MESOLABIUM ARCHITECTONICUM THAT IS, A most rare, and singular Instrument, for the easy, speedy, and most certain measuring of Plains and Solids by the foot: Necessary to be known of all men whatsoever, who would not in this case be notably defrauded: Invented long since by Mr. Thomas Bedwell Esquire: And now published, and the Use thereof declared by Wilhelm Bedwell, his nephew, Vicar of Tottenham. LONDON, Printed by J. N. for William Garet 1631 TO THE ILLUSTRIOUS, Righthonourable, Right-worshipful, and dear beloved, the Nobility, Gentry, and Commons of Great Britain, and Ireland. GOd, saith the wise man, hath ordered all things by measure, number, and weight. And man, the image of God, aught, as the Philosophers teach, to order all his life according to the same directions. And yet who knoweth not, 〈◊〉 little they are of all men regarded! To pass by the general, and to come to that which concerneth our commerce, What smatterer in the Mathematics is he, who knoweth not, what neglect or ignorance there is, even in those artists, whom all men, the Rich aswell as the Poor, do, and must daily trust, in matters of measuring! I accuse no man of wilful fraud or malice. But this I say There is no man whatsoever, that is not some piece of a scholar, that can measure 〈◊〉 truly: And those who are most 〈◊〉 in both, cannot do it either speedily, or readily. All which, Illustrious, Righthonourable, Right-worshipful, and Dear beloved, I promise in this short treatise, by the ordinary Instrument, in this 〈◊〉 used, to teach the meanest of understanding, though wholly unlearned, to do, with that speed, facility, and certainty, that may not be bettered. This as a prodromus, begun and ended, in the midst of many and great troubles, I thought good to premise and send out, before a larger discourse of the Fabric, and more ample Use thereof, which, God willing, shall follow, so soon as Figures and Diagrams may conveniently be cut, for that purpose, with all possible speed: In the mean time the Author, wholly devoted to his Country's service, resteth Your H. H. H. in all observancy, Wilhelm Bedwell. MESOLABIUM ARCHITECTONICUM. CHAP. 1. Of the Mesolabe: And of the use of it in general. 1 To measure by this Rule, is by two known lines, to find out the third unknown. THe Instrument whose use at this time we intent to 〈◊〉, is no other, in respect of matter and form, in general, but the Carpentars' rule, by them used in the measuring of Timber, and Board by the Foot square: For it is a flat Ruler, or oblong parallelogram, of two foot, or a foot & half long: Two inches and an half, or there about, broad: And of such convenient thickness as shall at cuery man's discretion be thought most fit. Again, as theirs, so this on the one side, containeth a Scale of equal divisions, First of 〈◊〉, Halfe-ynches, Quarters, Half quarters, and so forth: Then again, on the 〈◊〉 side, you havean Ynche divided into Seven, Eleven, Thirteen, 〈◊〉, 〈◊〉, and Three and twenty, and such other 〈◊〉 parts, as 〈◊〉 man for his 〈◊〉 〈◊〉 shall think 〈◊〉 〈◊〉, and the 〈◊〉 hand shall be able to perform. More over, on the other side, as on theirs also, you have a Scale of unequal divisions, serving for the measuring of Board and Timber: But after a far different manner For their divisions are only marks or small strokes, in one of the limbs of that side, determining from the Fore-end of the ruler in inches, and parts of inches, the Square measure of solids or Timber. Whereas this of ours consisteth of two for●es of strait lines, the one Bevelling o● sla●ting, drawn as ●●e from side to side: The other parallel that is equidistant one from another running along the Ru●ar, from the one end toward the other: And therefore cutting those former, and dividing them into unequal portions, whereby not only their said Quadrate or square measure is performed: But also all other whatsoever, and that with great facility, speed, and certainty. Lastly here, as also there, you must make a distinction between end, and end; For that end we call the For-end of the ruler, from whence the divisions of it into inches, on both sides are begun to be reckoned: And that the Backer-end where they do end and determine: Or, contrariwise, the For-end is that from whence the numbers ascribed to the Bevelling linnes are less and less. But the distances between them are greater and greater. Thus much of the Ruler, and the Parts thereof. Mensura, innuit Aristoteles, in quolibet mensurabili genere, est quippiam minimum: A measure, as Aristotle seemeth to intimate, is some small portion in every thing that is to be measured: And it is commonly termed of the Geometricians Famosa mensura: Aknowne, or set measure generally agreed upon amongst all men: As in measuring by hand-breadths, feet, and passes, one hand breadth, on foot, one pass. And in deed it is an old saying of Protagoras, as Aristotle recordeth, That man is the measure of all things. And true it is, That Vi●●●●us, and Hero the mechanic or engineer, do show, That generally all measures are taken from the parts of Man's body, as a Finger, an Inch (Pollex) an Hand, or Hands breadth, a Span, a Foot, a Cubite, a Pass, an Elne, a 〈◊〉. But who knoweth not, What great difference there is between man & man? And not only between men of divers Country's and climates: But e'en between those of one and the same province; Nay of one and the same family, children of the same parents? And, the limbs of men being proportional to their bodies, what 〈◊〉 must there needs be, between the measures taken from them? And in deed heerupon it came to pass, That the Measures, not only of divers Nations: But e'en of one and the same, are, and always have been much different, as doth manifestly appear by the diligent comparisons made of them by divers and sundry learned men, and especially by that hopeful Willebrordus Snellius, as we 〈◊〉, Godwilling, shortly teach in Ramus' Geometry, which we purpose to set out in English, for the benesite of such of our Country men, as delight in these study's, yet are ignorant of those languages where in they are written. This difference was in this our kingdom complained of in all ages: For from hence arose many grievous 〈◊〉 and suits in the Law, which our worthy Kings, and state in their Parliaments, in all ages have laboured to appease, by reducing all to an uniformity: For thus we find in our Statutes: It is ordained, That 3 grains of Barley, dry and round, do make an Inch: Twelve ynohes do make a Foot: Three foot do make a Yard: Five yards and half do make a 〈◊〉. And 40 〈◊〉 in length, and 4 in Breadth do make an Acre. 33 of Edward the first, De 〈◊〉 mensurandis Item, De Compositione 〈◊〉 et Perticarum. Again in a 〈◊〉 held in the 25 〈◊〉 of Queen Elizabeth, you have an Act, thus entitled: An Act for the restrainte of New-buildings, etc. in & ne'er the cities of London & Westminster Be it enacted by the authority afor'said, That a Mile shall be taken & reckoned in this manner, & no otherwise: That is to say, a Mile to contain 8 Furlongs. And every Furlong to contain 40 lugges or poles: And every Lugge or Pole, to contain 〈◊〉 foot and an half. Although this same our Rule may be fitted for sundry other sorts of measures: Yet we have here nothing to do, But with the Foot, and his parts, which are Ynches, Halfe-ynches, Quarters, Half-quarters, and such other sensible parts of the same. 2 Things to be measured by this Rule, are magnitudes. 3 A magnitude is a continual quantity. Amagnitude, or a bigness is that which hath one, or more dimensions: Now dimensions are in number three, to weet Length, Breadth, and Thickness. 4 A magnitude is of one 〈◊〉, or many. 5 The measure is of the same nature with the thing to be measured. 6 A magnitude of one dimension is called a Line. Aline, is a magnitude of length only. Or, Aline is a magnitude only long. Such are ways, or distances 〈◊〉 place and place. Such a magnitude, saith Proclus out of Apollonius, is conceived in the measuring of journeys. And by the difference of a lightsome place, from a darksome. Such are Lenghts, heights, Depths, and Breadths. Therefore here 7 The measure used is a line. Here therefore there is no further skill required in the measurer than a due application of the measure given: And therefore here in this case there is not any use of this our Instrument. CHAP. II. Of the measuring of Plains by the foot square. 1 A magnitude of many dimensions, is of two or three: That is called a Surface: This a Solid. 2 If a dimension given, be either greater, or lesser, than any of the numbers upon the ruler, you must take some lesser, or greater, which is proportional unto it. 3 A surface is a magnitude long and broad. That is, a surface is a magnitude which hath two dimensions, to weet Length and Breadth. Such magnitudes, saith Apollonius, are the shadows upon the ground, which overspread the fields far and wide, but do not enter into, or pierce the earth: Neither have they any thickness at all. The Greek word Epiphania, is here more significant. For this word intimateth no more but, The outward appearance of any thing. For of a magnitude nothing is to be seen but the surface. Such are bourds esteemed to be by the Carpentars': Wainscotte, by the joiners: Glass, by the Glasiers: Cloth, both linen & Woollen, by the Drapers: Land, Medowe, & Wood, by the Surueighers: For in the measuring of these, there is only Breadth & Length considered, with out any respect at all had to the Thickness. Therefore 2 Here the measure is a Surface. Surfaces, according to their divers natures, are measured with divers and sundry kinds of measures: Wood, Land, & Meadow, are measured by the Rod or Perch: Cloth, Painting, Paving, & Wainscotte, by the Yard: Board and Stone, by the Foot. Although this our Instrument may be fitted to all these, or any other like measure, Yet we at this time intent to meddle with no other but the last, to weet With the Footesquare. 4 A surface is either Plain or Uneven. 5 A Plain surface is a surface, which lieth equally between his bounds. A surface, the learned knowe is geometrically made of Lines: Therefore as lines are either strait or Crooked: So from hence are all surfaces Strait or Crooked: Or, to speak more properly, e'en or uneven, Plain or Rugged: Yea & by a strait line are surfaces tried, whether they be e'en, or uneven. For if a rightline applied to a surface every way, do touch it in all places, it is e'en: Otherwise, it is uneven. 9 Plains, as we said, are measured by the Foot square, That is the quadrate of 12 inches. A 〈◊〉 of plain or flat measure is the quadrate of 12 inches, or that which is equal unto it. That is, it 〈◊〉: 44 〈◊〉 Ynches: For 12 times 12, are 144. Having therefore a plain given of 12 inches broad, there is no question but 12 inches of that breadth shall make a Foot. But if the breadth given be greater or less than 12, there is a question, What length, with the breadth given, shall make a plain 〈◊〉 to the square 144. Here 7 Of the two lines ' given, the one is the breadth assigned, the other is always the bevelling line 12. Here again it must be remembered, That only those plains are to be measured which are Rightangled parallelogramms, Or to speak in their own Language, which are comprehended of a, Base, and Heigh which are rational between 〈◊〉: Ramus 9 e FOUR Those plains therefore which are not such, must be reduced unto these kind of figures. 1 An example or two shall make all plain. A board of 16 inches broad and 18 inches long, (And so a stock of 13 bourds) is to be measured. Here I find 16, the line answering to the Breadth, to 〈◊〉 the beveller 12, at 9 inches from the fore-end of the ruler. Therefore I say every 9 inches of that length shall make a Foot of board: Or which is all one, shall be 〈◊〉 to 144, the square of 12 inches. Now 9 inches I find to be contained in 18 foot, the Length, 24 times: Therefore I say, The board assigned doth contain 14 foot of board. Lastly, there being in the stock 13 such bourds, I say the whole stock doth contain 312 foot of board. TWO A Table of 36 inches broad, and 28 foot long, is to be measured. Here 36 is greater than any of the parallels found upon the ruler: Therefore by the 2 e of this, I take ●8 the half of it, which I find to meet with 22, the bevelling line, 〈◊〉 8 inches from the for'end of the ruler: Therefore every 8 inches of length, of the breadth 18, shall contain a root of board: But the breadth given is 36 inches: That is twice 18: Therefore every 8 inches in length, of that Table shall be 2 foot of board. Now again I find 8 inches, in 28 foot 42 times: Therefore the Table containeth twice so many foot: That is 84 foot of board. III A pane of Glass, 7 inches broad, is to be measured. Here 7 is lesser than any of the parallels: Therefore by the 2 e of this, I take 14, the double thereof: Which I observe to meet with 12, at 10 inches and 2 seaventh parts of an inch from the fore-end: Therefore every 10 inches and 2 seaventh parts of an inch, of 14 inches breadth, shall be a foot of Glass: But the breadth given is but 7 inches: Therefore every 10 inches, and 2 seaventh parts of an inch shall be but half a foot of glass. Of the measuring of Triangles, and all other Rightlined plains. 8 A triangle is nothing else but the half of a quadrangle, or parallelogramme: And if it have one right angle, it is the half of a rightangled parallelgramme. Therefore 9 It is to be measured as the Rightangled-parallelogramme, only conceive that the number found, shall be the double of that which is sought. Here therefore it must be conceived, That of the two sides encluding the Rightangle, the one is to be understood to, be the Breadth, the other the Length. I Suppose a Rightangled-triangle, whose sides including the Rightangle, are 18, and 24, are to be measured. Here I take 18 for the Height, or Breadth of the parallelogramme which also I find to meet with the bevelling line 12, precisly at 6 inches from the fore end of the Ruler: Again 6, the said line found, I find just 4 times in 24 the Length given: Therefore I aver the Triangle given to contain the half of 4 foot, that is 2 foot of board. 20 If the triangle given be not rightangled, then is it by a perpendicular, let fall within the triangle, from one of the corners unto the base, to be reduced unto two rightangled triangles. How this is to be done, Euclid teacheth at the 11 & 12 propositions of his ay, book; And P. Ramus, at the 9 & 10 elements of his V. book of Geometry. It is also to be done by the squire. Or by a triangled level, and otherwise. TWO An Obtuseangled triangle, whose three sides are 25, 40, and 42, is to be measured. here by one of those above named ways, I find the perpendicular or plumbline, falling from the greater corner, unto the opposite line, to be ●8. And 24 I find upon the Ruler to meet with the line of 12, at 6 inches from the fore-end of the same: Again 6 I find in 42 seven times: Therefore the Triangle given doth contain half so many foot, That is 3 foot and an half of board. 11 From hence it is manifest how any Rhombus, Rhomboides, Trapezium, or irregular rightlined multangles are to be measured. To weet, that they are to be measured by parts, or by the particular triangles, which every such figure doth contain. Examples you may have in the XIIII book of Ramus' Geometry, or in any others, which have written of Geometry. Of the measuring of any ordinate multangle figured. 12 Ordinate multangled plains are measured by their half Perimeter, and the plumbline from the centre, unto the midst of any one side. These sorts of plains may be measured, as the former were, by dividing them into their several Triangles. But this last is far shorter: And therefore to be embraced & rather to be used in practice. Here the half of the perimeter, or bout-line, 〈◊〉 to the Length in a 〈◊〉 〈◊〉: And the plumbline here, is in stead of the Height or Breadth there. 1 An 〈◊〉 Pentangle, whose sides are 24 inches a piece; And the 〈◊〉 from the centre, to the midst of any one of the sides 16, is to be measured. Here 10 the Plumbline or Height, doth, upon the ruler, meet with the 〈◊〉 line 12, at 9 inches from the oft named end: And 9 is contained in 60, the half of the perimeter, 6 times and two thirds: Therefore the Pentangle given containeth 6 foot, and two third parts of a foot of Board. TWO A Sexangled ordinate figure, whose sides are 12 inches broad a piece, is to be measured. Here the Plumbline from the centre to the midst of any one side, is 10 inches, and 8 one and twentyths of an inch: The double of 10 (that is 20.) and 〈◊〉 one & twenty parts of one inch, I observe to meet with the beveller 12, about 7 inches, & one quarter of an inch, from the fore end of the ruler. Which 7 and a quarter, is contained in 44 six time, and two twenty nineth parts. Therefore I say the Sexangled figure given doth contain 6 foot of board, and some small quantity more. The Circle, or Circular form is in like manner measured: For 13 The Circle is measured by the Ray, and the half of the perimeter. For, sauth the Geometrician; Planus è radio & peripheriae 〈◊〉 est area 〈◊〉. The plain of the ray, and half of the circumference is the content of the circle. A Round table, whose diameter is 4 foot, and 8 inches, (or 56 inches) is to be measured. The half of 59 is 28: And the half of the circumference is 88 Now ●8 being geater than any of the 〈◊〉, I take 14 the half thereof Wh ch I find to meet with the bevelling line 12, at 10 inches, and a quarter, from the for'end of the ruler: Therefore I say every 10 inches, and a quarter of an inch of that Table shall be 2 foot of board. And because 88 doth contain 10 and 1 quarter, 8 times, and 20 forty ones; Therefore I say, the whole doth contain 16 foot of of board, and 144 inches. CHAP. III. Of the measuring of Bodies or Solids by the Foot. 1 A Body is a magnitude of three dimensions. A Body or Solid is a magnitude which hath Length, Breadth, and Thickness. 2 Here the measure is also a body, to weet the Cube of 12 that is 1728. This is our opinion: Yet if any shall think it a paradox, or shall gain say it, or maintain the contrary, we will not contend. And 3 Of the three dimensions, two are given, the third is sought. 4 Bodies are of divers sorts: But we will at this time meddle only with such as are comprehended of parallelogrammes, or with Cylinders. True it is, that this our instrument may be fitted, and applied to the measuring of many other sorts of Solid bodies: But because we see no great use of it in the measuring of any other then of these two sorts: Therefore we will declare the use of it, in the measuring of these two only. Of these the first is the Parallelepipedum, which is a plain Solid, whose opposite sides are parallelogramme. I A rightangled parallelepipedum (or a squared 〈◊〉 log) of 12 Ynches 〈◊〉, 18 broad, and 16 foot long, is to be measured. Here the 〈◊〉 and Breadth are given: The Length is sought. These I find upon the ruler to meet at 8 inches from the 〈◊〉 named fore-end: Therefore I Say, Every 8 inches of that Log in 〈◊〉 shall make a 〈◊〉 〈◊〉 of timber. And because I find 8 Ynches, in 16 foot, 24 times: Therefore I say in the Tymbersticke given, there is 24 foot of solid measure. TWO A squared stone of 14 Ynches thick, five 〈◊〉 (or 60 inches) broad, and 10 foot long, is to be measured. Here 60 is greater than any of the parallels upon the ruler: Therefore I take 12 the 5th part of it: And I observe 12 and 14, to meet at 10 inches, and 2 seaunth parts of an inch, from the Fore-end of the ruler. Therefore I say, That every 10 inches, and 2 seaunth parts of an inch in length of that stone shall be 5 foot of solid measure. And because that 10 foot containeth 10 inches, and 2 seaunth parts of an 〈◊〉, 11 times and 5 seau'nty twoos: Therefore I say the whole stone containeth 58 foot, and one third part of a Foot of solid measure. III A rightangled Prisma, both whose sides, 〈◊〉 I mean, containing the rightangle, are 18 inches 〈◊〉 the whole being in length 16 foot, is to be 〈◊〉 〈◊〉 understand that, as before was showed, as a Triangle was but the half of a quadrangle: So a Prisma is nought but the half of a Parallelepipedum, sawn longways from 〈◊〉 to corner though the midst: And hence in 〈◊〉 it hath the name: This known I enter with the numbers given, and I find 18 to meet with 18, at 5 inches and one third part of an inch from the oft named end of the ruler: Therefore I say, That 〈◊〉 5 inches, and 〈◊〉 third part of an inch in length of that stick shall be but half a foot of solid measure. Now because 5 inches, and 1 third of an inch is contained in 16 foot, 67 times and 14 sixteen parts, that is almost 68 times: The fore I say, The 〈◊〉 given doth contain 〈◊〉 68 half foot's, or 34 foot of 〈◊〉 measure. IIII A sispaned solid, all whole sides are 6 inches broad a 〈◊〉 and 16 foot long, is to be measured. Here the two lines given are, as above was taught, the Plumblin: from the centre, unto the midst of any one of the sides: And the half of the compass; That, as before was taught, is 5 inches, and 2 〈◊〉 parts of an inch: This is, as you see 18. Now 5 and 2 eleu'nths doth meet with 18, at 19 inches and 〈◊〉 fifth part of an inch from the fore-end: Therefore I say, That every 19 〈◊〉, and one 〈◊〉 part of an inch, shall be a 〈◊〉 of solid measure. Lastly, because 16 inches, and 1 fifth part is 〈◊〉 in 16 foot, 10 times, and 2 fifteen pates, I say that the timber stick given doth contain 10 foot of solid measure, and some small quantity more. Lastly a Round column, or Cylinder, of 44 inches about, & 12 foot long, is to be measured. Here according to that above taught, the two lines given are, The half diameter, & the half circumference: This is 22: That 7. Now these two do meet upon the ruler at 11 inches, and 17 seventy two parts, of an inch, from the fore-end there of; Therefore the stick containeth about 13 foot of timber or solid measure. FINIS. AN APPENDIX TO THE MESOLABIUM. TWo things, for the further illustration of the Instrument, we have thought good here to annex unto the former. The one is a collation of this manner of measuring, with 〈◊〉 commonly taught and practised. 〈◊〉 is of the Measuring of Land by the 〈◊〉 A foot of solid measure, as all do generally 〈◊〉 Cube of 12 inches; that is, a square, or 〈◊〉 〈◊〉, all whose dimensions, to wit, Thickness, Breadth, and Length are equal: And the content, in numbers, is found by a continual 〈◊〉 of 12, 12 and 12, thus: 12 〈◊〉 12, are 〈◊〉: and 12 times 〈◊〉, are 〈◊〉. Therefore a foot of solid measure doth contain 1728 several cubes of an 〈◊〉, thickness, breadth, and height. This is 〈◊〉 it were the standard, whereby this kind of measuring is to be examined. All artificers generally do measure by a Table of square numbers. And therefore if the body given to be measured, be square, that is, if the thickness and breadth be 〈◊〉, they 〈◊〉 give the just content. But where these two dimensions do differ, there by their rules they do it not without some error. For to bring it to the use of their Table, they must first make the thickness and breadth equal; which they do, by taking the excess from the greater, and adding of it to the lesser, or, which is all one, by girding of the body about, and by taking of the quarter of the compass. An example or two will make all plain. Suppose a body, given to be measured, were 10 inches, thick, and 14 broad. Here they take 2 from 14, and add it unto 10, and so do make all the sides equal: Or by girding of it they find the compass to be 48. And the quarter of 48 to be 12. And their table for the square of 12 inches, doth give 12 inches for the length required to make a foot of solid measure. If this be 〈◊〉, than 10, 14, and 12, continually multiplied between themselves, shall be equal to 12, 12, & 12, continually multiplied between themselves. But 10, 14, and 12, do give for the product 1680. And 12, 12, and 12, as in former we saw, gave 1728. 〈◊〉 the difference between 〈◊〉, and 1680, is 48. Therefore the loss in every foot of 〈◊〉 body, by their measure, is 48 inches. 〈◊〉 upon our ruler, you see 14 the parallel, to meet 〈◊〉 〈◊〉 bevelling line at 12 inches, and 12/35 of an inch 〈◊〉 end of the ruler: And by multiplication you 〈◊〉 10, 14, and 12 12/35 do make 1728: Therefore our 〈◊〉 is true. If the body given be 8 inches thick, and 16 broad, they likewise take 4 from 16, and do add it unto 8, and so, as afore, do 〈◊〉 it to be equal to 12 inches square. Which, if it be true, than the product of 8, 16, and 12, shall be equal to the product of 12, 12, and 12. But the product by the continual multiplication of 8, 16, and 12, is but 1536: And the product of 12, 12, and 12, is 1728: and the difference between 1728, and 1536, is 192: Therefore by that their measure there is lost in every foot 192 inches. Now upon the 〈◊〉 we find 8 and 16 to cross one another at 13 ½: And 8, 16, and 13 ½ continually multiplied one by another, do give the product 1728: Therefore this kind of measuring by the ruler is exact. If it were 6 inches thick, & 18 inches broad, by the same reason, every 12 inches in length should make a foot of solid measure. For the sum of all the sides added together, is 48: And the quarter of 48 is 12: And their Table for the square of 12, doth assign 12 inches for the length. 〈◊〉 6, 18, and 12, multiplied continually do make but 1296, which differeth from 1728 by 432. Therefore by this their measure, in every foot of solid measure, there is lost 432 solid inches. That is just one quarter of a foot. Upon this our instrument 6, and 18, are observed to meet at 16 inches from the said fore end: and therefore it alloweth for a foot 16 in length. And 6, 18, and 16 continually multiplied do make 〈◊〉: therefore our rule is true. Of the measuring of Plains, or Land, Meadows, and Woods by the Acre. ALthough this instrument, as the title specifieth, be fitted only for the measuring of Plains and 〈◊〉 by the foot: Yet, as before is mentioned, it may easily be applied to other like sort of measuring. Now, among others, there being none of more frequent 〈◊〉 amongst us, than the Rod or Perch for the measuring of Land, Meadow, and Wood by the Acre: And this being either not easy to be done by the unlearned; or not speedily to be performed by any, I have thought it not amiss, for the further declaration of the use & excellency of this invention, and for the benefit of others, to add unto the former, something of this also. An Acre of Land is, as before we heard, an oblong parallelogramme, whose breadth is 4 Poles, and length 40. Therefore an Acre containeth 160 square Rods, of what figure or form soever it be. For 4 times 40, are 160. This here in this case is, as 144 was in Board, and 1728 was in solid measure, as it were the Standard, whereby this kind of measure is to be examined. Land, Meadow, or Wood, is to be measured by this instrument, in all respects, as Board or Glass was measured. Only two things are here first to be known: The first is, That as there the bevelling line of 12, was always given for the breadth, as appropriate to that kind of measure: So here another, peculiar to this manner of measure, is in like sort to be drawn overth wart the parallel lines from 6 〈◊〉 of an inch from the fore-end of the ruler, in the parallel 24, unto 13 and ⅓ in the parallel 12. The second is, That 〈◊〉 there lines and spaces did answer, and were denominated of inches and parts of inches; so here the same lines and spaces must be supposed to signify Rods and Perches, and parts of the same. An example or two will make all plain and manifest. 1 Suppose a right angled square field of 16 Pole broad, and 30 in length were to be measured: Here I find 16, to meet with the line of Land measure at 10 inches from the fore end of the ruler: Therefore I say, that every 10 rod in length, of the breadth of 16 rods shall make an 〈◊〉 of Land. And again, because the said 10 is found in 30 the length given 3 times. Therefore I say, the field assigned doth contain 3 Acres. 2 A square Meadow rightangled of 40 Poses in breadth, and 60 in length, 〈◊〉 to be measured. Here 40, the lesser number of the two given, is greater than any upon the ruler: Theresore I take 20, the half of 40: And I find 20 to meet with the line of land measure, at 8 inches from the said fore-end of the ruler. Therefore 〈◊〉 I say. That every 7 Pole in length of the breadth of 40 Pole, shall 〈◊〉 2 Acres of Meadow. Again because 8 is contained in 60 seven times and ½: therefore I aver, that the Meadow assigned to be measured, doth contain 15 acres. 3 Admit that a Wood to be measured were 160 rods square: that is, that every side of the same were 160 Poles in length. Here 160 is far greater than any number upon the ruler: Therefore I take 16, the tenth part of 160: which I 〈◊〉 to cross the line of land measure at 10 inches from the fore end of the ruler: wherefore first I affirm, that every 10 pole of the breadth of 160 pole, shall contain 10 Acres of Wood: Again, because 160 doth 10 sixteen times, I say, that the Wood doth contain 160 Acres. Or, which is all one, that every Pole in breadth of that length, doth contain an Acre. But some man may object and say, this is not a matter worth the learning, or of so many words, seeing that it is well known, that there are many men wholly unlearned, yea, and some of no extraordinary parts of capacity or understanding, which can measure Land, Meadow, or Woods, so that they be square, or of any ordinary form. I confess I have known diverse such. And yet is not this our labour in 〈◊〉 or altogether unprofitable: For first, what they do with much study or contention of mind, and are long in doing of it, we teach to do with great facility and speed. For they, although the field be a rightangled parallelogramme, must first measure at the least two sides comprehending one or other of the rightangles. And then multiply these two sides the one by the other: And lastly, the product found, they must divide by 160: And so by the quotient now found, answer the question propounded. We save a great deal of this labour. For we only measure the breadth of the field; then we seek upon the ruler, how much in length of this breadth doth make an Acre. Lastly, we apply this last number found unto the whole length, and so witho ut either multiplication or division, do speedily answer the demand. Another thing there is wherein this invention doth go far beyond the reach of the unlearned, which is thus. These 〈◊〉 can sooner measure and cast up the whole content of the field, than they can set you 〈◊〉 one, two, or three Acres of the same. Here it is all one to give the content of the whole, or any part, or parts of the same. And that which to those is most hard, here to us is most easy: I mean to set out any one or more Acres of the 〈◊〉; and that on which side or end of the field you shall think good.