure ynches about : So then a Boordes end of a foote broad will hardly goe into one of those bands ; but if you cleaue that Boordes end into a dosen or more péeces ; you shall well perceiue that you may put foure and more of such Boordes being so clest , into that band being round . This haue I written , not carpingly , but as desierous that the buyers would not deceiue themselues by their eye , neither that they should be deceiued by others .

CHAP. 7. The error in measuring timber that is waynie .

I Told you before in the second chapter , that there comes great losse to Carpenters in buying of waynie Timber after the ordinarie way : But here I shew the losse that comes ( indéede not to the buyer but the seller ) in measuring of waynie Timber .

It hath béen vsed ( as I am informed ) that if Timber were waynie , the buyer would measure it as by girting it about , and taking one quarter of the comqasse for the square , as in the case of round Timber before shewed , chapter the sixth . But it séemeth to me , that the seller grew wearie of this way , so that in my time to my knowledge , it hath not béen vsed in or about this Citie .

diagram

How much this losse is , doth not appeare by this demonstration . But in this figure ( if you will take my word ) there is lost a sixt part full . But I had rather you would trie the losse your selues , by the rule taught in the fift chapter of the second part of this booke , then to beleeue me in this case .

CHAP. 8. That in measuring of Timber that is taper growne , it is false to take the square of the middle thereof , and so to giue vp the content by the common Rule .

IT is so vsuall a thing in measuring of Timber that is taper growne , or narrower at one end then at another , to take the square of the middle of the péece , and so to measure it by the common Rule ; as that I shall be not onely not beléeued , but also reputed to haue apparantly erred in affirming the contrary : for it will be said , that that which it hath too much in the greater halfe , it wanteth in the lesser halfe . Indéede this is true in Boords or other superficies that taper , whose sides are straight : but in a solid or péece of Timber it is simply false .

How to demonstrate the error on a paper , so as that it may be easily conceiued , is very hard : but in a solid it may plainely be declared to the vnderstanding of the simplest .

If a Pyramis ( that is , a pyked péece of Timber , sharpe at one end , like a pyked stéeple ) were so measured , there would be lost very much . And alwayes the lesser the péece doth taper , the lesser is the losse . In generall , there is lost , ( by certaine pyramides ) a péece of Timber whose length is the sixt part of the length of the whole péece , and whose end or base is contained vnder halfe the difference of the sides of both the ends of such a tapered péece : that is , take for the breadth of the lost péce , halfe the difference of the breadth of both ends ; and for the thicknes of the lost péce , halfe the difference of the thicknes of both ends . As for example , if the péece be twentie ynches broad and sixtéene ynches thicke at one end , and fiftéene ynches broade , and twelue ynches thicke at the other end , and twentie foure foot long : then the last péece is foure foote long , two ynches and a halfe broade , and two ynches thick : for two and a halfe is the halfe of fiue , which is the difference of the breadth of both ends ; and two is the halfe of foure , which is the difference of the thicknes of both ends .

The losse that commeth by this error is to the seller : and yet in truth it is such as yéeldeth no profit to the buyer . For we lose more by such Timber , being imployed to any ordinarie vse , then that aduantage by the measure doth come vnto . But yet I would not haue workemen to take that way for a true measure which is false ; neither to be ignorant how such Timber is to be measured , if occasion doe offer , or if men shall require it .

In the second and third part of this booke , where I shew how to measure Timber , I will shew how tapered Timber may be measured : from which I will no longer detaine you , taking that to be sufficient which I haue written alreadie , to shew the grossest of the errors , generally committed by Carpenters and others , in measuring of Timber .

The end of the first part .
THE SECOND PART OF THE CARPENTERS RVLE , Wherein is set downe sundrie plaine wayes , truly to measure ordinarie Timber , and Boord .
CHAPTER . 1. How Boords and planks are ordinarily measured .

IN the former part of this booke , I told you that I had not obserued any maine error in measuring of ordinarie Boord and planke , though great fault in buying of wainie boord and planke , and measuring it as square : And therefore it shall not be needefull to inuent or set downe any new wayes of measuring of them : onely it shall suffice to repeate the wayes alreadie in vse ordinarily , which are thrée . Of which , the first is troublesome , and therefore rather to be knowne for varietie , then vsed in measuring . The second is true , but longer in doing then the third ; and therefore I would aduise the third to be followed and vsed . Only looke that your Rule be truely made , and that diligence be vsed in measuring therewith : else there will follow error of necessitie , bee the wayes to measure by , neuer so exact . The first way to measure boord .

The first way let be this . And first note that for this purpose , I call an ynch that which is an inch broad and twelue ynches long , of which ynches , twelue doe make a foote . Now if the Boord to be measured be vnder a foote in breadth , then reckon how many times twelue of the said ynches you can haue in the whole Boord , and so many foote is there in it . If the Boord be iust a foote broade , then it containes so many foote as it is féete in length . But if it be more then a foote broad ; then measure it as a foresaid as if it were two Boords , the one of a foote broade , the other so broad as the odde inches remaining : then adde their contents together for the content of the whole boord .

This rule , to them that vnderstand it , may be fitlier set downe in other words thus . Multiplie the number of inches in the breadth , by the number of feete in the length : then diuide the product by twelue , and the quotient shewes the number of féete in the boord .

The second way is thus , Measure the length of the boord : The second way to measure boord . Then ( if the same boord be vnder a foote brode ) draw or set the breadth thereof so many times on the boord , as it cōntayneth féete in length , and there make a pricke : And then so many foote as there are from the beginning to that pricke , so many foote doth the boord contayne .

And if the boord be aboue a foote broade , then you may take out the euen féete , and measure the remainder as aforesaid ; and then adde both those together for the content of the whole .

The third way , is this . Take the breadth of the boord ; The third way to measure boord . then finde by the Rule what length , maketh a foote , at that breadth ; and then sée how many times that length is in the boord , and so many foote are in it . These I stand not vpon , because they are well enough knowne to all men .

CHAP. 2. How to measure Timber by a certaine way called Drawght-measure .

THis way may fall out so tedious , as that I would not haue troubled either my selfe to write it , or you to reade it , had it not béen that I would not haue you ignorant of that way which some doe vse in some places of this Land . The way is thus .

diagram

This rule aforesaid will hold true for any Timber that is vnder a foote square . But if it be aboue a foote square , and not aboue two foote square ; then must you imagine that one quarter thereof is to be measured ( and therefore take but halfe of each of the two sides ) and when you haue so cast it vp as if it were but a quarter , accoumpt foure times so much for the content of the whole péece .

But if the péece of Timber be aboue two foote square , and not aboue foure foote square ; then may you imagine that a sixtéenth part thereof is to be measured . Which done , accoumpt sixtéene times somuch for the content of the whole péece : now to take the sixtéenth part , you may take a quarter of the breadth , and a quarter of the thicknes .

But this kinde of measure if it fall out in a péece of Timber aboue a foote square is so tedious , as that I doe wonder any man would content himselfe with it . And yet because I haue set it downe , take one note more , which is this . If the péece to be measured haue any od ynches in the length , either at the first or second working : then must you , hauing drawne the want so many times as there is féete in length , adde thereunto such part of the want , as the od measure of the length is part of a foote . And thus much of Drawght-measure .

CHAP. 3. How Timber may be measured by Boord measure onely .

WHen I considered the readines of many men in measuring of boord by one of the thrée former wayes , and how spéedily they will cast vp the whole stocke hauing measured one boord , by the obseruing of skores , tens , and fiues therein ; and withall considering the great error that those men run into in measuring of Timber : This moued me to thinke of some course , how these men that are so expert in measuring of boord , might by the same way measure any square péece of Timber .

This way , though it be true , yet it is troublesome ; and therefore I would wish them rather to vse some of the wayes following : But if their readines in casting vp a stocke of boords shall cause them to accept of this way , I will not be their let . Now the way is this .

Suppose the péece of Timber ( to be measured ) to be a stocke of boordes , consisting of so many boordes as there be inches in the thicknes of the péece . Then measure one of those supposed boordes : that done , cast vp the whole stocke . Now this being performed , you are to know that euery twelue foote of boord is a foote of Timber , as euery man may perceiue ; and sixe foote of boord is halfe a foote of Timber ; and thrée foote , is a quarter of a foote of timber . Also , euery thréeskore foote of boord is fiue foote of Timber , and so , sixeskore foote of boord is ten foote of Timber , nineskore foote of boord is fiftéene foote of Timber , twelueskore is twentie foote , fiftéeneskore or thrée hundreth , is twentie and fiue foote , sixe hundreth foote of boord is fiftie foote of Timber , and so foorth . Now if this be obserued , it is easie for any man to cast vp the content of any ordinarie péece of Timber truly .

There is farther to be noted , that if there be any od halfe ynches in the thicknes of the péece to be measured , that then it be accoumpted as halfe of one of the supposed boordes of the stocke : And if there be a quarter of one inch , then it is a quarter of a boord : And if there be an od thrée quarters of an inch , it is thrée quarters of a boord ; and if there be any other od part of an inch , it is the same part of a boord , and must be counted with the whole stocke .

This way is so plaine that it néedeth no example . I suppose the simplest may vnderstand it , and therefore I will say no more of it : Onely this you may obserue , that you may measure the péece , by imagining it to be diuided into two inch , thrée inch , or foure inch planke ; so that you search out how many foote of such planke is contayned in a foote of Timber .

CHAP. 4. How Timber may be measured after the ordinarie way , by deducting the lost square .

IT was obserued in the third chapter of the first part of this booke , that when any péece of Timber is measured after the most common way , which is by adding the breadth and thicknes together , and taking the halfe thereof for the square of the péece , &c. that then there is alwayes lost a square péece of Timber of the whole length of the péece to be measured , whose square is halfe the difference of the two sides of the péece . Now by obseruing of this lost square , any ordinarie péece of Timber may be truly measured as followeth .

Measure the péece , by taking for the square thereof , the halfe of the summe or totall of the two sides , as is aforesaid , and as is vsually accustomed . This done , note the difference of the two sides : then take the halfe of that difference for the side of the lost square , which is all the length of the péece . Then measure that supposed or lost square , and what it amounteth vnto , deduct out of the measure of the whole péece being measured after the said common way , and the remainder is the true content of the péece .

diagram

And because the lost square may haue in it an odde halfe inch , as some times it hath , as thrée inches and ahalfe or such like : I haue therefore added vnto the common rule , what length maketh a foote of Timber from an inch to ten inches , noting still euery halfe inch : As you may behold in the table of Timber measure hereafter following in the latter ende of this second part . For to take halfe the difference betwixt the two next adioyning numbers of inches added to the lesser of them ( as commonly men doe ) is not true : As for example ; at an inch square one hundred forty foure foote in length maketh a foote of timber : at two inches , thirtie six foote doth it : so that if the halfe betwixt thirtie sixe and one hundred and forty foure were taken and added to thirtie sixe for the length of a foote at an inch and a halfe square , it would be nintie foote , whereas in truth it is but sixtie foure foote : so in this example it erreth twentie sixe foote in length of a péece of Timber of an inch and a halfe square . And how so euer it be not so much in other numbers aboue two inches square , yet it alwayes erreth more or lesse .

CHAP. 5. How wainie or canted Timber may be measured .

ALthough there be diuers wayes to measure wainie or canted Timber , yet I will obserue onely this one plaine way . Measure then the péece as if it were not wainie , out of which deduct the waines , and the remainder is the true content of the péece .

How to deduct the waines is better taught by an example then by a precept : and therefore suppose the figure A were the ende of a péece of Timber of ten inches square , hauing one corner thereof wainie : this péece measure as if it were not wainie . That done , measure what the waine is either way from the corners , which I call the wants , because the péece

diagram
wanteth so much of square ; which want in this example is foure inches either way , as you may sée by the two prickt lines with the figure of foure standing by either of them . This measure as if it were a péece of Timber of foure inches either way , and as long as the whole péece . That done take halfe the content thereof ; the which halfe deduct out of the measure of the whole péece as it was before measured ; and the remainder will be the true content thereof . And as you doe with this corner , so must you doe with the rest , if they be wainie also .

And here I would haue you note , that where I bid the halfe of the content of the said square to be deducted , that the reason hereof is , for that in Geometry , all paralletograms and parallelipipidons , are deuided into two equall partes by their diameters , as it is demonstrated in Euclide , booke , 1. proposition 34. and booke 11. proposition 18. That is to say in playner termes for our purpose , a boord or a péece of Timber which haue their opposite or contrarie sides equally distant , being cut into two péeces by a straight line passing by the opposite corners , as you may sée in the figure B , I say , that boord or péece of Timber , is cut into two equall parts . So that it is manifest by this ,

diagram
that the said waine is halfe so much as the péece whose sides are equall to the said wants .

CHAP. 6. How round Timber may be measured .

I Told you before , that I would not trouble you with Geometricall workes . And howsoeuer I doe hereafter set downe some more artificiall wayes to measure by , yet my indeauor especially is to correct the common errors by that knowledge which most men haue alreadie : and therefore you may measure round Timber plainly in this sort . Girt the piece about with a line or thred : then take a quarter thereof for the square of that piece , ( though indeede it be not the square ) and so measure it after the common way . But now , note that which I told you in the first part of this booke , namely that this piece is more then it is thus measured to be . See a more exact way in the third part of this book , chap. the 10. And therefore to finde the true content , adde to that which you haue measured the piece to be , a quarter thereof , and also after the rate of a foote in a loade , and then the totall will shew the content of the piece , though not exactly , yet so neere , as that the losse will not bring any dammage to any man , that is worth looking after .

Take an example for more plainnes . Suppose the tree being measured after the former way to containe twentie foot : then must you adde thereto a quarter thereof which is 5 , and that makes 25 foote . Then adde to that , halfe a foote , which is after a foote in a loade , and the content of the piece will amount to 25 foote and a halfe .

And here is to be noted , that if the piece bee not exactly round , that then this measure will not hold , as was shewed in the seuenth Chapter of the former part of this booke . But how such Timber is to be measured , see in the latter end of the eleuenth Chapter of the third part .

CHAP. 7. How Timber may be measured by Arithmeticke .

THough the former waies for measuring of Timber be sufficient , yet because varietie giueth pleasure , and these that follow admit of lesse error in the working then some of the former , I haue therefore thought good to set them downe as followeth . And if any man shall obiect that I spend this labour in vaine , because Carpenters commonly haue not Arithmetick . I answere , that some haue more knowledge therin , then they know how to applie to the present purpose ; for whom chiefly I haue written this chapter and some others following : as also I could wish that others which haue not , would learne ; which they may do by bookes which are extant : or at least forbeare to be common Rule-makers , vntill they haue knowledge therein , or well vnderstand the tables which are for that purpose .

Here first I should haue shewed how to measure boord by Arithmeticke : but because it little differeth from measuring of paralellograms , I will therfore referre it to the third part of this booke , where it is taught how to measure them .

Now to measure Timber by Arithmeticke , doe thus . Reduce the length of the piece to be measured into inches , by multiplying the number of féet therein contained by twelue . Then multiplie the breadth by the thicknes , and the product or summe thereof multiplie by the length , being also reduced into inches , as aforesaid . Then diuide that product or of come by 1728 ( which is the number of inches in a foote of timber ) and the quotient will giue you the number of feete contained in that piece of timber .

Here it shall not be vnnecessarie to giue two examples ; the one where the lengeth , breadth and thicknes are euen inches ; and the other , where they haue odde parts of an inch .

Suppose the piece of timber to be measured to be nine foote in length , eightéene inches in breadth , and sixe inches in thicknes . Now multiplie nine foote the length by twelue , and that yéelds 108. inches , which being multiplied by 18. the breadth , makes 1944. and that being multiplied by sixe the thicknes , yéeldeth 11664. which is the content of the whole piece in inches . Then diuide this 11664. by 1728. and the quotient will be sixe foote and 1296/1728 parts of a foote , which being reduced to his lowest denomination , is ¾ of a foot . So that the whole piece is sixe foote and three quarters . And if you would know how many quarters of a foote is in any such fraction , you may diuide it by 432 the number of inches in a quarter of a foote of timber , & the quotient will tell you . But suppose the piece haue in the breadth or thicknes , an od quarter , or halfe , or thrée quarters of an inch : then reduce the length , breadth and thicknes into quarters of inches : after that , multiplie them being so reduced one into another , that is , the length by the breadth , and that of come or product by the thicknes . Then diuide that totall by 110592 ( which is the number of quarters of inches contained in a foote of timber ) and the quotient will shew you how many foote is contained in the piece . As for example , let the piece be 20 foote long , 8 inches and a halfe broad , and 11 inches and three quarters thicke : First I reduce 20 foote the length into quarters of inches by multiplying 20 by 12 , which is 240 , which brings 20 foote into inches : then I multiplie that 240 by 4 , to bring them into quarters of inches , which is 960. Secondly , reduce 8 inches and a halfe the breadth , into quarters of inches , which is 34. Thirdly , reduce 11. inches and thrée quarters the thicknes into quarters of inches , which is 47. Then multiplie 960 , and 34 , and 47 , together , and they make 1534080. Lastly , diuide these by 110592 , and the quotient will be 13 , and 96384 odde quarters of inches : which being diuided by 27648 , the number of quarters of an inch in a quarter * * Note here that I mean not by a quarter of a foote , a square of 3. inches , which is a quarter of a foote one way : but the fourth part of a square foot of timber . of a foot of timber , the quotient will be three quarters of a foote , and somewhat more , but it is not to be regarded in a whole piece of timber : so that the content of the piece is 13 foote , three quarters of a foote and somewhat more . This is more tedious to teach then to practise . Behold the worke on the other side of the leafe .

Note , that where there is no odde parts of an inch in the breadth or thicknes , then you may more briefly cast vp the content thus : Multiplie the number of inches in the breadth by the number of inches in the thicknes , and the product thereof multiplie by the number of féete in the length . Then diuide that totall by 144 , and the quotient will giue the content of the péece in feete . As in the first example behold .

CHAP. 8. A second way how Timber may be measured by Arithmetick .

MVltiplie the breadth in the thicknes , and by the product or ofcome thereof diuide 1728 , and the quotient will shew you how much in length maketh a foote of that piece ; according to which you must measure the whole piece . And if there be any odde quarters , or halfe inches in the breadth or thicknes , then reduce the two sides into quarters of inches , as is taught in the last Chapter . Then hauing reduced them , multiplie the breadth by the thicknes , and by their product diuide 27648. and the quotient shewes you the length of the foote in inches , which you may bring into féete by diuiding them by 12. Behold an example of either kind in the worke following .

Example 1.

Example 2.

And if you would know how many quarters of an inch any fraction of an inch doth containe : or for example , how many quarters of an inch either of the said fractions of 160/224 and 382/1598 doe amount vnto , you néede but diuide their numerators by a quarter of their denominators , or the product of the numerator being multiplied by 4 , by all their denominators , and either quotient is your desire .

CHAP. 9. How Timber of vnequall sides , may , by Arithmetick , be brought into a square , and so measured by the common way .

MAster Digs in his Tectonicon made a Table of Squares , or a table to shew the squares of any ordinarie péece of timber of vnequall sides , that so they might thereby truly measure the péece according to the Rule . Herein Master Digs took great paines , worthie to be commended . But men being loth to carrie a table in their pockets for trouble ( as it seemeth vnto me ) did altogether leaue it , and framed to themselues the said false way of haluing the summe of the two sides for the square , which turned to the great losse of the buyers . Now my desire is , that seeing men will not vse that Table because of trouble , that they would learne to carrie the ground thereof in their heads , which is contained in this Chapter . Wherein though there be some greater measure of Arithmetick required then in the former , namely extraction of the square roote , yet the fruite of that kind of Arithmetick will repay any mans trauaile , that either hath , or will take paines to learne it .

But to performe the question , doe thus : Multiplie the breadth and thicknes of the péece together . Then take the square roote of the product thereof , which square roote is ( as we call it ) the square of that péece , Then hauing the square giuen , you are not ignorant how to measure the péece by the rule . As for example : if the péece be 16 inches broade and 4 inches thicke : you must multiply 16 by 4 , which makes 64. the square roote whereof is 8 ( for 8 times 8 is 64. ) Therefore 8 is the square of that péece , and not 10 as your common way tels you .

If the roote cannot be exactly extracted , as oftentimes it cannot , take the néerest that you can , and which you shall thinke requisit , by such wayes as are alreadie extant in diuers authors .

CHAP. 10. A second way to finde the square of an vnequall sided peece of Timber , namely , by the Scale and compasse .

BEcause this way is more easie and spéedie then the former , and that which may also be done by those that haue no Arithmeticke : therefore I haue added this vnto the other .

What is meant by the compasse , that is , a paire of compasses , all men know : but what is meant by the terme Scale , many peraduenture are ignorant of : therefore it shall not be vnnecessarie to describe it , as in this manner .

A Scale is any line ( and in ordinarie matters , any right line ) diuided into any number of equall parts , be they greatter or lesser , wider or narrower , of the distance of an inch-halfe ynch , quarter , halfe quarter , or any other distance greater or lesser . This line thus diuided , is , I say , called a Scale ; whereof euery part or diuision may stand for a mile , a rod , a yard , a foote , an inch , or any other kinde of measure what you will , or haue vse of .

But to the matter . On the backe-side of the Rule ( as we call it ) there is ( you know ) a voide roome or place , about the middell thereof . Now in this voyde place , draw a line , as néere to the edge of the Rule as conueniently you can , some sixe or seauen inches long , as here I haue drawne the line AB . Then from it , ( and not from the edge of the Rule ) and at the bottom of it , draw another line thwart or crosse the Rule , which shall stand square or perpendicular to the former line , as is this line BC. Then must these two lines of AB and BC be diuided into quarters of inches , beginning from the angle or corner at B : and euery one of those quarters of inches , diuide into foure equall portions or diuisions , as here I haue diuided them .

Here I pray you note once for all , that I call all those quarters of inches , Parts of the Scale : euery one of which parts in this matter , is , or standeth for an inch ; and halfe a part is halfe an inch ; a quarter of a part is a quarter of an inch ; three quarters of a part is thrée quarters of an inch , and so forth . This done , your Scale is prepared .

diagram of ruler

Now to finde the square of any péece of Timber : first adde the breadth and thicknes together . That done , note the halfe thereof : Or , ( which is all one ) note what either of the two sides doth differ from the said halfe of the sum of both sides . note also what is halfe the difference of the two sides of the péece . Then reckoning how many inches and what od quarters of an inch there is in the said halfe of both the sides added together , you must open your conpasses ( being small or sharpe in the points ) to so many parts and such od quarters of a part in the Scale . That don , set one foote of the compasses being so opened , vpon the line BC , so many parts and od quarters of a part from B , as there be inches and od quarters of an inch in halfe the difference of the sides : And extend the other foote of the compasse vpon the line AB , and note where it cuts it , or lights vpon it . Now I say , that so many parts and od quarters of a part of the said line AB , as are from B to the place where the compasses did cut or light , so many inches and od quarters of an inch is there in the square of the péece .

This is far sooner wrought then spoken , and is better explained by an example , then taught by a rule . Therefore suppose the péece to be measured , be 7 inches one way , and 11 inches and a halfe the other way . First I adde 7 to 11 ½ , which makes 18 ½ . Then I note the halfe thereof , which is 9¼ . I note also halfe the difference of the two sides , which is two and ¼ . Lastly I open my compasses as wide as nine parts and one quarter of a part , and set one foote at D , which is two parts and ¼ from B , and I extend the other foote to the line AB , and it cuts or lights at the point E. Then I say that the square of that péece , is very néere nine inches , because there are from B to Calmost nine parts .

And here note , Note . that whereas I suppose the Scale to be quarters of inches , you may make them to be halfe quarters of inches . And truly where the lines AB or BC will not admit or receiue all the inches they should ( which yet I thinke will rarely happen ) then you must take for your Scale , halfe quarters of inches , and so procéede ( as is before shewed ) as if it were of quarters of inches .

Note also that this Scale may as fitly be made on your squire ; or else on the foreside of the Rule , taking for the line AB , the line next to the edge of the Rule , as it is commonly diuided into quarters of inches , without any subdiuisions into foure , for you may well enough guesse at them without any error to be respected : take also for the line BC , any one of the thwart strokes which serues to diuide the Rule into inches ( for I suppose them to stand square to the said other line ; ) which thwart stroke you must also diuide into quarters of inches , as before is taught . And thus if one of the thwart strokes should be worne out with pricking ( which will not be in short time , if you worke lightly and with small pointed compasses ) then you may take another of the thwart strokes , and diuide it into quarters , and so procéede as before .

CHAP. 11. How that measure which is called boord-measure is gathered .

BOord-measure , or the table of boord-measure , is nothing els but that which sheweth the length of a foote of boord or planke , at any breadth : though commonly we gather this measure , but for euery breadth from one inch to thirtie sixe inches , and so set them downe in order , as in the table of boord-measure following , which is from one to thirtie . Now you may finde the length of a foote at any breadth thus . Diuide one hundred forty soure ( which is the number of inches in a foote of boord ) by the breadth that giuen , and the quotient will shew you how many inches makes a foote in length at that breadth . As for example . Diuide one hundrech fortie foure by the breadth fouretéene , and the quotient will be 10 4/14 or 2/7 , that is , the length of a foote at 14 inches broade , is 10 inches and two seauen parts of an inch . So diuide 144 by the breadth 4 , and the quotient will giue you 36 inches for the length of a foote at that breadth of 4 inches . This 36 diuided by 12 , will giue you in the quotient the number of féete therein , which is iust thrée .

But if you will gather your table of boord-measure , for boordes that haue odde quarters of an inch in their breadth : then diuide 576 ( which is the number of quarters in a foote of boord ) by the number of quarters of inches in the breadth , and the quotient will giue you in inches , the length of a foote . As for example , let the breadth be 7 inches and a halfe : This is in all 30 quarters of an inch . By this 30 I diuide 576 , and the quotient is 19 6/30 or ⅕ , that is , the length of a foote is 19 inches and a fift part of an inch .

CHAP. 12. How that measure which is called Timber-measure is gathered .

TImber-measure , or the Table of Timber-measure , is that which giueth the length of a foote of Timber at any square . But our ordinarie Timber-measure is gathered , onely from the square of one inch to the square of 36 inches . It is thus gathered . Multiply the number of inches in euery square , by it selfe : then by the ofcome or product , diuide 1728 ( which is the number of inches contayned in a foote of Timber ) and the quotient will giue you in inches , the length of a foote of Timber at that square . As for example : let the square be 18. This 18 multiplyed in it selfe , makes 324 : by which , diuide 1728 , and the quotient will be 5 108/324 or ⅓ , that is , the length of a foote , is 5 inches and one third part of an inch .

But if you will gather your table of Timber-measure , for squares that haue odde halfes or quarters of an inch ; then multiply the number of quarters of inches in the square , by it selfe ; and by the product or ofcome thereof , deuide 27648 ( which is one fourth part of the quarters of inches contayned in a foote ) and the quotient will giue you the length of a foote in inches : which you may reduce into féete , by diuiding them by 12. As for example : let the square be 16 and ¾ . First I bring them into quarters , which is 67. This 67 I multiply in it selfe , and it makes 4489. By this 4489 I doe diuide 27648 , and the quotient will be 6 714/4489 which is 6 inches , and some what more then a sixt part of an inch .

The Table of Boord-measure .

THough Master Digges in his said Tectonicon , hath set downe two Tables , the one for boord-measure , and the other for Timber-measure : yet because those that here follow , doe differ , both in matter and fourme ; I haue therefore set them downe , with this explication of them insuing .

But first note ; that whereas Master Digges in both Tables , hath procéeded from one inch to 36 inches : I haue gon no farther then to 30 inches : and indéede it were no great matter , if they went but to 24 inches . For when the breadth of boord , or the square of Timber , doe excéede 24 inches , but especially 30 inches ; then they giue so little for the length of a foote , as that the oft drawing of that length vpon the boord or Timber , cannot but bréede apparant error , except great héede be taken . Therefore for your boord , if it excéede 30 inches in breadth , you may measure it as if it were halfe so broade , and count euery foote for two foote . And as for your Timber , if the square thereof excéede 30 inches , or rather 24 inches ; then measure it as if it were halfe that square ( as for example , if the square be 32 inches , then accoumpt it as 16 inches ) and then accoumpt euery foote for foure foote .

Note also that I haue left out of this Table of boord-measure ( of that which Master Digges hath set downe in his Table ) what maketh a foote at any number of inches and an od quarter or three quarters of an inch , both because they are not , neither can conueniently be set on the Rule , as also for that no losse of any moment comes by gessing at them . And therefore I haue contented my selfe to set downe onely inches and halfe inches . But if any man like to obserue iustly what maketh a foote of boord at any number of inches , and a quarter of an inch or thrée quarters of an inch , then either he may repaire to Master Digges his Table , or else let him finde it out by the way taught in the eleuenth chapter of this second part . But to returne to the meaning of this Table of boord-measure .

In the row AB , is set downe one vnder another , the breadth of boords from one inch to 30 inches , as you may plainly perceiue by looking on it . Euery one of these numbers from one to 30 , I will call a Breadth . In the colum vnder the letter C , is set downe against euery breadth , the number of féete , inches , and parts of an inch , that make a foote in length at that breadth . ( The féete in either of the colums , stand in the narrow row , at the left side , and the inches and parts of an inch in A C D ½ 1 12 8 2 6 4 9 ⅗ 3 4 3 5 1/7 4 3 2 8 5 2 4 ⅘ 2 2 2/11 6 2 1 10 : 1/7 7 1 8 4/7 1 7 ⅕ 8 1 6 1 4 16/17 9 1 4 1 3 3/19 10 1 2 ⅖ 1 2 ¾ 11 1 1 1/11 1 : ½ 12 1 11 : ½ 13 11 1/13 10 ⅔ 14 10 2/7 9 : ⅞ 15 9 ⅗ 9 : 2/7 16 9 8 : ¾ 17 8 ½ 8 : ⅕ 18 8 7 : ⅘ 19 7 : 4/7 7 : ⅓ 20 7 ⅕ 7 1/41 21 6 6/7 6 : 5/7 22 6 : ½ 6 : ⅜ 23 6 : ¼ 6 : ⅛ 24 6 5 : ⅞ 25 5 : ¾ 5 11/17 26 5 : ½ 5 : 3/7 27 5 ⅓ 5 : 2/9 28 5 : 1/7 5 1/10 29 5 4 ⅞ 30 4 ⅘ 4 : 5/7
the broder row at the right side thereof . Note also , that where as you sée in the fractions , sometimes one pricke , sometimes two prickes , that when one pricke stands before the fractions it signifies a littellesse ; and two prickes a little more . )

Now if that the breadth of the boord be a certaine number of inches and one halfe inch , then is the length of a foote ouer against those number of inches , in the colum vnder D.

But because an example will giue more light then many words otherwise : Therefore suppose the boord to be 11 inches broade : then finde 11 in the row AB , and ouer against it in the colum C you shall sée 1.1 and 1/11 , which is one foote , one inch , and one eleuenth part of an inch .

But if the boord were 11 inches and ½ broad : then séeke the length of a foote ouer against that 11 , in the colum D , and it is 12 inches and ½ . So if the breadth of the boord were 25 inches and ½ : looke for 25 in the row AB , and ouer against it in the colum D , you shall sée fiue inches and fiue seauenth parts of an inch . And let this suffice for the Table of boord-measure .

The Table of Timber-measure .

IN the row AB is set downe the squares of Timber from one inch to 30 inches . Ouer against euery one of these , in the colum vnder C , is set downe the length of a foote , in féete and inches , and parts of an inch . The féete are set downe in the row at the left hand of this columas farre as the squares doe yeld a foote , and the inches and parts of an inch are set downe in the row towards the left hand of the same colum .

But if the squares of the péece of timber be any number of inches betwéene 1 and 10 , and an od halfe inch more : then ouer against those number of inches in the colum vnder D , you shall finde the length of a foote , in féete , inches and parts of an inch : the féete standing ( as in the other colum ) in the row at the left side thereof , and the inches & parts of an inch , in the row at the right side . The reason why I added those halfe inches from 1 to 10 , is , because in measuring of Timber by deducting the lost square , as is set down in the fourth chapter of this second part of this booke , you shal haue occasion oft times to séeke out the length of a foote for a square of one inch and a halfe , two inches and ½ , thrée inches and ½ , and so forth to ten inches .

Here farther it is to be noted , that whereas you shall finde for the most part before the numerafor of the fractions , sometimes one pricke , sometimes two prickes : that one pricke doth signifie somewhat lesse , and two prickes doth signifie somwhat more . This I haue noted , as Master Digges hath also don , though it matters not whether such more or lesse , were obserued or not , séeing no sensible error comes by omitting them .

A C D ½ 1 144 64 2 36 23 ½ 3 16 11 9 1/16 4 9 7 1 ⅓ 5 5 9 3/25 4 9 : ⅛ 6 4 3 4 : 8/9 7 2 11 : 2/7 2 4 : ¾ 8 2 3 1 11 : 12/13 9 1 9 ⅓ 1 7 1/7 10 1 5 : 2/7 11 1 2 : 2/7 12 1 13 10 : ⅕ 14 8 : 13/16 15 7 : ⅔ 16 6 ¾ 17 6 18 5 ⅓ 19 4 : 25/32 20 4 : 5/16 21 3 : 11/12 22 3 : 4/7 23 3 : ¼ 24 3 25 2 : ¾ 26 2 : 9/16 27 2 : ⅜ 28 2 : ⅕ 29 2 : 1/16 30 1 : 11/12

An example or two in this case shall not be vnnecessarie . If the square of the péece giuen be fiue inches , finde fiue in the row AB , and ouer against it in the colum of C , you shall finde fiue foote nine inches and thrée twentie fiue parts of an inch . If the square giuen be thirtéene , then ouer against thirtéene you shall finde on the right side of the colum C , 10⅕ : which is 10 inches and somewhat more then one fift part of an inch , if the square giuen be sixe inches and a halfe : then ouer against sixe , you shall finde the colum D for the length of a foote , thrée foote foure inches and somewhat lesse then eight ninth parts of an inch : more examples néedes not . Enough hath béen said to any that is but of meane capacitie and desirous to learne .

The end of the second part .
THE THIRD PART OF THE CARPENTERS RVLE , Containing sundrie true waies to measure superficies and sollids , or ( as wee call them ) Boords and Timber , of extraordinarie formes .
CHAPTER . 1. The meaning of certaine tearmes of Geometrie generally vsed in this third part .

WHen I had written the former part of this booke concerning measuring of ordinarie Timber and Boord , and did consider , that besides the pleasure , there would come some good to Carpenters , if they could also measure extraordinarie fourmes : I haue therefore thought good to adde this part vnto the other two . True it is , that Master Digs in his Tectonicon hath not béen silent of the most of these things . But because he applies them to measuring of land , few or none doe thinke that they belong also to measuring of Timber : and therefore my labour I hope is not vnnecessarie , though I should but haue repeated the same thing and apply them to our vse , without adding any other thing .

I haue knowne some that would buy whole frames readie wrought by measure : but sure I am that no Carpenter could haue measured it for them , without the knowledge of that which is written hereafter .

And because I shall haue occasion to vse many tearmes of Geometrie , by which I may with more ease deliuer , and you with more plainnes perceiue my minde in these things : I haue therefore set downe the meaning , as plainly as I can , of some Geometricall tearmes , which most serue for our present purpose . And in this Chapter I explaine onely those tearmes that generally I vse throughout this whole part , and which doe not properly belong to any one chapter . The rest I will declare in the beginning of euery chapter , as the matter thereof giues occasion . But to the matter .

1. An Angle is nothing els but a corner , made by the méeting of two lines ( for I speake not of sollid Angles ) .

2. A right Angle ( which wee call a squire or a square Angle ) is that whose two lines comprehending or making the Angle , stand perpendicular or plumbe the one to the other .

3. An oblique Angle ( which we call beuell or skew ) is euery angle not being a right angle , whether it be greater or lesse , or ( as we say ) whether it spread or clitch .

4. A Superficies , is that which hath onely length and breadth , and no thicknes at all . Here note , that whereas we call boords , superficies , and Timber sollids , it is not because a boord is not a sollid ( for it hath length , breadth and thicknes ) but because we respect not ( in measuring of them ) but only their length and breadth .

5. A Sollid ( or a bodie ) is that which hath length , breadth , and thicknes .

6. Parallels are those lines , superficies , or sollids , that differ euery where alike , or are not néerer together in one place then in another .

7. A Figure , is any kinde of superficies or sollid that is bounded about , as Triangles , Squares , Circles , Globes , Cones , Prismes , and the rest .

8. The Base of a Figure , is any side ( as wee may say ) thereof , vpon 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 plumme line falling from the top thereof , to the base , ground , or bottome thereof . And whether this Perpendicular or plum line fall within or without the figure , it makes no matter , so as it be neither higher nor lower then the base or bottome .

CHAP. 2. How to raise and let fall a Perpendicular .

A Perpendicular line is that which stands plumme vpright vpon another , leaning neither the one way nor the other . A Perpendicular is said to be raised , when a point is giuen in a line , from which it must rise . It is said to bée let fall , when a point is giuen aboue the line , from which it must fall .

Now besides that you may both raise and let fall a Perpendicular by a squire or square : there are many waies Geometrically to doe both the one and the other . And because there is especiall vse in this third part , of letting fall a perpendicular or plumme line from a point giuen , as also from an angle in a figure to the base : I would not haue men ignorant how to doe the same without a squire , which is not alwaies at hand , when a rule & paire of compasses are . And yet I will set downe onely one way of many , to auoide tediousnes , which is thus .

diagram

Let the point giuen be A , the line giuen BC. Open the Compasses to any distance conuenient , and setting one foote in the point A , make an arke or péece of a Circle with the other foote , till it cut the line BC twice . Those two places of cutting we call Intersections , and are here in this example at B and C. Then finde the middle betwéene those two Intersections , and from that middle draw a line to the point A ( which is the point giuen ) and that line shal be perpendicular or plumbe from the point A to the line B C , as was required .

CHAP. 3. Of a Triangle and a Prisme , what they are , and how they be measured .
diagram

Triangles are diuers , both in respect of their sides and angles , and may bee measured diuers waies : But let this one way serue for all . Multiplie halfe of the base by all the height or perpendicular : Or ( which is all one ) multiplie all the base by halfe the height or perpendicular ; and either of the products giues the content of the Triangle .

Here I pray you remember what I meane by the base and height or altitude , according as was she wed in the former Chapter . In this figure you may suppose the longest line to be the base , and then the prickt line is the height or perpendicular .

Now note , that a Prisme is that whose two opposite plaines or ends be equall , like , and parallell , the other sides being parallelograms , that is , figures whose opposite sides are equall , and whose angles be all right angles . So that euery péece of timber may be called a Prisme , not being broder at one end then at another , of what fashion soeuer it be , whether the base or end thereof be of thrée sides , as cants ; or of foure , fiue , sixe , or more sides , as other timber ; and whether the sides be of equall length or of vnequall .

How to measure any Prisme or péece of timber of whatsoeuer fourme or fashion the base or end thereof is ; you néede but multiplie the content of the base or end by the altitude or height , or ( as wee call it ) the length of the péece , and the product giues the content thereof .

This might serue once for all , as sufficient to measure sollids or timber , whose bases or ends are like and equall : so that it néeded not but to teach how to measure euery kinde of base , as Triangles , Quadrangles , &c. But yet for plainnes sake , I will giue examples of euery forme , with some varietie of measuring them , when I speake of measuring plaine figures as being their bases .

1. Therefore if the base of your Prisme or péece of timber were the said figure A , multiplte the content of the Triangle A by the length of the péece , and the product giues the content thereof .

2. Or els you may measure that Prisme or péece of timber thus : Take the whole perpendicular , and suppose it to be one side of a squared péece of timber ( as we call it ) and take halfe the base for the other side : and so measure it by any of the waies taught in the second part of this booke .

This being vnderstood which is here written , the vse hereof is very generall . For besides that the Carpenter may measure hereby any canted péece of timber , as steps for staires , and canted railes , and such like : the plaisterer also , who often worketh by the yard , may hereby measure gable-ends , and such like fourmes . The Glasier hath likewise vse hereof : and also it may stand the Mason oft in stead .

CHAP. 4. What a Parallelogram is , and how it is measured .

A Parallelogram is a figure whose opposit sides are equall , Note that whereas I say all right angles , it is but to distinguish betwixt this figure , and the two next following , for they are Parallelograms also . and whose angles or corners are all right or square angles : such as is the figure B.

diagram

It is measured thus : Multiplie the length by the breadth , and the product or ofcome giues the content .

The prisme or péece of timber whose bases or ends are parallelograms , as is the said figure B , you may measure thus : Multiplie the content of the base by the length , and the product yéelds the sollid content , as is taught in the seuenth Chapter of the second part of this booke .

CHAP. 5. What a Rombus is , and how it is measured .

A Rombus , other wise called a Diamond forme , is a figure of foure equall sides & oblique angles : Or , the sides are all of one length , and none of the angles are square , such as is the figure B.

The measure of a Rombus is thus . Multiplie the length of one of the sides , by the perpendicular plumme line , drawne from that side to the opposite side , which in this figure is the prickt line , and the product yéelds the content .

diagram

1. The prisme or péece of timber whose bases are a Rombus , is measured thus . Multiplie the content of the base or end by the length , and the product giues your desire .

2. Or els ( as in the third Chapter of this part ) take all the perpendicular for one side of a squared péece , and one side of the figure B for the other side of the squared péece , and so measure it by any of the waies taught in the second part of this booke .

CHAP. 6. What a Romboides is , and how it is measured .

A Romboides ( called a Diamond-like figure ) is that whose opposite sides and opposite angles are equall . But all the sides are not equall , neither is any of the angles right : such is this figure C.

The measure differeth nothing from the measuring of a Rombus in the former Chapter , and might well haue bin soyned together : so that he that can measure one can measure both . But to the former waies you may adde this , if you please . First diuide it into two Triangles , as here this figure is by the prickt line . Then measure one of those Triangles by the way set downe in the third Chapter of this part : the double whereof is the content of this figure . The like is also to be done with a Rombus or a Parallelogram .

diagram

The Prisme or péece of timber whose bases are a Romboides , is measured as a péece whose base is a Rombus , as in the former Chapter . But if you measure the base or end by two Triangles , as in this Chapter , then take as for the sides of a squared péece of timber , all the perpendicular of one of the Triangles , and all the base of the same Triangle , and so measure it by any of the waies taught in the second part .

CHAP. 7. What a Trapezium is , and how it is measured .

A Trapezium is a figure of foure sides , not hauing all the sides and angles equall , as is the figure D.

It is measured thus . First diuide it into two Triangles , by drawing a line from one angle to the opposite angle ; as here the figure D is diuided by the prickt line . Then measure by the way taught in the third Chap. of this 3. part , both those triangles , seuerally : and adde both their contents for the content of the whole Trapezium .

diagram

1. The prisme or péece of timber , whose end or base is a trapezium , as the trapezium D , may be thus measured . Multiplie the content of the base or end , by the length of the péece , and the product giues the content thereof .

2. Or els , take the square roote of the content of the base or end D ; and so measure it by any of the wases taught in the second part .

3. Orels , suppose the péece to be cut into two Cants , according as the base or end thereof D is diuided . Then measure both those Cants , by the way taught in the third chapter of this part , and take the summe of those two Cants for the content of the whole péece .

CHAP. 8. What a regular mani-sided figure is , and how it is measured .

A Mani-sided figure , is that which hath more sides then foure . They are either regular or irregular . A regular mani-sided figure is that whose sides and angles are al equal , as is this figure marked with A.

diagram

An irregular , is that whose sides or angles are vnequall .

A regular mani-sided figure is thus measured . First draw a perpendicular from the center of the figure to the middle of one of that sides , as is the short prickt line in the said figure A. Then seeke the length or compasse about of all the sides : the halfe whereof you must multiplie by the length of the said perpendicular , and the product giues the content of the figure .

Now the center of a regular mani-sided figure is thus found : Raise a perpendicular vpon the middle of any two of the sides not being directly opposite , and where they intersect or cut one another , there is the center of the figure .

By this may you finde the center of a Circle , or of any péece of a circle , namely , if you draw any two right lines , either equall or vnequall , in the circle or péece of the circle , and from their middles doe raise perpendiculars : I say , the méeting of those perpendiculars , is the center of the circle or péece of a circle .

But if the figure haue an euen number of sides , as 6 , 8 , 10 , and so foorth ; then you néede but draw a line from any two opposite angles , and another line from any two other opposite angles : and where those two lines doe cut , there is the center of the figure . Thus haue I drawne the two long prickt lines to finde the center of this figure A.

1. The prisme or péece of timber whereof the figure A , or any other regular mani-sided figure is the base or end , is measured by multiplying the content of the base by the length of the péece , and the product giues the sollid content .

2. Or els , take the halfe of all the sides or compasse about as for one side of a squared péece of timber , and the said perpendicular for the other side , and so measure it by any of the waies taught in the second part .

CHAP. 9. How irregular mani-sided figures are measured .

IRregular mani-sided figures , as also regular mani-sided figures , may be thus measured . Diuide the figure into triangles , as here I haue diuided the figure B by certen prickt lines . Then measure each triangle by it selfe , as is taught in the third Chapter of this part , and adde the contents of all the Triangles together ; and their summe or totall is the content of the figure .

1. The prisme or péece of timber , whose base or end is the figure B , or any other vnequall mani-sided figure , is measured by multiplying the content of the base or end in the length of the péece , and the product giues the sollid content .

diagram

2. Or els , take the square roote of the number of inches in the content of the base , and so measure it by any of the waies taught in the second part of this booke .

3. And because for the measuring of euery sort of timber I haue vsed to set downe some way by which it may be done without Arithmeticke : Therefore for a third way ; as euery triangle in the figure B is measured , so measure so many sollids whereof the triangles be the bases , and whose lengths are that length of the irregular péece of timber , as is taught in the third Chapter of this third part ; and then adde all their contents for the content of the whole péece .

CHAP. 10. What a Circle , a Semicircle , and a Sector are , and how they be measured .

1. A Circle is a figure , plaine and round .

diagram

2. The round line the makes the Circle is called the Circumference .

3. The point in the middle of the circle is called that Center .

4. The line ( and indéed euerie line ) passing by the Center to both sides of the Circumference , is called the Diameter , as that black line in this circle C.

5. Halfe the Diameter , and indeede euery line passing from the Center to the Circumference , is called the Semidiameter .

6. Al lines in the Circle ( except the Diameter ) which are drawne from side to side , are called Cords . Such is the prickt line in the circle C.

7. The Diameter cuts the circle into two equall parts ; each of which parts is called a Semicircle .

8. A cord cuts the circle into two vnequall parts ; each of which parts is called a Section of a Circle . The one is called the greater section , because it is greater then that Semicircle . The other is called the lesser section , because it is lesse then the semicircle .

9 A Sector is any part of a circle , either greater or lesser then a semicircle , contained vnder two semidiameters méeting in the center , and a péece of the circumference .

10 Note that the circumference of a circle is triple the diameter , and almost one seauenth part more . But we count it as iust triple and one seauenth part , or as 22 is to 7. Therefore to finde the circumference , the diameter being knowne ; multiply your diameter by 22. and diuide the product by 7 , and the quotient shewes you the circumference .

Now to measure a circle doe thus . To measure a Circle . Multiplie halfe the diameter by halfe the circumference ; or all the diameter by a quarter of the circumference ; or all the circumference by a quarter of the diameter ; and any of the products giues the content of the circle .

The Semicircle is thus measured . To measure a Semicircle . Multiply halfe the diameter in halfe the arke or arch thereof ; or all the diameter in a quarter of the arke thereof , &c. And either of the products giues the content of the semicircle .

A Sector is measured as a semicircle , To measure a Sector . namely by Multiplying halfe the semidiameter in halfe the arke or arch thereof , &c. the product whereof is the content of the sector .

1 The sollid or péece of Timber whose end or base is a circle , a semicircle , or a sector , may be thus measured . Multiply the content of the base or end in the length of the peece , and the product giues the sollid content .

2 Or , take the semidiameter for one side of a squared péece of Timber , and halfe the arke ( be it of a circle , semicircle or sector ) for the other side , and so measure it by any of the wayes taught in the second part .

Here it were not vnnecessarie to shew how to finde the quantitie of an arke or péece of a circle . But of the diuers wayes that are found , you may vse this . First hauing taken the length of your semidiameter in any measure , which let be for more conueniencie in * * For the greater measure the greater error in this kinde of worke . inches , then open your compasses to an inch , and sée how many of them is contained in your arke .

CHAP. 11. How to measure Sections of a Circle .

I Told you in the last chapter that there be two sorts of sections , a greater which is more then a semicircle ; and a lesser , which is not so much as a semicircle . They are measured diuersely .

A greater section is thus measured , To measure a greater Section . draw two semidiameters from either end of the arke to the center , as the two prickt lines in the figure A. Then Multiply halfe the arke in the semidiameter , and to the product thereof adde the content of the said triangle , made of the cord and the two semidiameters , and the totall will be the content of the whole section .

diagram

The sollid or péece of Timber , whose base is the figure A or any greater section of a circle , is found as before is oft said , by multiplying the content of the base in the length of the péece .

Or take halfe the arke as one side of a squared péece of Timber , and the semidiameter as the other side , and so measure it by any of the wayes in the second part : then adde thereunto the content of the sollid , whereof the triangle before named is the base , and the totall will be the content of the sollid , whose end is the said section .

diagram

The sollid or péece of timber , whose base or end is the figure B , or any lesser section of a circle , may be measured as before is taught , by multiplying the content of the figure in the length of the sollid .

Or els , take halfe the arke as one side of a squared péece of timber , and the semidiameter for the other side , and so measure it by the waies in the second part of this book . Then deduct out of the content thereof , the content of the sollid or péece of timber , whereof the triangle made of the corde and the two semidiameters , is the base or end : and the remainder is the content of the sollid , made vpon the said lesser section B.

In the 6. chapter of the second part of this book , To measure an Ouall . I told you that I would shew you in the end of this Chapter , how to measure an Ouall : which in a word is thus . First finde ( by that which is taught in the 8. Chapter of this part ) the foure centers of the foure arkes or péeces of circles , whereof the Ouall is made ( for euery Ouall is made , by ioyning together foure seuerall arkes or péeces of circles , whereof those at the two ends are equall , and those at the two sides are also equall ) . Then hauing found those arkes , draw their corde vnder them : so shall you see the whole Ouall to be diuided into foure lesser sections of a circle and a parallelogram . Therfore measure all those foure sections , as afore is taught in this Chapter ; and measure also the parallelogram , by the way taught in the third Chapter of this third part : and the content of them all ( that is , of those foure sections and the parailelogram ) is the content of the Ouall .

Nete , that when you haue measured one section at the end of the Ouall , you haue also the content of the other section at the other end ( because they are equall ) : Therefore you néede but to double the one , and so you haue the content of them both . In like sort may you doe with the sections at the sides , that is ; hauing measured the one , doe but double it for the content of them both .

If a péece of timber were Ouall fashion , that is , had his base or end like an Ouall : then it may be measured by multiplying the content of the base by the altitude or length of the péece , as hath béen often mentioned .

Note further , that by this which is here taught , and by that which is afore written of triangles and mani-sided figures , you may measure any kinde of figure , either plaine or sollid , that is made of straight lines and circular lines together , if you doe but vse diligence , and diuide or imagine a diuision of the whole figure into his proper sections and other figures as becommeth ; and so measure those sections and other figures seuerally , and then adde their contents together .

CHAP. 12. What a Pyramis is , and how it is measured .

A Pyramis is a sollid figure , contained vnder many plaine superficies set vpon one plaine superficies , and gathered together into one point , as this pyramis A , Pyramids are diuers according as their bases be diuers , as of 3 , 4 , 5 , or more sides , as is at large shewed in the 10. definition of the 11. booke of Euclides Elements of Geometrie .

But of what sorts soeuer the pyramids be , To measure the sollid content of a Pyramis . they are measured by one rule , which is thus : Multiplie the altitude or height by the third part of the content of the base : or contrarily , multiplie the content of the base by the third part of the altitude or height , and either of the products is the content sollid of the pyramis .

Here note , that the altitude or height is not the length of the side of the pyramis , but ( as I told you in the first chapter of this third part ) it is the length of the perpendicular or plumme line , falling from the point or top thereof , to the base or bottome .

diagram

But if you would measure the superficies or outside of a pyramis : To measure the superficiall content of a Pyramis . then ( in so much as all the sides are triangles ) measure euery triangle by it selfe , and to all their contents adde the content of the base , and the totall or summe is the superficiall content of the pyramis .

CHAP. 13. How to measure parts of a pyramis , or timber that is taper growne .

IF a part of a pyramis doe taper much , you may lay it on a paper by a scale , and make it out to a perfect pyramis . Hauing so done , measure the whole pyramis . Measure also that part thereof which was added ; which substract or take from the content of the whole pyramis , and the remainder giues the content of the péece of a pyramis . This way is not conuenient in ordinarie tapered timber , and therefore take rather the way here following .

You know that in the last chapter of the first part of this booke , I told you , that if a tapered péece were measured by the square of the middle , there was lost to the seller a péece of timber , whose length is the sixt part of the length of the whole péece ; and which hath for his breadth , halfe the difference of the breadth of both ends ; and for his thicknes , halfe the difference of the thicknes of both ends . Therefore hauing measured the peece by the square of the middle , deduct the said lost péece , and the remainder is the content of the tapered péece . This lost péece you may measure ( and so deduct ) thus . Take the halfe of the difference of the breadth of both ends of the tapered péece , for the breadth of your lost péece ; and halfe the difference of the thicknes of both ends , for the thicknes of the lost péece : & with that breadth and thicknes measure it according to the length ; which ( as I said ) is the sixt part of the length of the whole péece . And note , that by the length , I meane not the side of the tapered péece , but the plumme or height thereof .

CHAP. 14. What a Cone is , and how it is measured .

A Cone is a round pyramis ( as it were ) which hath for the base a Circle , as is this figure . It is largely described in the 18. definition of the 11. booke of Euclides Elements . To measure the sollid content of a Cone .

The sollid content of a Cone is thus measured : Multiplie the content of the base in the third part of the height , and the product is the sollid content of the Cone . To measure the superficiall content of a Cone .

diagram

Note here againe , that the side of the Cone is not the height , plumme or perpendicular line from the top to the base , but the length of the ridge or sloping of the Cone .

There are many other kinds of plaines and sollids , but I may not stand to write of them . If any man , either for pleasure or profit , shall desire to know them , or their measure , let him looke into Euclides Elements , Master Digs his Pantametria , Master Lucars Solace , and other good books of Geometrie , which are extant in English .

FINIS .