CHAP. I. The meaning of certain terms of Geometry used in this Book.

BEcause all Carpenters or other Artificers, in their Trade or Calling, doe in a manuer (and according to their fa­shion) use some kind of Geometry, although themselves be [Page 2] ignorant thereof, therefore I did con­sider that they might bee sooner brought to measure Board and Tim­ber by that art of Geometry (seeing they have their Rule and Compasses by them) then by Arithmetick, being but few of them can write, and there­fore uncapable of that art: and of them few which can write, not one in ten that hath Arithmetick, which is the only cause (as I suppose) that most of them are so ignorant in this art (which doth so much concern them) notwith­standing, all those excellent Rules which have been formerly delivered by the learned: But now to our in­tended purpose.

SEeing I shall have occasion in this Work to use some terms of Geometry, by which I may with more ease deliver, and you with more plainnesse perceive my minde in these things: I have therefore set down the [Page 3] meaning, as plainly as I can, of some Geometricall terms, which most serve for our present purpose.

• 1 An Angle is nothing else but a corner, made by the meeting of two lines (for I speak not of solid angles.)
• 2 A right Angle (which we call a square angle) is that whose two lines comprehending or making the angle, stand perpendicular or plumb the one to the other.
• 3 A Perpendicular line is that which stands plumb upright upon another, leaning neither the one way nor the other.
• 4 A Superficies is that which hath only Length and Breadth, and no Thicknesse at all.
• 5 A Solid, or a Body, is that which hath Length, Breadth, and Thick­nesse.
• 6 Parallels are those lines that dif­fer every where alike, or are not nee­rer together in one place then another.
• [Page 4]7 A Figure is any kinde of Su­perficies or Solid that is bounded a­bout, as Triangles, Squares, Circles, Globes, Cones, Prismes, and the rest.
• 8 The Base of a Figure is any side thereof upon which it may be suppo­sed to stand; or if you take any side of a Figure for the Ground or Bottome, or lower part thereof, that same is the Base.
• 9 The height of a Figure is the length of a Perpendicular or plumb line, falling from the top thereof to the Base or bottome thereof.

CHAP. II. How to raise a Perpendi­cular on any part of a right line given.

LEt AB be a right line given, and let C be a point therein, whereon I would raise a perpen­dicular, open the Compasses to any convenient distance, and setting one foot in the point C, with the other mark on either side thereof, the equall distances CE, and CF: then opening your Compasses to any convenient wider distance, with one foot in the points E and F, strike two arch lines, crossing each other as in D, from whence draw the line DC, which is [Page] [...] [Page] [...] [Page] [...] [Page] [...] [Page] [...] [Page] [...] [Page] [...] [Page 1] [...] [Page 2] [...] [Page 3] [...] [Page 4] [...] [Page 5] [...] [Page 6] a perpendicular to AB, or as we call it, a square line to the line AB.

Or you may from the given point C, prick out any five equal distances, and opening your Compasses to 4 of them, with one foot in C, strike an arch or piece of a Circle towards N, then opening your Compasses to all 5 divisions, with one foot in 3, cross the same arch line in N, from whence draw the line NC, which is a perpen­dicular to the line AC, as before, for if 3 lines be joyned together, so they be in such proportion, as 3, 4, and 5; they will make a right angle.

CHAP. III. How to let fall a Perpendi­cular from a point assign­ed, to a line given.

LEt the point given be D in the former Chapter, and let the line whereon it should fall be AB, open the Compasses to any conveni­ent distance, & setting one foot in the point D, make an arch or piece of a Circle with the other foot, till it cut the line AB twice, that is at E and F, then finde the middle between those two Intersections, and from that mid­dle, draw a line to the point D (which is the point given) and that line shal be perpendicular or plumb from the point D to the line AB, as was required.

CHAP. IV. To a line given, to draw a parallel line, at any di­stance required.

SUppose the line given to be AB, unto which I must draw a paral­lel.

Open your Compasses to the di­stance [Page 9] required, and setting one foot of your Compasses in the end A, strike an arch on that side the given line whereon the parallel is to be drawn, as the arch C, then doe the like in the end B, as the arch line D, then draw the line CD, so as it may but touch or be a touch line to these two arches C and D, and this line so drawn shall be parallel to the line AB, as was re­quired.

CHAP. V. To perform the former pro­position at a distance re­quired, and by a point li­mited.

ADmit AB in the former Chap­ter, to be a right line given, whereunto it is required to have a parallel line drawn at the di­stance, and by the point C.

Place therefore one foot of your Compasses in C, from whence take the shortest distance to the line AB, as CA, at which distance, with one [Page 11] foot in the end B, with the other strike the arch line D, by the extream part of which arch line D, and the point C, draw the line CD, which is parallel to the given line AB, which was required.

CHAP. VI. Having two lines given, to finde a third proportionall line to them.

THe two lines given are A and B, and it is required to finde a third line, which shall bee in such proportion to A, as A is to B. Make any angle whatsoever, as the angle HEC.

Here note, that an angle is always represented by three letters, whereof the middle letter represents the angle intended.

Then place the line A, from the angle E to D, and the line B from E to F, and draw the line DF.

Place also the line A from E to H, and lastly, by the 4 Chapter, from the point H, draw the line HC parallel to FD.

So shall EC be a third proportio­nall line to the two given lines, as was required.

CHAP. VII. Having three lines given to finde a fourth proporti­onall line to them.

THE three lines given are A B and C, and let it be requi­red to finde a fourth line, which shall have such proportion to A, as B hath to C, make any angle, as DGK, now seeing the line C hath the same proportion to the linne B, as the line A to the line sought for, therefore place the line C from G to H, and the line B from G to F, then draw the line FH, now place the line A from G to I, by which [Page 15] point I draw the line EI parallel to FH, till it cutteth DG in E, so have you EG the fourth proporti­onall line required, which is 24.

For as the line 12, is to the line 16, so is the line 18, to the line 24, which is the length of the line we sought for.

These two last Chapters, would I have you diligently to consider, and throughly to learne, because it is the ground-work of that which I [Page 16] intend to deliver in this Booke: which being well understood, will bring much pleasure and profit to the unlearned Artificer, for whose sake this was written.

CHAP. VIII. The making of a Rule or Scale, for the measuring of Board and Timber.

BEfore we can give the true con­tent of any Board or piece of timber, we must have some Rule or Scale whereby we may measure the length, bredth, & thickness thereof, & for this purpose, the common Rules, commonly divided into inches, half in­ches quarters, and half quarters, will very wel serve for our purpose; both for to measure lengths, bredths, and thicknesses withall: and likewise for to [Page 18] give the true content thereof, by the Rules to that purpose delivered, only herein, I would have you take no­tice, that this Scale thus divided,

I would have drawn upon both edges a­like, but differently figured, only for rea­dinesse, after this manner; that sidethat you would have the readiest for the mea­suring of lengths, bredths, and thick­nesses, I would have you number them with 1, 2, 3, 4, 5, 6, and so unto 24, dividing the whole two-foot Rule, into 24 equall parts or inches; and the o­ther [Page 19] edge, which hath the same divi­sions, I would have you number with 4, 8, 12, 16, 20, 24. unto 96: dividing the whole two foot into 96 equall parts; that is every inch into 4 parts, only for readinesse sake, in the work which doth follow.

This line thus divided, is called a Scale; which is no other thing, but a right line divided into any number of equall parts, be they greater or lesser, wider or narrower, so they be equall: every part, or division of which line, may stand for a mile, a rod, a yard, a foot, an inch, or any other kinde of measure what you will, or have use of: and this line I would have you use, in giving up the Content of either Board, or Timber.

I have described this Scale but to 4 inches; but you may thereby per­ceive what I mean by the whole Rule.

CHAP. IX. How any Board may be measured Geometrically.

IF you do well understand that which hath been delivered in the seventh Chapter, you may thereby measure any Board with ease and de­light; for as there is three numbers, or three lines given, whereby the fourth proportionall is found: so in every Board, there is three lines, or numbers (which you will) which be given us, whereby, we may by the seventh Chapter finde a fourth proportionall line or number: which is the number of feet, conteined in the whole board.

The first of the three Numbers given, is always 12, which is the side of a square foot of Board, or the side of a cubicall foot of Timber;

The second number is alwayes the number of feet, conteined in the length of the Board, the third number is alwayes the number of inches con­teined in the breadth of the Board.

And the fourth number which is here sought for, will alwayes be the number of square feet, conteined in the whole Board: the proportion will be always thus.

• As 12 to the length in Feet:
• So the bredth in inches,
• To the Content in Feet.

And seeing examples teacheth better then many words: therefore let us suppose the three lines given us in the seventh Chapter, to be three such numbers as here we have spoken of.

And therefore let the first line C, be a number of 12, taken from some Scale with your Compasses, and placed from G to H, which 12 doth signifie 12 inches, which is the side of a square foot of Board.

And let the second line B, be a number of 16, taken from the same Scale, and placed from the angle at G unto F, and draw theline FH, and this number of 16 doth signifie 16 feet, the length of a supposed Board.

And so let the third line A, be a number of 18, taken from the same Scale and placed from G to I, and this number of 18 doth signifie 18 inches; the breadth of the sup­posed board: now from the point I, by the fifth Chapter, draw the line IE parallel to FH, till it cutteth EG in E: So have you EG the fourth proportionall line required; which being taken between your Compasses, and aplyed to your [Page 23] Scale, will shew it to be 24, and so many square feet are in that Board, whose length is 16 foot, and breadth is 18 inches.

Let us take one Example more, for plainnesse sake, let the figure A, be a Board to be measured, whose length is 9 foot, and breadth 16 inches:

now draw two lines, so as they make any angle, as the angle ABC, which being done, first, take [Page 24] 12 from your Scale, and place it from your Scale, and place it from B to D, then take 9 of the same divisions, which doth signifie 9 foot, the length of the Board, and place them from B to E, and draw the line ED, then take 16 of the same divisions, (which doth signifie 16 inches, which is the breadth of the Board,) and place them from B to C: and lastly, by the fifth Chapter, draw the line C, A, parallel to DE, till it cutteth AB in A, so shall AB be the fourth proportionall number; which being applyed unto your Scale, will reach unto 12; and so many square feet are in that Board.

• For, as 12 is to 9 foot, the length:
• So is 16 the bredth,
• To 12 foot the Content.

And here note, that when you have any odde parts of an inch in the breadth of your board you must take the like parts of one divisi­on from your Scale, more then your even parts was: and so must you doe when you have odde parts of a foot in the length of your Board: as for example, suppose a Board to be eight foot, and three quarters long; now for to set down this length you must take from your Scale eight whole divisions, and three quarters of one; and so apply them to your use: and this must be noted throughout this Booke.

And here note also, that if your Board be taper grown, that is, wider at one end then at the other; then measure the breadth thereof in the middle, and with that wideness pro­ceed according to your Rules given:

And this may very well suffice for Timber that doth taper also.

CHAP. X. How Timber may be mea­sured geometrically.

THE measuring of Timber doth little differ from mea­suring of Board, by the last Chapter, but only in measuring of Timber we have a double work; but the last Chapter well understood, will give light sufficient hereunto.

Therefore by the last Chapter, first, measure how many square feet of flat measure there is in one of the sides of your Timber, as if it was a board by it self, which being done, you have three numbers given you, whereby you may by the seventh Chapter finde a fourth in propor­tion unto them, which fourth num­ber, [Page 27] is the number of Cubicall feet conteined in that piece of Timber.

The First number is always 12.

The Second is always the number of square feet conteined in one of the sides, I mean, of flat measure.

The Third, is always the number of inches conteined in the thickness of the Timber: and this will be always the proportion for this work.

First,

• As 12 is to the length in feet, so is the breadth in inches,
• To the superficiall content, of that same side.

Secondly,

• As 12 to this superficiall content,
• So is the thickness in inches, to the solid content in feet,

As for example, suppose the fi­gure A to be a piece of Timber to be measured, whose length is 8 foot, and breadth 18 inches, and thickness 14 inches.

Now draw two lines, so as they may make any angle as the lines BE, and HE, meeting in the angle E, this being done, first, place 12 (which is the side of a Cubicall foot, of Timber) from E to F, then place 8 (the length of your piece in feet) from E to D, and draw the line DF, [Page 29] and then place 18 (the bredth of your piece in inches) from E to H, and then by the 5 Chapter, draw the line HC, parallel to DF, till it cut­teth BE, in C: So shall CE be the number of feet of flat measure, con­teined in the broadest side of the piece of Timber.

Thus far we have proceeded ac­cording to the last Chapter; and now we have three numbers more given us, whereby we may finde a fourth proportionall unto them.

Wherefore, first, we have 12 already placed, from E to F, secondly, we have the superficial content of the broadest side, already placed from E to C: there fore draw the line FC, and thirdly, we have 14 the number of inches con­teined in the thickness, which we must place from E to G: and lastly, from the point G, by the fifth Chap. draw the line GB parallel to FC, till it cutteth BE at B: So shal BE be the number of [Page 30] cubicall feet conteined in that piece of Timber noted with the Letter A, which being taken between your Compasses, and applyed unto your Scale, will reach unto 14, and so many feet is in that piece.

Now here I will give you one ex­ample, of a piece of Timber hewed just square: Let the figure B be a piece of Timber so hewed, whose length is 9 foot. And let it be 8 inches square, now having made any angle, as the angle ADC, first place 12 from D to C.

And 9 foot the length, from D to A, and draw the line CA, then place 8, the thicknesse of one of the sides, from D to F, and by the 5 Chapter, draw the line FE parallel to CA, till it cuteth DA, at E, so shall ED, be the superficiall content, of one of the sides, thus far according to the 9th. Chapter, as if it had been a board.

And now hear we have three num­bers, [Page 31]

already placed, whereby we may finde a fourth, after this manner. First, we have 12 here placed from D to C, and we have the superficiall content of one of the sides, placed from D to E, and therefore draw the line CE, and hear we have also, 8 the thicknesse of one of the sides, already placed from [Page 32] D to F, from which point F, draw the line FG, parallel to CE, till it cutteth AD, in G, so shall GD, shew you the solid content of that piece, which being applyed unto your Scale, will reach unto foure and so many Cubi­call feet is there in that piece, marked with the letter B.

This Chapter would I have you well to consider, because I do not in­tend to repeat, what I have heare de­livered, but only describe unto you, the end of some pieces, according to their formes, and so give you some Rules, for to measure them by this Chapter.

CHAP. XI. Of round Timber

CHAP. XII. How triangled Timber, or Timber which hath but three sides, may be mea­sured

TRiangles, are made of straight lines, o [...] crooked, or of both to­gether, but I speake only of Right lined Triangles, which is no­thing else, but a figure made of three right lines, as the figure ABC.

Triangles are divers, both in re­spect of their sides and angles, and may be measured divers wayes, but let this one way serve for all: take half of the base, and suppose it to be one side of a squared piece of Timber; & take [Page 43] the whole hieght, or perpendicular, for the other side of the same piece, and so measure it by the former part of the tenth Chapter, in all respects, as there is shewed.

Let the Triangle ABC, be the end of a piece of Timber to be mea­sured, which hath but only three sides.

Now seeing the Base BC is 16, I take one halfe thereof, which is eight inches, for on side of a squared piece of Timber, & I take 10, the whole length of the perpendicular, (which is the

pricked line AD) for the other side of the same piece: and so as if it were a piece of 10 inches broad, and eight inches thick, you may cast up the con­tent, by the former Rules.

CHAP. XIII. How Timber, whose end is a Rhombus, (or Diamond form) is to be measured.

A Rhombus (or Diamond) is a fi­gure of foure equall sides) but no right Angles, such as is the figure ABCD, for the measuring whereof, observe this example. Let the said figure ABCD, be the end of a piece of Timber to be measured: now taking the length of the side or base AB, which is 14 inches for one of the sides of a squared piece of tim­ber, and the length of the perpendicu­lar DE, which wil be found to be 12, [Page 46] and something better then the eighth part of one more, for the other side

of the same piece, with which two sides, as if it were an unequall squared piece of timber, proceed in all things, according to the former part of the tenth Chapter.

CHAP. XV. How Timber whose end is a Rhomboides (or Dia­mond-like) is measured.

A Rhomboides (or Diamond-like) is a figure, whose opposite sides, and opposite Angles, are only equall, and it hath no right Angles. Such as is the figure FGHI, and may

[Page 48] be measured after this manner: take the length of the side HI, or FG, which is 16 inches for one side of a squared piece of timber, and take the perpendicular FL, which is 10 inches, for the other side of the same piece, & so you may measure it by the former part of the tenth Chapter, as if it were an unequall squared piece of 16 inches broad, and 10 inches thick.

All other four sided figures besides the true Square, and the unequall Square in the tenth Chapter, and the Rhombus in the last Chapter, and the Rhomboides in this, are called Trape­zias or Tables.

CHAP. XV. How to measure Timber, whose end is a Trapezi­am.

A Trapeziam is any irregular four sided figure of what fashion so­ever, as the figure ABCD is a Trapeziam, and may be cast into two

Triangles, by drawing the Diagonall [Page 50] line AC, and so each Triangle mea­sured as is before shewed, which being done, adde the contents of them both together, and you shall have the con­tent of the whole Trapeziam ABCD. Or you may more readily measure it thus: Take one half the Diagonall line AC, which in this example will be 8 inches, for one side of your piece, and take the two perpendiculars BF and DE, and joyn them both toge­ther in one sum, so shall you have in this example 10 inches for the other side of your piece, with which two sides, (as if it were an unequall squared piece of Timber) proceed as before, in the former part of the tenth Chapter.

CHAP. XVI. How to measure Timber, whose sides are many, as 5, 6, 7, 8, 9, 10, or more, so they be all equall.

MAny sided figures are those which have more sides than foure, and are generally cal­led Pollygons.

A piece of timber whose end shall have more sides than foure, may be measured after this manner, adde all their sides together, and take halfe that number for one side of an unequal squared piece of Timber, then let fall a perpendicular from the centre or midst of the figure, to the midst of some one [Page 52] side, and take that length for the other side of the same piece, with which two sides proceed as before is shewed.

Suppose the figure A to be the end of a piece of Timber of five sides, be­ing all equall, and each side conteining 12 inches, which being added toge­ther into one sum will make 60, the half whereof wil be 30 for the breadth of your piece, then take the length of the perpendicular (falling from the [Page 53] center A to the midst of one of the sides,) which here is 8 inches, for the thicknesse of the same piece, with which breadth and thicknesse proceed in all things according to the former part of the tenth Chapter.

This rule is generall in all kinde of regular Polygons, how many sides soever they have.

Here I might have proceeded to have shewed by what means Pyramidall or picked Timber, or Steeples may be mea­sured: but considering how little this appertaineth to Carpenters, and how sufficiently they be handled by Master Diggs in his Geometricall works, I for­beare here to write of them.

CHAP. XVII. How to finde the length of a Foot of Board, at any breadth given.

THe breadth of a Board being given, with the number of 12, (the side of a square foot of Board,) you may by the sixt Chapter finde how much in length will make a foot at any given breadth, by finding a third proportionall number, which shall be to 12, as 12 is to the given breadth.

As suppose a Board to be 16 inches broad, and I would know how much in length will make a foot thereof.

First, make any angle, as ABC, [Page 55] then place 16 inches, the breadth of your Board, from B to D, & 12 inches from B to E, and draw the line DE:

Then again place 12 from B to G, frō which point G (by the fifth Chapter) draw the line GF parallel to DE, till it cutteth AB in F, so shall FB be the length of a foot of Board at 16 inches broad, which being applyed to your Scale, will reach unto 9, which doth shew, that at 16 inches breadth, 9 inches in length doth make a foot of square Board: For, [Page 56]

• As 16 is to 12;
• So is 12 to 9.

CHAP. XVIII. The breadth and thickness of a piece of Timber given, to find how much in length shall make a foot of square Timber at that breadth and thicknesse.

SUppose a piece of Timber to be 18 inches broad, and 14 inches thick. First, make any angle, as DFB, and place 18 inches from F to A, this is the supposed breadth of your piece; then place 12, the side of a Cubicall foot of Timber from F to E, [Page 57] and draw the line AE: So likewise place 12 from F to G, from which point G, draw the line GH parallel to AE, till it cutteth FD in H, so shall HF be the length of asquare foot of flat measure at the former breadth gi­ven: thus far according to the last Chapter.

Now to proceed, place 14 the thick­nesse of your piece from F to C, and draw the line CG; and lastly, from the point H, draw the line HI parallel [Page 58] to CG, till it cutteth FB in I, so shall IF be the length of a foot required, which being applyed to your Scale, will reach almost unto 7 inches, it wanteth but one seventh part of an inch, and such is the length of a foot of Timber whose breadth is 18 inch­es, and thicknesse 14 inches.

CHAP. XIX. Of the Table for Board and Square Timber, and also for round Timber.

CHAP. XX. How to finde a mean propor­tionall line between two lines given.

THe Tables of Timber measure servs for such timber as is just square or round, it will not be unnecessary to shew you how to finde a mean proportional line, between two lines given, or to bring an unequall squared piece of Timber, to a true square, and so to apply it to your Ta­ble.

Let a piece of Timber to be mea­sured, be 9 inches broad, and 4 inches thick. Now: because it is not just square, it cannot be measured by the Table, therefore we must finde a mean between the two given sides, after this manner.

First, take 9 from your Scale, and place it on the line AC frō C to D, & place 4 of the same divisions, from D to A, upon which line AC describe the Semicircle ABC, and one the

point D (where the two lines are joyn­ed) by the second Chapter, raise a perpendicular to cut the Circumfe­rence in B, so shall BD be the mean proportionall line desired, which be­ing applyed to your Scale, will reach unto 6.

So that it doth appear, that a piece of Timber that is 9 inches broad, and 4 inches thick, is equall to a piece of 6 inches square.

And hereby doth another of our Carpenters errours appear, which is this, they do put both sides together, and then they take half of that num­ber, for the true square of that piece, which is meerly false. For in this ex­ample, joyn 9 and 4 together, and they will make 13, the halfe whereof is 6 and a halfe, which indeed, according to our Rule should be but six.

Having found the mean proportio­nall number, you may enter the Ta­ble of Timber therewith, as hath bin [Page 69] formerly shewed, concerning square Timber. By this Rule the ingenious Practicioner may bring any of the for­mer pieces, of what fashion soever, to be measured by the Table of square Timber.

And hee that hath Arithmetick, may apply the proportions given in this Book, to the Rule of Three, and thereby he shall finde the contents as before.

CHAP. I. Of the Scale, and the gradu­ations or divisions there­of, and how they are to be used.

THe Scale here mentioned in this second Part, is a two foot Rule made with a joynt, having a line of e­qual [Page 72] parts issuing frō the center there­of, and divided into 100 equall parts, upon the flat or edge thereof, may be made the other Rule according to the directions and figures in the preceding Part, so that any Artificer having his Rule and a pair of Compasses about him, may measure his Board, Timber, or Stone, two severall ways.

And before I come to any perticu­lar Propositions of the measuring of any Timber, or Board, I wil first shew the use of this opening Scale, in find­ing of proportionall Lines, which is the ground of the way of measuring in the former Book, and also apply the use of them to some other Conclusi­ons following.

CHAP. II. To divide a line gi­ven, into any num­ber of equall parts.

THe line given is AB, and it is required to di­vide the same into five equall parts.

Take with your Compas­ses the given line AB, and fit that in any number that may be equally parted into five without any remainer, as fit it in 100, there let the Scale rest, then take it over in one fifth part thereof, viz. in 20, and that distance set from A to 1, [Page 74] so is A 1 one fifth part of the given line AB, which was required.

CHAP. III. To take any part or parts of a line.

THe line given is AB, now of the same line it is re­quired to cut off three eight parts thereof.

Take the given line AB, and apply it to some number that may be parted in 8, which 8 is the Denominator, and take it o­ver in the Numerator, or fit it in so many times the Numera­tor.

As fit AB in 80, which is 10 [Page 75] times 8 the Denominator, there let rest the Scale, then take it over in ten times the Numerator, viz. in 30, and that distance set from A to C, so is AC three eight parts of AB given: then must CB be the rest, which is five eights. For if you take it over in 50, which is five eights of 80, (as the Scale stood wherein the line AB was fitted) and CB will appear to be the rest.

CHAP. IV. A line containing any part or parts of a line, thereby to finde the whole line.

SUppose that AB be three eight parts of some line, and let it be required to find the whole line.

Take both Numerator and Deno­minator [Page 76]

so many times as you please, as take each ten times, makes 30 and 50, then take the line AB and fit that in 30, and there let rest the Scale, then take it over in 50, & that distance lay down for the line CD, and so shall CD be the whole line, whereof AB was three eights.

CHAP. V. A line being given, conteining any number of equall parts, to cut off from it so many as shall be required.

AS let AB be a line given, conteining 52 parts upon some Scale, and let it be required to cut off from it 24 of the same parts.

Take with your Com­passes the given line AB, and set that in 52 of the e­quall parts, there let rest the [Page 78] Scale, then take it over in 24, the parts required, and that distance set from A to C, so is AC 24 of the same parts whereof AB is 52, which was required to be done.

CHAP. VI. To lay down sodainly 2, 3, or more lines in proportion required.

IT is required to lay down foure lines in proportion one to another, as these foure numbers following. The numbers given

• 60 A
• 50 B
• 32 C
• 23 D

Open by chance your Scale, and [Page 79] there let it rest, then take it over in 60 and in 50, and lay them both down, also take it over in 32 and 23, and lay them down, and so have you four lines A, B, C, D, in proportion, ac­cording to the four numbers given.

CHAP. VII. In a Map or Plot, the length of any line being known, thereby to find the length of all or any of the rest.

AS in the Plot ABCDEF, let the length of the line AB be known to be 47 parts, on some Scale, now it is required to finde the length of the line CD.

Take the known line AB, and fit that in 47, and let the Scale rest, then take CD, and bring it along the e­quall parts, till it be equally fitted on each side, which is 73 parts, so is CD 73 of the same as AB is 47, the like of all the rest.

But if you desire the distance from C to F, take it, and bring it along

your Scale as it standeth, so finde you 90.

CHAP. VIII. Ʋnto two lines given to find a third in proportion.

CHAP. IX. Ʋnto 3 Lines given, to find a fourth in proportion, that is to perform the Rule of Three in Lines.

AS let A B C be the 3 lines, unto the which it is required to finde a fourth in proportion, that is, as the first is to the second, so is the third to the fourth.

Take the three lines one after ano­ther with your Compasses, and apply them to any Scale of equall parts to know their length, and suppose you find them as the numbers which stand by them.

Then take the second line B, and apply it to some Scale of equall parts,

and fit that over in the length of the first line 48, there let the Scale rest, then take it over in the third line C, viz. in 40, and that distance lay down for D, which is 30 parts of the same Scale, where the line A is 48, and is the fourth proportionall required: for as A is to B, so is C to D, &c.

CHAP. X. To divide a line given into two such parts, bearing proportion one to the other, as two numbers given.

AS let it be required to di­vide the given line AB into two such parts, bear­ing proportion one to the other, as 28 to 21, viz. that AC may be to CB, as 28 to 21.

Adde your two given num­bers together, viz. 28 and 21, make 49, then take with your Compasses the given line, AB [Page 87] and fit it in 49, there let the Scale rest, then take it over in 28, which set from A to C, so is AC to BC, as 28 to 21, which was required to be done.

CHAP. XI. To measure flat Measure.

A Board being 16 inches broad, now it is required to finde how much in length makes one foot.

Take on any Scale of equall parts 12, the number of inches in one foot, and fit that in the breadth of the board which is 16, there let the Scale rest, then take it over in 12 alwayes, and that apply to the same Scale of equall parts where the 12 was taken, and it sheweth 9, and so many inches in length make a foot of board required, [Page 88] for if a board have 16 inches in length and 9 in breadth, these two numbers multiplyed together make 144 inches, the number of square inches contein­ed in a foot of square board, or glass, &c.

Let a board be seven inches & three quarters broad, now it is required to finde how much in length makes one foot.

Take (as before) 12 of some Scale of equall parts, and fit it on seven three quarters, the breadth thereof, and then take it over in 12, as before, but to fit it in seven three quarters, would open the Scale too wide, there­fore take four times seven three quar­ters, which is 31, & fit 12 in that, and take it over in four times 12, which is 48, and that distance applyed to the same Scale where the 12 was taken, sheweth 18 three fifths, and so many in length shall make one foot.

If a board be two inches broad, [Page 89] how much in length shal make a foot?

Multiply two the breadth of the Board, and 12 the inches in the foot by 10 makes 20, and 120, then take of some small Scale 120 (which may be done upon some Scale placed upon your Rule) and fit that on 20 on the Scale, and take it over in 12, and fit it in 2, 3, or 4 times 20, and take it over in so many times 12, and that apply to the same Scale, where the 120 was ta­ken, and it sheweth 72 inches, and so many in length is a foot of Board, the Board being two inches broad.

Let a Board be three inches and three quarters broad, now you desire to know how much in length maketh a foot.

Bring three inches and a quarter into quarters, and it maketh 13 quar­ters, then multiply 12, the inches in a foot, into quarters, and it maketh 48: take then 48 parts of some small Scale and fit that in 13, then let the Scale [Page 90] rest, and take it over in 12, and apply that to the same Scale where the 48 was taken, sheweth 44 and one third part, and so many inches in length is required to make a foot.

But having taken your 48 on some small Scale, and are to fit it on 13, now if it open your Scale too wide, you may fit it over in two or three times 13, and take it over in so many times 12, as fit 48 in four times 13, that is in 52, and take it over in four times 12, that is, in 48, and it sheweth, 44 and one third, as before, being apply­ed to the same Scale where the 48 was taken.

Again, let a Board be 5 inches, and three eight parts of an inch broad, and it is required to finde how many inches in length make a foot.

Bring five and three eights into eights, makes 43, and 12 into eights make 96, then take 96 and fit it in 43, or in twice 43, there let the Scale rest, [Page 91] and take it over in 12, and also apply it to the same Scale where 96 was ta­ken, and it sheweth 26 and three quarters, and so much in length makes a foot of Board, the breadth being 5 inches, and three eight parts of an inch, which is the thing desi­red.

CHAP. XII. To measure Board that is broader at one end then at the other.

SUppose a Board be broad at one end 20 inches, and at the other 16: now it is required to finde how much in length makes one foot throughout the whole Board.

Adde the breadth at both ends to­gether, [Page 92] and take halfe thereof for a mean breadth, so finde you 18, then is it all one as if your Board were 18 inches, and you would know how much in length makes a foot.

Take 12 and fit it in 18, and take it over in 12, and so much makes a foot.

Let a board be broad at one end ten inches and a quarter, and at the o­ther seven and a halfe, now the desire is to know how much in length makes a foot.

Adde both the numbers together and take halfe, which maketh 8 inch­es and seven eight parts of an inch for the common breadth; then bring 8 inches and seven eights of an inch & 12 inches into eights, and it maketh 71 eights, and 96 eights. Take then 96 in some Scale, and fit that in 71, then let the Scale rest, then take it o­ver in 12, and that apply to the same Scale where the 96 was taken, and it sheweth 16 and a quarter, and so ma­ny [Page 93] inches in length make one foot of Board.

CHAP. XIII. To finde how many square feet any whole Board con­taineth, without finding how much in length makes a foot.

IMagine a Board be 15 foot long, and 16 inches broad, and it is re­quired to finde how many square foot of Board it containeth.

Take the length of 15 on some Scale of equall parts, and fit that in 12 the inches in a foot (alwayes) there let the Scale rest, then take it over in 16 the breadth, and apply it to the same Scale where the length was taken, it [Page 94] sheweth 20, and so many square foot is found to be therein contained.

Let a Board be 17 foot and a quar­ter long, and 16 inches and a halfe broad, and the desire is to know how many foot it containeth.

Take 17 and a quarter the length, and fit it in 12, and take it over in 16 and a half, and that apply to the same Scale whe 17 and a quarter the length was taken, it sheweth 23 and two thirds, and so many foot it contain­eth.

Or you may bring 17 and a quarter into quarters, makes 69, and in like manner 12 into quarters, makes 48, and take it over in 16 and a halfe the breadth, so finde you 23 and two thirds as before.

CHAP. XIV. To measure Board that is broader at the one end then at the other, in the same manner.

SUppose a Board be broad at the one end 18 inches, and at the o­ther end 14, and long 21 foot, I demand how many square foot it con­taineth.

Adde the breadth at both ends to­gether, makes 32 inches, whose halfe is 16 inches for a mean breadth, then proceed as before, take 21 and fit it in 12, and take it over in 16, or fit it in five times 12, and take it over in five times 16, so finde you 28 for the area required.

Again, let a Board be broad at the one end 11 inches and a halfe, and at she other 7 and three quarters, and 15 foot and three quarters long, now the Area is required.

First, adde them both together, and take half, makes 9 five eight parts, for the mean breadth.

Then take 15 three quarters, the length on any Scale, and fit in 12, and take it over in 9 five eights, and that applyed to the same Scale where the length was taken, and it sheweth how many foot it containeth.

Or bring 12, and 9 five eights into eights, make 96 and 77, then fit fifteen three quarters the length, in 96, and take it over in 77, and that sheweth on the same Scale where the 15 three quarters was taken, twelve two thirds, the Area desired.

CHAP. XV. To measure Timber.

SUppose a piece of Timber be 18 inches broad, and deep 16 inches, it is required to finde how much in length doth make a foot.

Take twelve the inches in a foot on any Scale of equal parts, & fit that in the breadth eighteen, and take that over in twelve, alwayes. Again, set that distance in sixteen the depth, and take it over in twelve still, and that ap­ply to the same Scale where the twelve was taken shew six, and so many inches in length make a foot, the thing required.

Again, let a piece of Timber be broad sixteen inches, and deep thir­teen [Page 98] and a halfe, and it is required to finde how much in length make one foot.

As before, fit twelve in sixteen, and take it over in 12 still, & that apply to the same Scale where the twelve was taken, sheweth eight inches, and so many inches in length make one foot.

Again, let a piece of Timber be fifteen three quarters broad, and ele­ven three quarters deep: I demand how much shall make a foot?

Bring fifteen three quarters, and twelve into quarters, makes sixty three, and forty eight, then take twelve on some Scale of equall parts, and fit it in sixty three, and take it over in 48, and that distance fit in eleven one quarter, and take it over in 12: Or as before, bring eleven one quarter, and twelve both into quarters, makes for­ty eight and forty five, then fit it in forty five, and take it over in forty [Page 99] eight, and that applyed to the same Scale where the first twelve was ta­ken, sheweth nine foure fifths, and so many inches in length will make one foot.

If a piece of Timber be seven one quarter broad, and five & a half deep, it is required to finde how much in length shall make a foot.

Bring seven one quarter, and five an a half into quarters, makes twenty nine, eight hundred twenty two, like­wise twelve makes 48, then take twelve on any Scale of equall parts, and fit it on twenty nine, and take it over in forty eight, which distance fit again in twenty two, and take it over in forty eight, and that applyed to the same Scale where the twelve was taken, sheweth forty three one third part, and so many inches in length make a foot, which was required to be done.

CHAP. XVI. To measure Timber that is broader at one end than at the other.

SUppose a piece being broader at the one end than at the other, be given to be measured.

First, take some place neer the big­ger end for a meane part, then take the breadth and depth thereabout, which suppose to be twenty and fif­teen, then proceed as before, so finde you 5 three quarters, and so many inches in length make a foot.

CHAP. XVII. How Perpendicular heights may be found without ei­ther Instrument or Arith­metick.

TAke a trencher, or any simple boards end, of what fashion soever, such as you can get, & draw thereon a line towards one of the sides, as the line AB, and on the point A, raise a perpendicular, as AD, then in the line AB, knock in two pins, one at A, and the other at B, then on the point or pin at A, hang a thrid with a plummet, then lift up this board with the end A, towards [Page 102] the height required, till you bring the two pins into one straight line, with your eye, and the top of the height required, and directly where the thrid falleth, there mark it with a prick of your Compasse, as at E, and draw the line AE, now measure the distance between your standing, and the base of your altitude, which here wee will suppose to be 36 foot, as from F to G, and take 36 from your Scale, & set it down from A to D, from which point D, raise a Perpendicular, to cut the plumbe-line AE in E, so shall DE be the height required, which being applyed unto your Scale, will reach unto 32, and so many foot is the alti­tude GH.

But suppose you cannot come unto the base G, and therefore cannot mea­sure the distance GF, and yet it is re­quired to find the altitude GH, there­fore to perform this, first take your standing at F, as before, and lift up [Page 105] your board, till you bring the two pins A and B into one straight line, between your eye, and the top at H, and directly where the thred falleth,

[Page 106] there make a mark, as at E, and draw the line AE, then measure out so ma­ny foot as you think good, in a straight line towards the base G, and there take your other standing, as here sup­pose 16, this 16 take from your Scale, and place them from A to C, and in the point C hang your thrid and plummet, and lifting up your Board, till you take your sight as before, make a mark directly where your thrid crosseth the former plumb line AE, as at E, from which point E let fall a perpendicular to the line AD, as DE, so shall DE be 32 for the al­titude of GH as before.

Here note, that the altitude thus found, is from the levell of the eye upwards, and therefore the height frō the eye downwards is to be added thereto to make it compleat.

CHAP. XVIII. How to take the altitude or height of a building by a bowl of water.

PLace on the ground a Bowle of water, which done, erect your body straight up, and goe back in a right line from the building, till you espie in the center or middle of the water the very top of the altitude, which done, observe the place of your standing, and measure the height of your eye from the ground, together with the distance from your standing to the water, and the distance from the water to the base or foot of the al­titude, which being all exactly taken, will help you to the altitude required by the Rule of proportion.

Example, Let the altitude required be AB, the Bowle of water placed on the ground at C, then go back­ward from C (your body erected as

streight as may be) till your eye at E espie the top of the altitude AB in the water, which found, observe the place of your standing at D, and mea­sure the altitude from your eye to the [Page 107] ground, which is five foot, then mea­sure the distance from D to C, which is 6 foot, and likewise the distance from C to B which is 80 foot: these three distances being known, work by the Rule of proportion thus.

• As the distance CD, to the altitude ED;
• So is the distance CB, to the altitude AB:

Which will be found to be 66 foot and 8 inches.

CHAP. XIX. How to take the altitude of a Building by a line and plummet the Sun shining.

LEt the Building whose altitude you desire to know be AB, casting his shadow in a right line to C, at C [Page 110] let fall a line and plummet (whose length before you know in feet and inches) observing where the end of that shadow lighteth, which suppose

at D, then measure the length of the shadow of the string, and consequent­ly the shadow of the Building, both which being exactly taken, work thus by the Rule of Proportion.

If CD the shadow of the line and [Page 111] plummet 4 foot 5/11, give EC 7 foot in altitude; what altitude doth 14 foot give, which is the shadow of the altitude required.

Multiply and divide according to the Rule, and you shall finde in your Quotient 22 foot, which is the true altitude of the building required.

CHAP. XX. How to find the altitude of a Building by two sticks of one length joyned in a right angle.

CAuse two sticks to be joyned in a right angle, as is in the figure MN and OP, having at O a hole made wherein to hang a thrid & plummet.

The two sticks being thus prepared, [Page 110] come to the building whose altitude you require (which building let be AB) then apply the end of your crosse staffe (noted with D) to your eye, & hold it up and down till the third and

plummet hang just upon the perpendi­cular, then goe backward or forward till your eye at D looking over E, espy the top of the building at A, which [Page 111] found, marke well the place of your standing, which is at F, and measure the distance from your eye to the ground, which is DF, and set that same distance off from F to C, then measure the distance from C to B, for that is the true height of the building AB.

CHAP. XXI. To finde a Distance by the two sticks joyned square.

THis experiment is grounded upon the fourth proposition of the 6 Booke of Euclid.

Let the distance which you desire to know be AB, set up a staffe at A of four foot long, (or more or lesse at your pleasure) as the staffe AC, at the end of the staffe C place a thrid as CD.

Then hanging the angle of the square on the top of the staffe at C, move it up or down till you see the farthest part of your longitude: the square so remaining, and the staffe not [Page 113] removed, draw the string that is faste­ned at C, close by the side of the square, till it touch the ground at D, then measure how many times the di­stance DA is conteined in the Staffe, for so many times is the Staffe con­teined in the longitude.

Example.

The Staff supposed four foot high placed at A, and the Square being [...]

CHAP. XXII. How to describe a Town or City according to Cho­rographicall proportion, by the helpe of a plain glasse.

TO performe this conclusion, you must resort to some high place in the Town or Coun­trey you would describe, from whence you may behold all the Castles, Ports Harbours, Bays, Gates, Forts, and such other notable places as you in­tend to describe: which place being chosen, provide a plain glasse, which in the midst of the Platforme hang [Page 117] parallel to the Horizon, (in the doing of which you must be very carefull) so that moving up and down the plat­forme, you may in the Center of the Glasse, see all those notable places. The foundation being laid, let us now proceed to the worke; and first of all on your platforme, you must draw a Meridian line, which must passe just under the Glasse, so that if a perpendicular line were let fall from the Center of the Glasse to the plat­forme it might cut the Meridian line at right Angles, and by having this line drawn, you may draw the line of East and West at right Angles to the Meridian; and in like manner, the two and thirty points of the Com­passe, with Circles and Parallels, as is usuall in the projecting of Sea-charts; so that thereby you may know how all the chief places in the Town are situate, and how they bear from you: This done, move Circularly about [Page 118] the Glasse, observing always when you espie any marke in the Center of your Glasse to set up a staffe, writing thereupon the name of the place, whether it be Village, Port, Road, or such like, you shall in the end situate, as it were, the whole Countrey, in due proportion upon your platform, so that measuring the distance of eve­ry staffe set up from the Center of your platforme, and the distance like­wise of every staffe from other, you may by the Rule of Proportion, finde out the distance of every Town, Vil­lage, Fort, Haven, and the like, from your platform; and also the distance between any two places there descri­bed. This Experiment is marvei­lous pleasant to practise, and most exactly serving for the description of a plaid Champion Countrey, which when you have thus traced out upon the platform, you may, by the help of Scale and Compasses, project in pa­per [Page 119] or parchment with a Scale of Leagues, Miles, Furlongs, Pa­ces, or other measures, as liketh you best.