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 [Page 3]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 [Page 4]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 [Page 5]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.

Suppose therefore that this figure or picture marked with the letter A. were a péece of Timber, hauing for his length

foure foote: for his thicknes sixe ynches: and for the breadth eightéene ynches. Now this péece of Timber is indéede but thrée foote of Timber, where as by the former way it is foure foote: For eightéene ynches and sixe make twentie foure, the halfe whereof is twelue.

Now to proue and demonstrate the error, behould the figure marked with the letter B.

which suppose to be the end of that péece of Timber onely, hauing (as you may sée) for the breadth eigh­téene ynches; and for the thicknes sixe ynches. Then cut off by the prickt line, twelue ynches from the breadth, being eightéene ynches; which twelue ynches, is supposed to be the side of the square of the said péece. This done, there remayneth sixe ynches, which remaynder, sup­pose to be layd vnto the lower side of the figure B. and then there doth appeare a defect, or want of a péece, of sixe ynches square. For howsoeuer the remainder hath sixe ynches breadth, which being laid to sixe, the breadth of the whole [Page 6]figure, it makes twelue; yet hauing but sixe ynches for the length also, it wants sixe of twelue to which it is applyed. As by the beholding of the figure it doth plainly appeare.

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 [Page 8]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.

I will not stand to demonstrate this or any thing else in this booke geometrically, because it is beyond the common capacitie. But yet you

may perceiue the great­nes of this error, onely by beholding this figure A. where I haue made a square with prickes, in the circle: the foure fides of which square are e­quall to the length of the circumference or ring of the circle, which circle you must imagine to be the end of a round péece of Timber to be measured. Now this square, euery man can [Page 9]perceiue to be lesse then the circle, though he cannot tell how much.

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.

THose that doe their worke but by guesse, doe not sticke at small differences; and therefore they measure by one and the same way, both that Tim­ber that is perfectly round, and that also which is somwhat

flat, called Duall fashion, as is the figure marked with the letter B. I shewed in the last chap­ter, that it was false to measure a circle by taking it to be equall to a square, whose foure sides are equall to the circumference or ring of the circle. And to measure Duall fashion, or that which is not perfectly round but somwhat flat, by the same way, is also false. Onely herein is the difference, that in measuring a circle by the said false way, the error in all circles is alwayes alike; but in Ouals the error varies. For the flatter they be the lesse is the er­ror, vntill it be so flat, that it be equall to the square, whose foure sides are equall to the ring or circumference: and after [Page 10]that, the flatter they be, they are still the lesser, and alwayes lesser then the said square.

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 [Page 11]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.

To proue this course to be

erronious, behold the figure A. the which suppose to be the end of a péece of Timber that is waynie or canted into eight equall cants of fiue ynches a péece: Or, the figure is a figure of eight equall sides, whereof euery one is fiue inches. Now girt this péece about, and it will come to fortie inches, the quarter whereof is ten inches, [Page 12]for the square of that canted péece. According to these ten ynches, I haue made a square within the canted péece by certaine prickt lines: By the very beholding whereof you may sée that the canted péece is bigger then the said square of prickt lines is, and so that losse commeth to the seller.

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 [Page 13]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.

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 [Page 15]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.

Suppose this figure AB to be a péece of Timber being in length foure foote, in breadth nine ynches, and in thicknes foure ynches. Now take the breadth or thicknes, which you will (but here let it be the breadth nine ynches) and note what that wanteth of a foote, which is thrée ynches. These thrée ynches set on, or draw alongst the figure or péece of Timber foure times, that is, so many times are there is féete in the length of the péece, beginning at one end, which let be héere the end A; and where that endeth make a pricke, which let be the pricked line next vnto the end A: And this is the

first worke. The second followes. Then must you measure how many foote it is from that pricke vnto the end B, which in this erample is thrée foote. Then note what the other side of the péece, which was foure ynches, doth want of a foote, which is eight ynches. These eight ynches draw alongst the péece thrée times beginning at the end B, that is, so many times as there is féete from the said first pricke to the end B; and where that endeth make the second pricke, which let be [Page 17]the prickt line next to the end B. Then the space betwéene these two prickt lines shewes the content of the péece of Timber; (which space in this example is twelue ynches: So that this péece is one foote of Timber.) My meaning is, the péece of Timber to be measured, is so many foote as the space betwéene those two prickes is féete in length. And if there fall out any odde ynches in that space, they are so many twelfth parts of a foote, as there be ynches; so that sixe yn­ches is halfe a foote, and thrée ynches a quarter of a foote, and nine ynches thrée quarters of a footè, and so forth.

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 Drawght­measure.

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. [Page 20]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.

As for example, suppose the figure A to be a péece of Tim­ber, hauing for the length foure foote, for the breadth eigh­téene

inches, and for the thicknes sixe inches. This being measured after the most common way is foure foote of Tim­ber, and so let it be measured. Then note the difference of the two sides, which is twelue inches (for eightéene is more then sixe by twelue.) Then take the halfe of twelue, which is sixe; this sixe is the side of the lost square. Now measure that square of sixe inches being as long as the péece to be measured (which is foure foote) and you shall finde it to be a foote of Timber. This foote deduct out of the content of the péece as it was before measured after the common way, and there will remaine thrée foote, which thrée foote is the true content of that péece marked with the letter A.

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 [Page 21]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

wanteth so much of square; which want in this example is [Page 22]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,

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 [Page 23]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 [Page 27]more knowledge therin, then they know how to applie to the present purpose; for whom chiefly I haue writtē 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 Rule­makers, 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 [Page 25]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.

M Ʋltiplie 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 y ^e 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 [Page 29]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 great ter 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 [Page 30]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.

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 boord­measure 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 y ^e 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 [Page 33]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 Timber­measure 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 [Page 34] 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.

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 y ^e 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.

Let the point giuen be A, the line giuen BC. Open the Cō ­passes 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, [Page 42]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.

TRiangles are made of straight lines, or crooked lines, or of both together. But I speake only of right lined Triangles, which are thus defined. A Triangle is that which is com­prehended of thrée right lines, as is the figure marked with A.

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, [Page 43]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 gable­ends, 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.

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.

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.

The Prisme or péece of timber whose bases are a Rom­boides, is measured as a péece whose base is a Rombus, as in [Page 46]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.

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 hase 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.

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 y ^t 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 [Page 48]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 mani­sided 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.

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 enery sort of timber [Page 49]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 y ^t 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.

2. The round line y ^e makes the Circle is called the Circumfe­rence.

3. The point in the middle of the circle is called y ^t Center.

4. The line (and indéed eue­rie line) passing by the Center to both sides of the Circumfe­rence, is called the Diameter, as y ^t 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 y ^t Semicircle. [Page 50]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 she wes 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 [Page 51]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.

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.

A lesser section is thus measured. To mea­sure a lesser section. Draw two semidiame­ters from either end of the arke vnto the center of the Circle, whereof this section is a part, as be the two prickt lines without this figure B. Then multiplie halfe the arke in the semidiame­ter; and out of the product there­of,

take the content of the triangle made of the corde and the two semidiameters, and the remainder will be the content of the lesser section B.

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 [Page 53]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 thē 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.

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 y ^e third part of the height, and the product is the sollid content of the Cone. To mea­sure the su­perficiall content of a Cone.

The superficiall content of a Cone is thus found. Multiplie halfe the circuit or compasse of the base,

[Page 56]by all the side, and the product is the superficiall content of the spire; to which if you adde the content of the base, the to­tall will be the superficiall content of the whole Cone.

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.