THE CA●●●NTERS RULE, Or, A BOOK SHOWING MANY 〈◊〉 ways, truly to measure ordinary Timber, and other extraordinary solids, or Timber: WITH A DETECTION OF SUNDRY great errors, generally committed by Carpenters and others in measuring of Timber; tending much to the buyers great loss.  
PVBLISH●● ESPECIALLY FOR THE GOOD of the 〈…〉 Carpenter's in London, and others also; and is very 〈◊〉 for Masons, Shipwrights, joiners, and 〈…〉 to measure Timber and Board, and 〈…〉 other superficies and solids.  
By R●HARD MORE Carpenter.  
AT LONDON Imprinted by FELIX KYNGSTON. 1602.   

TO THE RIGHT WORSHIPFUL SIR HENRY BILLINGSLEY Knight, Alderman of the City of London.  
RIght worshipful, when I had often considered the great errors, and by error the great loss coming to this City, in measuring of Timber after the common ways; and had long expected some man's endeavour for redress, but saw none: I was in a sort provoked (though the unfittest of a number) to attempt this thing. And because it is not otherwise likely, but that I in this case, shall have many opposers: I have therefore emboldened myself, humbly to desire your worship's protection: craving also your favourable acceptance, of this my poor labour; that so it may be the better accepted for the common good.  
Neither is my desire, only to entreat your worship's favour and protection, but also to testify my dutiful thankfulness for your great pains and no small cost, in publishing in our English 〈…〉 Euclide● Elements of Geometry. A book, from which 〈…〉 well may, so myself needs must confess, that I have received all that little insight in Geometry, which I have attained unto. Yea such is that book (as I may well say) that if men's tongues should be silent, yet their actions would not spare, plainly to declare the worthiness thereof, to the praise of your labours. And thus in all duty, craving your worship's protection of 〈◊〉 small fruit of your own labours; craving also pardon for my boldness, I humbly take my leave: desiring God to bless and keep you in all good estate, to the glory of his name, the benefit of the Commonwealth, and the good of yourself, in this life and the next.  
Your Worship's 〈◊〉 all duty RICHARD MORE.   

TO THE WORSHIPFUL, THE MASTER, WARDENS, AND Assistants of the Company of Carpenters of the City of London: And to all other the courteous Readers.  
AFter I had long considered, and that not without some grief, the great loss that cometh to those of my Company and others, by errors ordinarily committed in measuring of timber; I did often think with myself what might be the readiest course to redress them. One while I thought to have declared to you the Master and Wardens, what errors I had observed in our ordinary measure, and so to have desired you in all duty to have provided a remedy. But when I perceived, that custom had caused error to be received as a truth, and that therefore men would not forsake them, unless they were plainly convinced to have erred: And when I did also see, that most men are very ignorant in true measure, though they seem and profess to know much therein; and did also remember that not only Carpenters in this City do in this sort err, 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 private, and such as was like to do but little good. The best course that I could think upon, was, to publish some book, wherein not only true measure should be set down, but also the common errors plainly laid open to the capacity of the simplest; that so all men might take knowledge thereof.  
Myself I knew to be very unfit for this work, both because of mine inability, as also by reason of some exceptions that divers might take against me, to their own hindrance and my discouragement. But when I saw none addressed hereunto then (remembering the saying of a Heathen, that men are 〈◊〉 for themselves, but for their Country) I did set aside all lets, and both boldly and rudely have put out this my simple book, in manner as you see.  
The book I have divided 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 show how ordinary timber may be measured, both by sundry plain ways, as also by ways more artificial. And in the third part I show, how extraordinary timber and solid forms may be measured.  
The errors ordinarily committed in measuring timber, are many. But of them all, especially two, bringeth loss to the buyer. The first is by buying of waynie timber and measuring it as square. The other is by taking half the breadth and thickness of a piece being added together, for the square thereof.  
These errors are such, as bring apparent damage: and therefore they are not to be respected lightly. I would they might not be said to be a great cause, if not of the overthrow, yet of the impairing of some men's estates.  
For as for wainie timber, who knows not, that as it is often measured and hewed, a load and a half will not go so far in use, as a load of good timber being well hewed and justly measured.  
And as for halving the sum of the two sides for the square, the damage hereby is exceeding much; not only to those that buy much Timber at the first hand, but also, and that especially (which is the more pity) to the poorer sort that buy at the second hand. For myself have abated of that which workmen themselves have measured a piece unto by the foresaid false way, after 3, 4, 5, yea 6 foot of timber in 20: and yet I have given them no more than their measure.  
I cannot stand to amplify these errors in this place: they are more fully set down in the book itself. By which (if they be well weighed and considered upon) it will plainly appear to be true as I have said, namely, that there cometh great loss by the ordinary measuring of timber.  
Now seeing the case so stands, my desire and request is, even to every man, but especially to you the Master, Wardens, and Assistants of our Company; that you would put your helping hand to reform these errors. Many reasons might be alleged to induce you hereunto, but for some reasons best known to myself, I omit them all, not doubting of your care and regard hereunto.  
I have heard some men wish, that we had an act of Parliament procured for redress of false measure. But truly that seemeth to be the law of our land already, though not the statute law: for the law intendeth that the buyer should have that which he buyeth. And when we buy a load of timber, do we not intend that we have bought 50. foot of solid timber, every foot being full 12. inches every way, that is, in length, breadth & thickness? Why then may we not require so much for a load, and why is not the seller bound to deliver it? And if a man that buyeth twenty bushels of wheat, will not take nineteen; or if he buy a yard of cloth, he will not take three quarters: Then why should we (buying a load of timber) take 40. foot for it, yea and less too (as oft times it falls out) whereas we should have 50. foot.  
I know that many men will object many things; and it is not for me at this time to refute them. Only whereas some may say, that what is wanting in the measure, is allowed in the price. I answer: 1. That is not always. 2. But where allowance is made, it is not according to the value of the loss coming by the false measure. 3. Again, it is a preposterous course, to raise or let fall a price by altering of a set measure to an uncertainty, whereas we use in all trades to raise or let fall the price according to the goodness or badness of the wares, keeping still the same measure for them both. And surely herein we are but deceived: For oft times we have twelve pence bated in the price of a load, and lose four or five shillings in the measure thereof.  
Therefore let us either have just measure, if we buy by measure; or else let us buy it by guess. For it is a shame to pretend to measure truly, and yet do nothing less.  
As touching myself, I desire that my endeavour may be favourably accepted: for I writ with no other mind then as tendering my Companies good. If any thing that I have said, be impugned as false, I will be ready whensoever you will call me, to prove my affirmations. If any man of knowledge shall except against me, for my rude writing and gross demonstrations: I pray them to understand, that I writ not for them, but for the simple; and therefore I demonstrate grossly, not being well able to do better: Also, that I am not a scholar but a Carpenter, and therefore could not but write rudely. Unto such also I do further say, that my desire is, that they would take knowledge, that this hath been one cause of my writing, even to provoke them (seeing our present error and loss, with which peradventure they were not so well acquainted as myself) to take some pains to reform the same, in better sort then either I have or can do.  
While this book was in printing I came to the sight of a Ruler, sometimes invented by one Master Bedwel; which as it is easy, so it is most speedy, and not less certain (being truly made) for the measuring of timber and board; which I expect and hope will be shortly published for the common good. But I will not trouble you with more words. Only let me put you in mind of this one thing: That in as much as nothing is more fit for Carpenters to make them ready, not only in measure but also in other things, than Geometry: that therefore such as are of reasonable capacity, would spend some part of their spare times to study the same, in some measure at the least. For your furtherance herein, there are especial good helps, both by the Lecture at Gresham College every Thursday in the Term times (if you were but a little entered) as also by Euclides Elements of Geometry, which the right Worshipful Sir Henry Billingsley Knight, to the great good of the Commonwealth, though to his own great travel and charge, hath translated and published in our English tongue. But I am over tedious: I pray you accept of this my poor endeavour, as a fruit of my hearty good will to my Company, and others whom it may concern. If it take none effect, I shall be sorry: but if it do any good, I have enough for my labour.  
RICHARD MORE.   

THE FIRST PART OF THE CARPENTER'S RULE, showing divers gross errors, generally committed by Carpenters and others in measuring of Timber, arising from the Rule, and the use thereof.  

CHAPTER. 1. The error in the Rules, and the reason thereof.  
THose that endeavour to amend things amiss, must first of all labour to prove the faults, especially when the parties deceived, are not yet persuaded, that they are out of the way. And because men little regard to return unto the right, when they do not perceive that they are far in the wrong: I have therefore thought good to spend this first part of my book, in showing 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 do arise, either by means of the Rule or Ruler, with which they measure Timber; or else by the misapplying or using thereof. I mean, the ordinary way of measuring Timber is very erroneous. But first of the faults to be observed in the Ruler itself.  
The ordinary Rule which Carpenters, Shipwrights, and others do use to measure Timber withal, was invented and published by Master Leonard Digger. 
Digges Tectonicon.  Now those Rules that agree with that book are so true, as that I deem them overcurious that except against them; and they will serve truly to measure Timber withal. But the most which have that kind of measure on them which Master Digges describeth, are very false. I mean, those divisions or strikes which are set on them for measuring of Board and Timber, are not in their right places.  
And as those strikes and divissons agree not with the truth, so upon divers Rules you shall find them to disagree one from another: Yea hardly shall you see two Rules that do every where agree. Neither is this error insensible, and so not to be respected; but apparently gross, and therefore not to be tolerated. For myself have seen some Rules that in some of their divisions have erred from the truth and from one another, after the rate of six foot in an hundredth of Board, and after four foot in a load of Timber. Neither is this error rare and in some alone, but so general, as that if a man would examine them, he should be forced to say, that true Rules are very scant.  
If I were demanded the cause of this fault or error in the Rule, I should answer, that in mine opinion it ariseth hence. Many will take upon them to make Rules, who have not Master Digges his book to make them by; or if they have it, understand 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 down in the eleventh and twelfth chapters of the second part of this book. But ordinarily men make them one by another, whereof hath grown the former error.  
Thus much for the Rule. Now should follow to show the errors in measuring Board and Plank: But because I have observed no main error in measuring of ordinary plank or Board (though great fault in buying of waynie Board, and measuring it as square) which I speak of in the next chapter following. I will therefore say nothing thereof now, but pass to errors in measuring of Timber.   

CHAP. 2. The great loss that cometh in buying of timber and Board that is wainie, and measuring it as square.  
EVery man that buyeth Board or Timber, intendeth to have for a foot as followeth, that is to say, in Board twelve inches in length; and twelve in breadth; and in timber, twelve inches in length, twelve in breadth and twelve in thickness. But who so buyeth Timber or Board that is waynie, and taketh it as if it were not waynie, hath not his measure for his money, but far less. This is a great fault, and general throughout the whole company, and amongst others also; yea and apparently known and noted by every man, and yet not amended nor endeavoured to be amended by any.  
This fault in Board seemeth to come from carelessness, and in Timber (as I take it) it did first grow from ignorance. For when men knew not now to measure a waynie piece of Timber, nor bow to deduct the veins, having measured it as not waynie; then they did carelessly overpass this thing as a necessary evil, or as an inconvenience that could not be avoided.  
But it may be said, that there is allowance in the price. I answer, what is it in a load of Timber or in a persell of Board to have twelve pence abated in the price, and to lose four or five shillings in the measure. This is to get in the hundredth and lose in the Shire.  
And besides this loss that cometh by waynie Timber, there is also another great loss thereby not to be forgotten. You know that when the Timber is very waynie, we are compelled to hue away a great part of it, after it is sawed to bring it to some reasonable squareness fit for use. If then you shall put this loss to the other, you shall find, that it is no small damage that cometh by buying of waynie timber. And therefore there should be an extraordinary care and respect had to the measure, that we lose not by it, seeing so great loss cometh by the stuff itself.  
My desire therefore is, that as other faults; so this may be amended. Not that I would have Timber hewed die square, for that were a spoil of much good Timber: nor that I would have girt measure used, for that (in this kind) is very false, as appeareth afterwards: But I would have the want or wainynesse deducted or taken out of the whole piece; which how it may be done, is showed in the second part of this book, Chapter 5.   

CHAP. 3. That it is very false, to take for the square of a piece of Timber, the half of the breadth and thickness, added together: by which square, should be found the length of a foot in that piece.  
IT is a common practice, and that of old, that who so of the ordinary and common sort of men, do measure Timber, they take this course. First they add or put together the breadth and thickness of the piece of Timber to be measured; then they take the half thereof for the square of the same piece; and then according to that square, they give the length of a foot by the Rule. As for example, if the piece be ten inches one way, and fourteen inches the other way, they add together ten and fourteen, which make twenty four; the half of twenty four is twelve, which (say they) is the square of this piece of Timber; which is very false. And yet this course is so generally received, and so commonly reputed and taken for good, as that he shall be thought overcurious of some, that shall but except against it. Howbeit if you will but mark that which followeth, you shall plainly see that it is an error, and that a great one.   diagram  diagram 
Thus then appeareth the error. If you will know the greatness thereof, you shall understand, that in the former example, and so in all Timber that have like differences of sides as that piece hath, there is loss always to the buyer, a quarter of that which he buyeth, or one load of Timber in four. If the difference of the sides be smaller, than is the loss smaller: but if it be greater, than is the loss greater. And in general take this for a rule. That there is lost by this kind of false measuring, a square piece of Timber, as long as the whole piece to be measured, and as broad and thick as half the difference of the two sides of the said piece of Timber to be measured. As for example, let the sides be ten and fourteen, than their difference is four (for fourteen  more than ten, by four) now the half of four is two, which two is the side of the said lost square piece of Timber throughout the piece.  
Now seeing this error of taking half the sum of the sides for the square, is very manifest; and seeing also it is an error tending to the great loss of the buyer, yea to the loss of three load in four, and more also as the differences may be: Therefore I desire that none would follow that corrupt custom, and most false way; but suffer themselves to be better instructed, in more righter ways. Which if they refuse to do, and shall buy much Timber after that rate, their purses are like to pay for it.  
But some may say unto me, this fault is by divers discerned, and a redress not altogether unthought of. I grant, many do see it, and do endeavour to reform it thus. If a piece of Timber be eighteen inches one way, and six inches the other way, they imagine it to consist of three squares of six inches a piece (as indeed it doth) the which they measure severally, and then add all their contents together, which they take for the measure of the whole piece.  
This is true both in this example, and in all others that will fall out in any number of just squares. But when the sides of the Timber to be measured, have not this answereablenes in them, as nine inches one way and fourteen the other way; or seven and nintéene, or such like: then they are to seek. For when they have taken the thickness out of the breadth as often as they may, then is the remainder unsquare: In measuring whereof, they fail and err, as a foresaid; and yet they profess, not to regard it. For (say they) if the difference of the sides be not past three or four inches, that piece may be measured according to the said erroneous way, and no great loss happen thereby. But what the loss is, I have showed before, which although it seem but small, yet in time it ariseth to no small matter.   

CHAP. 4. The error in measuring of timber not hewed square but bevell.  
THough this error be not of any great moment, because Timber hewed bevell or skew, cometh not often to be measured: yet because sometimes there is such Timber to be sold, I would not have workmen to be ignorant, that it is an apparent error to measure it, as if it were hewed square.  
When the corners of the Timber are hewed unsquare; I do not mean waynie; for of that I have spoken before in the second chapter: but when it is hewed bevell or diamond wise (as sometimes it is by the carelessness or unskilfulness of the hewer) then to take the square of such Timber, according to the breadth of the sides thereof, is untrue; for the sides are longer than if it were square, and so gives the piece to be bigger than it is.  
How great this error is, I stand not to show at this time: Only note that it doth increase and decrease according as the corners of the Timber do serve from square. But how such Timber is to be measured, see in the third part, Chapter she sixth.   

CHAP. 5. The error in the ordinary measuring of round Timber.  
THe former errors do always bring loss to the buyer, but these that follow do always bring loss to the seller. And though such round Timber cometh not often to be measured here in London (the worse for the buyer) yet would I have Carpenters know the error which is used in measuring the same; which is this.  
They girt the Timber round about with a line or thread, and then take one quarter thereof for the square (as we call it) of the piece to be measured, and so find out (as they think) the length of a foot in that piece of Timber or tree.  
This is a gross error, and yieldeth not the content of the piece or tree, by one fift part of it and more.   diagram 
But how round Timber is to be measured is taught in the sixth chapter of the second part, and in the ninth chapter of the third part of this book: which when you have learned and tried, than you will tell yourself, that in measuring round Timber after the former way, is very erroneous and intolerable.   

CHAP. 5. Error in measuring of Timber not fully round, but somewhat flat called oval fashion.   diagram 
So that to measure ovals by the former way, is so uncertain, that you may have peradventure more than you should, and peradventure less than you should by a fift part, a fourth part, a third part, and so forth. But because (ordinarily) trees or Timber that are Dual fashion, do not much vary from a round: Therefore the loss is always to the seller, being measured as a foresaid.  
Here I had purposed to have somewhat digressed from my general purpose, and to have showed the great loss that cometh by buying of faggots that are bound flat: But the Statute for fuel in the last Parliament hath prevented me: Where it is enacted, that bands of faggots shall be Round. But yet because I fear all men understand not what is meant by the word Round, or if they do, yet they will not regard to redress it: I pray you therefore take knowledge, that where as men do commonly buy the flattest faggots for the greatest, they are deceived; for indeed they are less than the round by far; and evermore the flatter they are, the lesser they are. And to speak no more than myself have tried; I have taken four faggots (indeed of the flattest) and have put the wood of them all into three of their own bands without straining. Lo than the great loss that comes to folk by this error. That which I say of faggots I intent also to be spoken of billets: For the flatter they are, the lesser they are, being no more in compass then the statute requires.  
I need not stand to demonstrate that this is an error, if I were persuaded you understood that which is said before of ovals. But for farther proof, take a quart pot of pewter that is round, and make it more flat by crushing in the sides thereof; and you shall find that it will not hold a quart of water as before it did, and yet the compass of the pot is as much as it was before. You may also prove the same by a Board thus. The faggot band ought to be by the Statute twenty four inches about: So than a boards end of a foot broad will hardly go into one of those bands; but if you cleave that boards end into a dozen or more pieces; you shall well perceive that you may put four and more of such Boards being so cleft, into that band being round. This have I written, not carpingly, but as desirous that the buyers would not deceive themselves by their eye, neither that they should be deceived by others.   

CHAP. 7. The error in measuring timber that is waynie.  
I Told you before in the second chapter, that there comes great loss to Carpenters in buying of waynie Timber after the ordinary way: But here I show the loss that comes (indeed not to the buyer but the seller) in measuring of waynie Timber.  
It hath been used (as I am informed) that if Timber were waynie, the buyer would measure it as by girding it about, and taking one quarter of the comqasse for the square, as in the case of round Timber before showed, chapter the sixth. But it seemeth to me, that the seller grew weary of this way, so that in my time to my knowledge, it hath not been used in or about this City.   diagram 
How much this loss is, doth not appear by this demonstration. But in this figure (if you will take my word) there is lost a sixth part full. But I had rather you would try the loss yourselves, by the rule taught in the fift chapter of the second part of this book, then to believe me in this case.   

CHAP. 8. That in measuring of Timber that is taper grown, it is false to take the square of the middle thereof, and so to give up the content by the common Rule.  
IT is so usual a thing in measuring of Timber that is taper grown, or narrower at one end then at another, to take the square of the middle of the piece, and so to measure it by the common Rule; as that I shall be not only not believed, but also reputed to have apparently erred in affirming the contrary: for it will be said, that that which it hath too much in the greater half, it wanteth in the lesser half. Indeed this is true in Boards or other superficies that taper, whose sides are strait: but in a solid or piece of Timber it is simply false.  
How to demonstrate the error on a paper, so as that it may be easily conceived, is very hard: but in a solid it may plainly be declared to the understanding of the simplest.  
If a Pyramid (that is, a piked piece of Timber, sharp at one end, like a piked steeple) were so measured, there would be lost very much. And always the lesser the piece doth taper, the lesser is the loss. In general, there is lost, (by certain pyramids) a piece of Timber whose length is the sixth part of the length of the whole piece, and whose end or base is contained under half the difference of the sides of both the ends of such a tapered piece: that is, take for the breadth of the lost péce, half the difference of the breadth of both ends; and for the thickness of the lost péce, half the difference of the thickness of both ends. As for example, if the piece be twenty inches broad and sixteen inches thick at one end, and fifteen inches broad, and twelve inches thick at the other end, and twenty four foot long: then the last piece is four foot long, two inches and a half broad, and two inches thick: for two and a half is the half of five, which is the difference of the breadth of both ends; and two is the half of four, which is the difference of the thickness of both ends.  
The loss that cometh by this error is to the seller: and yet in truth it is such as yieldeth no profit to the buyer. For we lose more by such Timber, being employed to any ordinary use, than that advantage by the measure doth come unto. But yet I would not have workmen to take that way for a true measure which is false; neither to be ignorant how such Timber is to be measured, if occasion do offer, or if men shall require it.  
In the second and third part of this book, where I show how to measure Timber, I will show how tapered Timber may be measured: from which I will no longer detain you, taking that to be sufficient which I have written already, to show 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 CARPENTER'S RULE, Wherein is set down sundry plain ways, truly to measure ordinary Timber, and Board.  

CHAPTER. 1. How Boards and planks are ordinarily measured.  
IN the former part of this book, I told you that I had not observed any main error in measuring of ordinary Board and plank, though great fault in buying of wainie board and plank, and measuring it as square: And therefore it shall not be needful to invent or set down any new ways of measuring of them: only it shall suffice to repeat the ways already in use ordinarily, which are three. Of which, the first is troublesome, and therefore rather to be known for variety, then used in measuring. The second is true, but longer in doing then the third; and therefore I would advise the third to be followed and used. Only look that your Rule be truly made, and that diligence be used in measuring therewith: else there will follow error of necessity, be the ways to measure by, never so exact. 
The first way to measure board.   
The first way let be this. And first note that for this purpose, I call an inch that which is an inch broad and twelve inches long, of which inches, twelve do make a foot. Now if the Board to be measured be under a foot in breadth, then reckon how many times twelve of the said inches you can have in the whole Board, and so many foot is there in it. If the Board be just a foot broad, than it contains so many foot as it is feet in length. But if it be more than a foot broad; then measure it as a foresaid as if it were two Boards, the one of a foot broad, the other so broad as the odd inches remaining: then add their contents together for the content of the whole board.  
This rule, to them that understand it, may be fitlier set down in other words thus. Multiply the number of inches in the breadth, by the number of feet in the length: then divide the product by twelve, and the quotient shows the number of feet in the board.  
The second way is thus, Measure the length of the board: 
The second way to measure board.  Then (if the same board be under a foot broad) draw or set the breadth thereof so many times on the board, as it conntayneth feet in length, and there make a prick: And then so many foot as there are from the beginning to that prick, so many foot doth the board contain.  
And if the board be above a foot broad, than you may take out the even feet, and measure the remainder as aforesaid; and then add both those together for the content of the whole.  
The third way, is this. Take the breadth of the board; 
The third way to measure board.  then find by the Rule what length, maketh a foot, at that breadth; and then see how many times that length is in the board, and so many foot are in it. These I stand not upon, because they are well enough known to all men.   

CHAP. 2. How to measure Timber by a certain way called Drawght-measure.  
THis way may fall out so tedious, as that I would not have troubled either myself to write it, or you to read it, had it not been that I would not have you ignorant of that way which some do use in some places of this Land. The way is thus.   diagram 
This rule aforesaid will hold true for any Timber that is under a foot square. But if it be above a foot square, and not above two foot square; then must you imagine that one quarter thereof is to be measured (and therefore take but half of each of the two sides) and when you have so cast it up as if it were but a quarter, account four times so much for the content of the whole piece.  
But if the piece of Timber be above two foot square, and not above four foot square; then may you imagine that a sixtéenth part thereof is to be measured. Which done, account sixteen times somuch for the content of the whole piece: now to take the sixtéenth part, you may take a quarter of the breadth, and a quarter of the thickness.  
But this kind of measure if it fall out in a piece of Timber above a foot square is so tedious, as that I do wonder any man would content himself with it. And yet because I have set it down, take one note more, which is this. If the piece to be measured have any odd inches in the length, either at the first or second working: then must you, having drawn the want so many times as there is feet in length, add thereunto such part of the want, as the odd measure of the length is part of a foot. And thus much of Drawght-measure.   

CHAP. 3. How Timber may be measured by Board measure only.  
WHen I considered the readiness of many men in measuring of board by one of the three former ways, and how speedily they will cast up the whole stock having measured one board, by the observing of scores, ten, and five therein; and withal considering the great error that those men run into in measuring of Timber: This moved me to think of some course, how these men that are so expert in measuring of board, might by the same way measure any square piece of Timber.  
This way, though it be true, yet it is troublesome; and therefore I would wish them rather to use some of the ways following: But if their readiness in casting up a stock of boards shall cause them to accept of this way, I will not be their let. Now the way is this.  
Suppose the piece of Timber (to be measured) to be a stock of boards, consisting of so many boards as there be inches in the thickness of the piece. Then measure one of those supposed boards: that done, cast up the whole stock. Now this being performed, you are to know that every twelve foot of board is a foot of Timber, as every man may perceive; and six foot of board is half a foot of Timber; and three foot, is a quarter of a foot of timber. Also, every thréeskore foot of board is five foot of Timber, and so, sixeskore foot of board is ten foot of Timber, nineskore foot of board is fifteen foot of Timber, twelueskore is twenty foot, fiftéeneskore or three hundredth, is twenty and five foot, six hundredth foot of board is fifty foot of Timber, and so forth. Now if this be observed, it is easy for any man to cast up the content of any ordinary piece of Timber truly.  
There is farther to be noted, that if there be any odd half inches in the thickness of the piece to be measured, that then it be accounted as half of one of the supposed boards of the stock: And if there be a quarter of one inch, than it is a quarter of a board: And if there be an odd three quarters of an inch, it is three quarters of a board; and if there be any other odd part of an inch, it is the same part of a board, and must be counted with the whole stock.  
This way is so plain that it needeth no example. I suppose the simplest may understand it, and therefore I will say no more of it: Only this you may observe, that you may measure the piece, by imagining it to be divided into two inch, three inch, or four inch plank; so that you search out how many foot of such plank is contained in a foot of Timber.   

CHAP. 4. How Timber may be measured after the ordinary way, by deducting the lost square.  
IT was observed in the third chapter of the first part of this book, that when any piece of Timber is measured after the most common way, which is by adding the breadth and thickness together, and taking the half thereof for the square of the piece, etc. that then there is always lost a square piece of Timber of the whole length of the piece to be measured, whose square is half the difference of the two sides of the piece. Now by observing of this lost square, any ordinary piece of Timber may be truly measured as followeth.  
Measure the piece, by taking for the square thereof, the half of the sum or total of the two sides, as is aforesaid, and as is usually accustomed. This done, note the difference of the two sides: then take the half of that difference for the side of the lost square, which is all the length of the piece. Then measure that supposed or lost square, and what it amounteth unto, deduct out of the measure of the whole piece being measured after the said common way, and the remainder is the true content of the piece.   diagram 
And because the lost square may have in it an odd half inch, as some times it hath, as three inches and ahalfe or such like: I have therefore added unto the common rule, what length maketh a foot of Timber from an inch to ten inches, noting still every half inch: As you may behold in the table of Timber measure hereafter following in the latter end of this second part. For to take half the difference betwixt the two next adjoining numbers of inches added to the lesser of them (as commonly men do) is not true: As for example; at an inch square one hundred forty four foot in length maketh a foot of timber: at two inches, thirty six foot doth it: so that if the half betwixt thirty six and one hundred and forty four were taken and added to thirty six for the length of a foot at an inch and a half square, it would be ninety foot, whereas in truth it is but sixty four foot: so in this example it erreth twenty six foot in length of a piece of Timber of an inch and a half square. And how so ever it be not so much in other numbers above two inches square, yet it always erreth more or less.   

CHAP. 5. How wainie or canted Timber may be measured.  
ALthough there be divers ways to measure wainie or canted Timber, yet I will observe only this one plain way. Measure then the piece as if it were not wainie, out of which deduct the wanes, and the remainder is the true content of the piece.  
How to deduct the wanes is better taught by an example then by a precept: and therefore suppose the figure A were the end of a piece of Timber of ten inches square, having one corner thereof wainie: this piece measure as if it were not wainie. That done, measure what the wain is either way from the corners, which I call the wants, because the piece  diagram wanteth so much of square; which want in this example is four inches either way, as you may see by the two pricked lines with the figure of four standing by either of them. This measure as if it were a piece of Timber of four inches either way, and as long as the whole piece. That done take half the content thereof; the which half deduct out of the measure of the whole piece as it was before measured; and the remainder will be the true content thereof. And as you do with this corner, so must you do with the rest, if they be wainie also.  
And here I would have you note, that where I bid the half of the content of the said square to be deducted, that the reason hereof is, for that in Geometry, all paralletograms and parallelipipidons, are divided into two equal parts by their diameters, as it is demonstrated in Euclid, book, 1. proposition 34. and book 11. proposition 18. That is to say in plainer terms for our purpose, a board or a piece of Timber which have their opposite or contrary sides equally distant, being cut into two pieces by a strait line passing by the opposite corners, as you may see in the figure B, I say, that board or piece of Timber, is cut into two equal parts. So that it is manifest by this,  diagram that the said wain is half so much as the piece whose sides are equal to the said wants.   

CHAP. 6. How round Timber may be measured.  
I Told you before, that I would not trouble you with Geometrical works. And howsoever I do hereafter set down some more artificial ways to measure by, yet my endeavour especially is to correct the common errors by that knowledge which most men have already: and therefore you may measure round Timber plainly in this sort. Gird the piece about with a line or thread: then take a quarter thereof for the square of that piece, (though indeed 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 book, namely that this piece is more than it is thus measured to be. 
See a more exact way in the third part of this book, chap. the 10.  And therefore to find the true content, add to that which you have measured the piece to be, a quarter thereof, and also after the rate of a foot in a load, and then the total will show the content of the piece, though not exactly, yet so near, as that the loss will not bring any damage to any man, that is worth looking after.  
Take an example for more plainness. Suppose the tree being measured after the former way to contain twenty foot: then must you add thereto a quarter thereof which is 5, and that makes 25 foot. Then add to that, half a foot, which is after a foot in a load, and the content of the piece will amount to 25 foot and a half.  
And here is to be noted, that if the piece be not exactly round, that then this measure will not hold, as was showed in the seventh Chapter of the former part of this book. But how such Timber is to be measured, see in the latter end of the eleventh Chapter of the third part.   

CHAP. 7. How Timber may be measured by Arithmetic.  
THough the former ways for measuring of Timber be sufficient, yet because variety giveth pleasure, and these that follow admit of less error in the working then some of the former, I have therefore thought good to set them down as followeth. And if any man shall object that I spend this labour in vain, because Carpenters commonly have not Arithmetic. I answer, that some have more knowledge therein, than they know how to apply to the present purpose; for whom chief I have written this chapter and some others following: as also I could wish that others which have not, would learn; which they may do by books which are extant: or at least forbear to be common Rule-makers, until they have knowledge therein, or well understand the tables which are for that purpose.  
Here first I should have showed how to measure board by Arithmetic: but because it little differeth from measuring of paralellograms, I will therefore refer it to the third part of this book, where it is taught how to measure them.  
Now to measure Timber by Arithmetic, do thus. Reduce the length of the piece to be measured into inches, by multiplying the number of feet therein contained by twelve. Then multiply the breadth by the thickness, and the product or sum thereof multiply by the length, being also reduced into inches, as aforesaid. Then divide that product or of come by 1728 (which is the number of inches in a foot of timber) and the quotient will give you the number of feet contained in that piece of timber.  
Here it shall not be unnecessary to give two examples; the one where the lengeth, breadth and thickness are even inches; and the other, where they have odd parts of an inch.  
Suppose the piece of timber to be measured to be nine foot in length, eighteen inches in breadth, and six inches in thickness. Now multiply nine foot the length by twelve, and that yields 108. inches, which being multiplied by 18. the breadth, makes 1944. and that being multiplied by six the thickness, yieldeth 11664. which is the content of the whole piece in inches. Then divide this 11664. by 1728. and the quotient will be six foot and 1296/1728 parts of a foot, which being reduced to his lowest denomination, is ¾ of a foot. So that the whole piece is six foot and three quarters. And if you would know how many quarters of a foot is in any such fraction, you may divide it by 432 the number of inches in a quarter of a foot of timber, & the quotient will tell you. But suppose the piece have in the breadth or thickness, an odd quarter, or half, or three quarters of an inch: then reduce the length, breadth and thickness into quarters of inches: after that, multiply them being so reduced one into another, that is, the length by the breadth, and that of come or product by the thickness. Then divide that total by 110592 (which is the number of quarters of inches contained in a foot of timber) and the quotient will show you how many foot is contained in the piece. As for example, let the piece be 20 foot long, 8 inches and a half broad, and 11 inches and three quarters thick: First I reduce 20 foot the length into quarters of inches by multiplying 20 by 12, which is 240, which brings 20 foot into inches: then I multiply that 240 by 4, to bring them into quarters of inches, which is 960. Secondly, reduce 8 inches and a half the breadth, into quarters of inches, which is 34. Thirdly, reduce 11. inches and three quarters the thickness into quarters of inches, which is 47. Then multiply 960, and 34, and 47, together, and they make 1534080. Lastly, divide these by 110592, and the quotient will be 13, and 96384 odd quarters of inches: which being divided by 27648, the number of quarters of an inch in a quarter 
* Note here that I mean not by a quarter of a foot, a square of 3. inches, which is a quarter of a foot 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 foot, 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 foot, three quarters of a foot and somewhat more. This is more tedious to teach then to practise. Behold the work on the other side of the leaf.  
 
Note, that where there is no odd parts of an inch in the breadth or thickness, than you may more briefly cast up the content thus: Multiply the number of inches in the breadth by the number of inches in the thickness, and the product thereof multiply by the number of feet in the length. Then divide that total by 144, and the quotient will give the content of the piece in feet. As in the first example behold.  
  

CHAP. 8. A second way how Timber may be measured by Arithmetic.  
MVltiplie the breadth in the thickness, and by the product or ofcome thereof divide 1728, and the quotient will show you how much in length maketh a foot of that piece; according to which you must measure the whole piece. And if there be any odd quarters, or half inches in the breadth or thickness, then reduce the two sides into quarters of inches, as is taught in the last Chapter. Then having reduced them, multiply the breadth by the thickness, and by their product divide 27648. and the quotient shows you the length of the foot in inches, which you may bring into feet by dividing them by 12. Behold an example of either kind in the work following.  
Example 1.  
Example 2.  
And if you would know how many quarters of an inch any fraction of an inch doth contain: or for example, how many quarters of an inch either of the said fractions of 160/224 and 382/1598 do amount unto, you need but divide 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 unequal sides, may, by Arithmetic, 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 show the squares of any ordinary piece of timber of unequal sides, that so they might thereby truly measure the piece according to the Rule. Herein Master Digs took great pains, worthy to be commended. But men being loath to carry a table in their pockets for trouble (as it seemeth unto me) did altogether leave it, and framed to themselves the said false way of halving the sum of the two sides for the square, which turned to the great loss of the buyers. Now my desire is, that seeing men will not use that Table because of trouble, that they would learn to carry the ground thereof in their heads, which is contained in this Chapter. Wherein though there be some greater measure of Arithmetic required then in the former, namely extraction of the square root, yet the fruit of that kind of Arithmetic will repay any man's travail, that either hath, or will take pains to learn it.  
But to perform the question, do thus: Multiply the breadth and thickness of the piece together. Then take the square root of the product thereof, which square root is (as we call it) the square of that piece, Then having the square given, you are not ignorant how to measure the piece by the rule. As for example: if the piece be 16 inches broad and 4 inches thick: you must multiply 16 by 4, which makes 64. the square root whereof is 8 (for 8 times 8 is 64.) Therefore 8 is the square of that piece, and not 10 as your common way tells you.  
If the root cannot be exactly extracted, as oftentimes it cannot, take the nearest that you can, and which you shall think requisite, by such ways as are already extant in divers authors.   

CHAP. 10. A second way to find the square of an unequal sided piece of Timber, namely, by the Scale and compass.  
BEcause this way is more easy and speedy than the former, and that which may also be done by those that have no Arithmetic: therefore I have added this unto the other.  
What is meant by the compass, that is, a pair of compasses, all men know: but what is meant by the term Scale, many peradventure are ignorant of: therefore it shall not be unnecessary to describe it, as in this manner.  
A Scale is any line (and in ordinary matters, any right line) divided into any number of equal parts, be they greatter or lesser, wider or narrower, of the distance of an inch-halfe inch, quarter, half quarter, or any other distance greater or lesser. This line thus divided, is, I say, called a Scale; whereof every part or division may stand for a mile, a rod, a yard, a foot, an inch, or any other kind of measure what you will, or have use of.  
But to the matter. On the backside of the Rule (as we call it) there is (you know) a void room or place, about the middle thereof. Now in this void place, draw a line, as near to the edge of the Rule as conveniently you can, some six or seven inches long, as here I have drawn 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 cross 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 divided into quarters of inches, beginning from the angle or corner at B: and every one of those quarters of inches, divide into four equal portions or divisions, as here I have divided them.  
Here I pray you note once for all, that I call all those quarters of inches, Parts of the Scale: every one of which parts in this matter, is, or standeth for an inch; and half a part is half an inch; a quarter of a part is a quarter of an inch; three quarters of a part is three quarters of an inch, and so forth. This done, your Scale is prepared.   diagram of ruler 
Now to find the square of any piece of Timber: first add the breadth and thickness together. That done, note the half thereof: 
Or, (which is all one) note what either of the two sides doth differ from the said half of the sum of both sides.  note also what is half the difference of the two sides of the piece. Then reckoning how many inches and what odd quarters of an inch there is in the said half of both the sides added together, you must open your conpasses (being small or sharp in the points) to so many parts and such odd quarters of a part in the Scale. That done, set one foot of the compasses being so opened, upon the line BC, so many parts and odd quarters of a part from B, as there be inches and odd quarters of an inch in half the difference of the sides: And extend the other foot of the compass upon the line AB, and note where it cuts it, or lights upon it. Now I say, that so many parts and odd 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 odd quarters of an inch is there in the square of the piece.  
This is far sooner wrought then spoken, and is better explained by an example, then taught by a rule. Therefore suppose the piece to be measured, be 7 inches one way, and 11 inches and a half the other way. First I add 7 to 11 ½, which makes 18 ½. Then I note the half thereof, which is 9¼. I note also half 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 foot at D, which is two parts and ¼ from B, and I extend the other foot to the line AB, and it cuts or lights at the point E. Then I say that the square of that piece, is very near 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 half quarters of inches. And truly where the lines AB or BC will not admit or receive all the inches they should (which yet I think will rarely happen) than you must take for your Scale, half quarters of inches, and so proceed (as is before showed) 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 divided into quarters of inches, without any subdivisions into four, for you may well enough guess at them without any error to be respected: take also for the line BC, any one of the thwart strokes which serves to divide the Rule into inches (for I suppose them to stand square to the said other line;) which thwart stroke you must also divide into quarters of inches, as before is taught. And thus if one of the thwart strokes should be worn out with pricking (which will not be in short time, if you work lightly and with small pointed compasses) than you may take another of the thwart strokes, and divide it into quarters, and so proceed as before.   

CHAP. 11. How that measure which is called boord-measure is gathered.  
BOord-measure, or the table of boord-measure, is nothing else but that which showeth the length of a foot of board or plank, at any breadth: though commonly we gather this measure, but for every breadth from one inch to thirty six inches, and so set them down in order, as in the table of boord-measure following, which is from one to thirty. Now you may find the length of a foot at any breadth thus. Divide one hundred forty sour (which is the number of inches in a foot of board) by the breadth that given, and the quotient will show you how many inches makes a foot in length at that breadth. As for example. Divide one hundrech forty four by the breadth fourteen, and the quotient will be 10 4/14 or 2/7, that is, the length of a foot at 14 inches broad, is 10 inches and two seven parts of an inch. So divide 144 by the breadth 4, and the quotient will give you 36 inches for the length of a foot at that breadth of 4 inches. This 36 divided by 12, will give you in the quotient the number of feet therein, which is just three.  
But if you will gather your table of boord-measure, for boards that have odd quarters of an inch in their breadth: then divide 576 (which is the number of quarters in a foot of board) by the number of quarters of inches in the breadth, and the quotient will give you in inches, the length of a foot. As for example, let the breadth be 7 inches and a half: This is in all 30 quarters of an inch. By this 30 I divide 576, and the quotient is 19 6/30 or ⅕, that is, the length of a foot 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 giveth the length of a foot of Timber at any square. But our ordinary Timber-measure is gathered, only from the square of one inch to the square of 36 inches. It is thus gathered. Multiply the number of inches in every square, by itself: then by the ofcome or product, divide 1728 (which is the number of inches contained in a foot of Timber) and the quotient will give you in inches, the length of a foot of Timber at that square. As for example: let the square be 18. This 18 multiplied in itself, makes 324: by which, divide 1728, and the quotient will be 5 108/324 or ⅓, that is, the length of a foot, is 5 inches and one third part of an inch.  
But if you will gather your table of Timber-measure, for squares that have odd halves or quarters of an inch; then multiply the number of quarters of inches in the square, by itself; and by the product or ofcome thereof, divide 27648 (which is one fourth part of the quarters of inches contained in a foot) and the quotient will give you the length of a foot in inches: which you may reduce into feet, by dividing 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 itself, and it makes 4489. By this 4489 I do divide 27648, and the quotient will be 6 714/4489 which is 6 inches, and some what more than a sixth part of an inch.   

The Table of Boord-measure.  
THough Master Digges in his said Tectonicon, hath set down two Tables, the one for boord-measure, and the other for Timber-measure: yet because those that here follow, do differ, both in matter and form; I have therefore set them down, with this explication of them ensuing.  
But first note; that whereas Master Digges in both Tables, hath proceeded from one inch to 36 inches: I have gone no farther then to 30 inches: and indeed it were no great matter, if they went but to 24 inches. For when the breadth of board, or the square of Timber, do exceed 24 inches, but especially 30 inches; then they give so little for the length of a foot, as that the oft drawing of that length upon the board or Timber, cannot but breed apparent error, except great heed be taken. Therefore for your board, if it exceed 30 inches in breadth, you may measure it as if it were half so broad, and count every foot for two foot. And as for your Timber, if the square thereof exceed 30 inches, or rather 24 inches; then measure it as if it were half that square (as for example, if the square be 32 inches, than account it as 16 inches) and then account every foot for four foot.  
Note also that I have left out of this Table of boord-measure (of that which Master Digges hath set down in his Table) what maketh a foot at any number of inches and an odd quarter or three quarters of an inch, both because they are not, neither can conveniently be set on the Rule, as also for that no loss of any moment comes by guessing at them. And therefore I have contented myself to set down only inches and half inches. But if any man like to observe justly what maketh a foot of board at any number of inches, and a quarter of an inch or three quarters of an inch, then either he may repair to Master Digges his Table, or else let him find it out by the way taught in the eleventh chapter of this second part. But to return to the meaning of this Table of boord-measure.  
In the row AB, is set down one under another, the breadth of boards from one inch to 30 inches, as you may plainly perceive by looking on it. Every one of these numbers from one to 30, I will call a Breadth. In the colum under the letter C, is set down against every breadth, the number of feet, inches, and parts of an inch, that make a foot in length at that breadth. (The feet in either of the columns, 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 brother row at the right side thereof. Note also, that where as you see in the fractions, sometimes one prick, sometimes two pricks, that when one prick stands before the fractions it signifies a littellesse; and two pricks a little more.)  
Now if that the breadth of the board be a certain number of inches and one half inch, then is the length of a foot over against those number of inches, in the colum under D.  
But because an example will give more light than many words otherwise: Therefore suppose the board to be 11 inches broad: then find 11 in the row AB, and over against it in the colum C you shall see 1.1 and 1/11, which is one foot, one inch, and one eleventh part of an inch.  
But if the board were 11 inches and ½ broad: then seek the length of a foot over against that 11, in the colum D, and it is 12 inches and ½. So if the breadth of the board were 25 inches and ½: look for 25 in the row AB, and over against it in the colum D, you shall see five inches and five seventh 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 down the squares of Timber from one inch to 30 inches. Over against every one of these, in the colum under C, is set down the length of a foot, in feet and inches, and parts of an inch. The feet are set down in the row at the left hand of this columas far as the squares do yield a foot, and the inches and parts of an inch are set down in the row towards the left hand of the same colum.  
But if the squares of the piece of timber be any number of inches between 1 and 10, and an odd half inch more: then over against those number of inches in the colum under D, you shall find the length of a foot, in feet, inches and parts of an inch: the feet 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 half 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 book, you shall have occasion oft times to seek out the length of a foot for a square of one inch and a half, two inches and ½, three inches and ½, and so forth to ten inches.  
Here farther it is to be noted, that whereas you shall find for the most part before the numerafor of the fractions, sometimes one prick, sometimes two pricks: that one prick doth signify somewhat less, and two pricks doth signify somewhat more. This I have noted, as Master Digges hath also done, though it matters not whether such more or less, were observed or not, seeing 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 unnecessary. If the square of the piece given be five inches, find five in the row AB, and over against it in the colum of C, you shall find five foot nine inches and three twenty five parts of an inch. If the square given be thirteen, then over against thirteen you shall find on the right side of the colum C, 10⅕: which is 10 inches and somewhat more than one fift part of an inch, if the square given be six inches and a half: then over against six, you shall find the colum D for the length of a foot, three foot four inches and somewhat less than eight ninth parts of an inch: more examples needs not. Enough hath been said to any that is but of mean capacity and desirous to learn.  
The end of the second part.    

THE THIRD PART OF THE CARPENTER'S RULE, Containing sundry true ways to measure superficies and solids, or (as we call them) Boards and Timber, of extraordinary forms.  

CHAPTER. 1. The meaning of certain terms of Geometry generally used in this third part.  
WHen I had written the former part of this book concerning measuring of ordinary Timber and Board, and did consider, that besides the pleasure, there would come some good to Carpenters, if they could also measure extraordinary forms: I have therefore thought good to add this part unto the other two. True it is, that Master Digs in his Tectonicon hath not been silent of the most of these things. But because he applies them to measuring of land, few or none do think that they belong also to measuring of Timber: and therefore my labour I hope is not unnecessary, though I should but have repeated the same thing and apply them to our use, without adding any other thing.  
I have known some that would buy whole frames ready wrought by measure: but sure I am that no Carpenter could have measured it for them, without the knowledge of that which is written hereafter.  
And because I shall have occasion to use many terms of Geometry, by which I may with more ease deliver, and you with more plainness perceive my mind in these things: I have therefore set down the meaning, as plainly as I can, of some Geometrical terms, which most serve for our present purpose. And in this Chapter I explain only those terms that generally I use throughout this whole part, and which do not properly belong to any one chapter. The rest I will declare in the beginning of every chapter, as the matter thereof gives occasion. But to the matter.  
1. An Angle is nothing else but a corner, made by the meeting of two lines (for I speak not of solid Angles).  
2. A right Angle (which we call a squire or a square Angle) is that whose two lines comprehending or making the Angle, stand perpendicular or plumb the one to the other.  
3. An obliqne Angle (which we call bevell or skew) is every angle not being a right angle, whether it be greater or less, or (as we say) whether it spread or clitch.  
4. A Superficies, is that which hath only length and breadth, and no thickness at all. Here note, that whereas we call boards, superficies, and Timber solids, it is not because a board is not a solid (for it hath length, breadth and thickness) but because we respect not (in measuring of them) but only their length and breadth.  
5. A Solid (or a body) is that which hath length, breadth, and thickness.  
6. Parallels are those lines, superficies, or solids, that differ every where alike, or are not nearer together in one place then in another.  
7. A Figure, is any kind of superficies or solid that is bounded about, as Triangles, Squares, Circles, Globes, Cones, Prismes, and the rest.  
8. The Base of a Figure, is any side (as we may say) thereof, upon which it may be supposed to stand. Or if you take any side of a figure for the ground or bottom or lower part thereof, that same is the Base.  
9 The height of a figure, is the length of a Perpendicular or plum line falling from the top thereof, to the base, ground, or bottom 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 than the base or bottom.   

CHAP. 2. How to raise and let fall a Perpendicular.  
A Perpendicular line is that which stands plum upright upon another, leaning neither the one way nor the other. A Perpendicular is said to be raised, when a point is given in a line, from which it must rise. It is said to be let fall, when a point is given above 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 ways Geometrically to do both the one and the other. And because there is especial use in this third part, of letting fall a perpendicular or plum line from a point given, as also from an angle in a figure to the base: I would not have men ignorant how to do the same without a squire, which is not always at hand, when a rule & pair of compasses are. And yet I will set down only one way of many, to avoid tediousness, which is thus.   diagram 
Let the point given be A, the line given BC. Open the Compasses to any distance convenient, and setting one foot in the point A, make an ark or piece of a Circle with the other foot, 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 find the middle between those two Intersections, and from that middle draw a line to the point A (which is the point given) and that line shall be perpendicular or plumb from the point A to the line B C, as was required.   

CHAP. 3. Of a Triangle and a Prism, what they are, and how they be measured.   diagram 
Triangles are divers, both in respect of their sides and angles, and may be measured divers ways: But let this one way serve for all. Multiply half of the base by all the height or perpendicular: Or (which is all one) multiply all the base by half the height or perpendicular; and either of the products gives the content of the Triangle.  
Here I pray you remember what I mean 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 pricked line is the height or perpendicular.  
Now note, that a Prism is that whose two opposite plains or ends be equal, like, and parallel, the other sides being parallellograms, that is, figures whose opposite sides are equal, and whose angles be all right angles. So that every piece of timber may be called a Prism, not being brother at one end then at another, of what fashion soever it be, whether the base or end thereof be of three sides, as cants; or of four, five, six, or more sides, as other timber; and whether the sides be of equal length or of unequal.  
How to measure any Prism or piece of timber of whatsoever form or fashion the base or end thereof is; you need but multiply the content of the base or end by the altitude or height, or (as we call it) the length of the piece, and the product gives the content thereof.  
This might serve once for all, as sufficient to measure solids or timber, whose bases or ends are like and equal: so that it needed not but to teach how to measure every kind of base, as Triangles, Quadrangles, etc. But yet for plainness sake, I will give examples of every form, with some variety of measuring them, when I speak of measuring plain figures as being their bases.  
1. Therefore if the base of your Prism or piece of timber were the said figure A, multiplte the content of the Triangle A by the length of the piece, and the product gives the content thereof.  
2. Or else you may measure that Prism or piece of timber thus: Take the whole perpendicular, and suppose it to be one side of a squared piece of timber (as we call it) and take half the base for the other side: and so measure it by any of the ways taught in the second part of this book.  
This being understood which is here written, the use hereof is very general. For besides that the Carpenter may measure hereby any canted piece of timber, as steps for stairs, and canted rails, and such like: the plasterer also, who often worketh by the yard, may hereby measure gable-ends, and such like forms. The Glazier hath likewise use 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 opposite sides are equal, 
Note that whereas I say all right angles, it is but to distinguish betwixt this figure, and the two next following, for they are Parallellograms also.  and whose angles or corners are all right or square angles: such as is the figure B.   diagram 
It is measured thus: Multiply the length by the breadth, and the product or ofcome gives the content.  
The prism or piece of timber whose bases or ends are parallellograms, as is the said figure B, you may measure thus: Multiply the content of the base by the length, and the product yields the solid content, as is taught in the seventh Chapter of the second part of this book.   

CHAP. 5. What a Rombus is, and how it is measured.  
A Rombus, other wise called a Diamond form, is a figure of four equal sides & obliqne 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. Multiply the length of one of the sides, by the perpendicular plum line, drawn from that side to the opposite side, which in this figure is the pricked line, and the product yields the content.   diagram 
1. The prism or piece of timber whose bases are a Rombus, is measured thus. Multiply the content of the base or end by the length, and the product gives your desire.  
2. Or else (as in the third Chapter of this part) take all the perpendicular for one side of a squared piece, and one side of the figure B for the other side of the squared piece, and so measure it by any of the ways taught in the second part of this book.   

CHAP. 6. What a Romboides is, and how it is measured.  
A Romboides (called a Diamondlike figure) is that whose opposite sides and opposite angles are equal. But all the sides are not equal, 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 have been soyned together: so that he that can measure one can measure both. But to the former ways you may add this, if you please. First divide it into two Triangles, as here this figure is by the pricked line. Then measure one of those Triangles by the way set down 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 Prism or piece of timber whose bases are a Romboides, is measured as a piece 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 piece 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 ways taught in the second part.   

CHAP. 7. What a Trapezium is, and how it is measured.  
A Trapezium is a figure of four sides, not having all the sides and angles equal, as is the figure D.  
It is measured thus. First divide it into two Triangles, by drawing a line from one angle to the opposite angle; as here the figure D is divided by the pricked line. Then measure by the way taught in the third Chap. of this 3. part, both those triangles, severally: and add both their contents for the content of the whole Trapezium.   diagram 
1. The prism or piece of timber, whose end or base is a trapezium, as the trapezium D, may be thus measured. Multiply the content of the base or end, by the length of the piece, and the product gives the content thereof.  
2. Or else, take the square root of the content of the base or end D; and so measure it by any of the wases taught in the second part.  
3. or else, suppose the piece to be cut into two Cants, according as the base or end thereof D is divided. Then measure both those Cants, by the way taught in the third chapter of this part, and take the sum of those two Cants for the content of the whole piece.   

CHAP. 8. What a regular mani-sided figure is, and how it is measured.  
A Mani-sided figure, is that which hath more sides than four. They are either regular or irregular. A regular mani-sided figure is that whose sides and angles are all equal, as is this figure marked with A.   diagram 
An irregular, is that whose sides or angles are unequal.  
A regular mani-sided figure is thus measured. First draw a perpendicular from the centre of the figure to the middle of one of that sides, as is the short pricked line in the said figure A. Then seek the length or compass about of all the sides: the half whereof you must multiply by the length of the said perpendicular, and the product gives the content of the figure.  
Now the centre of a regular mani-sided figure is thus found: Raise a perpendicular upon the middle of any two of the sides not being directly opposite, and where they intersect or cut one another, there is the centre of the figure.  
By this may you find the centre of a Circle, or of any piece of a circle, namely, if you draw any two right lines, either equal or unequal, in the circle or piece of the circle, and from their middles do raise perpendiculars: I say, the meeting of those perpendiculars, is the centre of the circle or piece of a circle.  
But if the figure have an even number of sides, as 6, 8, 10, and so forth; than you need but draw a line from any two opposite angles, and another line from any two other opposite angles: and where those two lines do cut, there is the centre of the figure. Thus have I drawn the two long pricked lines to find the centre of this figure A.  
1. The prism or piece 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 piece, and the product gives the solid content.  
2. Or else, take the half of all the sides or compass about as for one side of a squared piece of timber, and the said perpendicular for the other side, and so measure it by any of the ways 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. Divide the figure into triangles, as here I have divided the figure B by certain pricked lines. Then measure each triangle by itself, as is taught in the third Chapter of this part, and add the contents of all the Triangles together; and their sum or total is the content of the figure.  
1. The prism or piece of timber, whose base or end is the figure B, or any other unequal mani-sided figure, is measured by multiplying the content of the base or end in the length of the piece, and the product gives the solid content.   diagram 
2. Or else, take the square root of the number of inches in the content of the base, and so measure it by any of the ways taught in the second part of this book.  
3. And because for the measuring of every sort of timber I have used to set down some way by which it may be done without Arithmetic: Therefore for a third way; as every triangle in the figure B is measured, so measure so many solids whereof the triangles be the bases, and whose lengths are that length of the irregular piece of timber, as is taught in the third Chapter of this third part; and then add all their contents for the content of the whole piece.   

CHAP. 10. What a Circle, a Semicircle, and a Sector are, and how they be measured.  
1. A Circle is a figure, plain 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 Centre.  
4. The line (and indeed every line) passing by the Centre to both sides of the Circumference, is called the Diameter, as that black line in this circle C.  
5. Half the Diameter, and indeed every line passing from the Centre to the Circumference, is called the Semidiameter.  
6. All lines in the Circle (except the Diameter) which are drawn from side to side, are called Cords. Such is the pricked line in the circle C.  
7. The Diameter cuts the circle into two equal parts; each of which parts is called a Semicircle.  
8. A cord cuts the circle into two unequal parts; each of which parts is called a Section of a Circle. The one is called the greater section, because it is greater than that Semicircle. The other is called the lesser section, because it is less than the semicircle.  
9 A Sector is any part of a circle, either greater or lesser than a semicircle, contained under two semidiameters meeting in the centre, and a piece of the circumference.  
10 Note that the circumference of a circle is triple the diameter, and almost one seventh part more. But we count it as just triple and one seventh part, or as 22 is to 7. Therefore to find the circumference, the diameter being known; multiply your diameter by 22. and divide the product by 7, and the quotient shows you the circumference.  
Now to measure a circle do thus. 
To measure a Circle.  Multiply half the diameter by half 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 gives the content of the circle.  
The Semicircle is thus measured. 
To measure a Semicircle.  Multiply half the diameter in half the ark or arch thereof; or all the diameter in a quarter of the ark thereof, etc. And either of the products gives the content of the semicircle.  
A Sector is measured as a semicircle, 
To measure a Sector.  namely by Multiplying half the semidiameter in half the ark or arch thereof, etc. the product whereof is the content of the sector.  
1 The solid or piece 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 piece, and the product gives the solid content.  
2 Or, take the semidiameter for one side of a squared piece of Timber, and half the ark (be it of a circle, semicircle or sector) for the other side, and so measure it by any of the ways taught in the second part.  
Here it were not unnecessary to show how to find the quantity of an ark or piece of a circle. But of the divers ways that are found, you may use this. First having taken the length of your semidiameter in any measure, which let be for more conveniency in 
* For the greater measure the greater error in this kind of work.  inches, then open your compasses to an inch, and see how many of them is contained in your ark.   

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 than a semicircle; and a lesser, which is not so much as a semicircle. They are measured diversely.  
A greater section is thus measured, 
To measure a greater Section.  draw two semidiameters from either end of the ark to the centre, as the two pricked lines in the figure A. Then Multiply half the ark in the semidiameter, and to the product thereof add the content of the said triangle, made of the cord and the two semidiameters, and the total will be the content of the whole section.   diagram 
The solid or piece 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 piece.  
Or take half the ark as one side of a squared piece of Timber, and the semidiameter as the other side, and so measure it by any of the ways in the second part: then add thereunto the content of the solid, whereof the triangle before named is the base, and the total will be the content of the solid, whose end is the said section.   diagram 
The solid or piece 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 solid.  
Or else, take half the ark as one side of a squared piece of timber, and the semidiameter for the other side, and so measure it by the ways in the second part of this book. Then deduct out of the content thereof, the content of the solid or piece of timber, whereof the triangle made of the cord and the two semidiameters, is the base or end: and the remainder is the content of the solid, made upon the said lesser section B.  
In the 6. chapter of the second part of this book, 
To measure an oval.  I told you that I would show you in the end of this Chapter, how to measure an oval: which in a word is thus. First find (by that which is taught in the 8. Chapter of this part) the four centres of the four arks or pieces of circles, whereof the oval is made (for every oval is made, by joining together four several arks or pieces of circles, whereof those at the two ends are equal, and those at the two sides are also equal). Then having found those arks, draw their cord under them: so shall you see the whole oval to be divided into four lesser sections of a circle and a parallelogram. Therefore measure all those four 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 four sections and the parailelogram) is the content of the oval.  
Neat, that when you have measured one section at the end of the oval, you have also the content of the other section at the other end (because they are equal): Therefore you need but to double the one, and so you have the content of them both. In like sort may you do with the sections at the sides, that is; having measured the one, do but double it for the content of them both.  
If a piece of timber were oval fashion, that is, had his base or end like an oval: than it may be measured by multiplying the content of the base by the altitude or length of the piece, as hath been 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 kind of figure, either plain or solid, that is made of strait lines and circular lines together, if you do but use diligence, and divide or imagine a division of the whole figure into his proper sections and other figures as becometh; and so measure those sections and other figures severally, and then add their contents together.   

CHAP. 12. What a Pyramid is, and how it is measured.  
A Pyramid is a solid figure, contained under many plain superficies set upon one plain superficies, and gathered together into one point, as this pyramid A, Pyramids are divers according as their bases be divers, as of 3, 4, 5, or more sides, as is at large showed in the 10. definition of the 11. book of Euclides Elements of Geometry.  
But of what sorts soever the pyramids be, 
To measure the solid content of a Pyramid.  they are measured by one rule, which is thus: Multiply the altitude or height by the third part of the content of the base: or contrarily, multiply the content of the base by the third part of the altitude or height, and either of the products is the content solid of the pyramid.  
Here note, that the altitude or height is not the length of the side of the pyramid, but (as I told you in the first chapter of this third part) it is the length of the perpendicular or plum line, falling from the point or top thereof, to the base or bottom.   diagram 
But if you would measure the superficies or outside of a pyramid: 
To measure the superficial content of a Pyramid.  then (in so much as all the sides are triangles) measure every triangle by itself, and to all their contents add the content of the base, and the total or sum is the superficial content of the pyramid.   

CHAP. 13. How to measure parts of a pyramid, or timber that is taper grown.  
IF a part of a pyramid do taper much, you may lay it on a paper by a scale, and make it out to a perfect pyramid. Having so done, measure the whole pyramid. Measure also that part thereof which was added; which subtract or take from the content of the whole pyramid, and the remainder gives the content of the piece of a pyramid. This way is not convenient in ordinary tapered timber, and therefore take rather the way here following.  
You know that in the last chapter of the first part of this book, I told you, that if a tapered piece were measured by the square of the middle, there was lost to the seller a piece of timber, whose length is the sixth part of the length of the whole piece; and which hath for his breadth, half the difference of the breadth of both ends; and for his thickness, half the difference of the thickness of both ends. Therefore having measured the piece by the square of the middle, deduct the said lost piece, and the remainder is the content of the tapered piece. This lost piece you may measure (and so deduct) thus. Take the half of the difference of the breadth of both ends of the tapered piece, for the breadth of your lost piece; and half the difference of the thickness of both ends, for the thickness of the lost piece: & with that breadth and thickness measure it according to the length; which (as I said) is the sixth part of the length of the whole piece. And note, that by the length, I mean not the side of the tapered piece, but the plum or height thereof.   

CHAP. 14. What a Cone is, and how it is measured.  
A Cone is a round pyramid (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. book of Euclides Elements. 
To measure the solid content of a Cone.   
The solid content of a Cone is thus measured: Multiply the content of the base in the third part of the height, and the product is the solid content of the Cone. 
To measure the superficial content of a Cone.    diagram 
Note here again, that the side of the Cone is not the height, plum 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 plains and solids, 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 look into Euclides Elements, Master Digs his Pantametria, Master Lucars Solace, and other good books of Geometry, which are extant in English.   
FINIS.