The LINE of PROPORTION or NUMBERS, Commonly called Gunters Line, Made Easie. By the which may be Measured all manner of Superficies and Solids, as Board, Glass, Pavement, Timber, Stone, etc. ALSO, How to perform the same by a Line of Equal Parts, drawn from the Centre of a Two-Foot-Rule. Whereunto is added, The Use of the Line of Proportion Improved: Whereby all manner of Superficies and Solids, may both exactly and speedily be measured, without the help of Pen or Compasses, by Inspection, looking only upon the Ruler. By WILLIAM LEYBOURN. London, Printed by J. S. for G. Sawbridge, at his House on Clerkenwell-green. 1667. LICENCED, Nou. 9 1666. Roger L' Estrange. To the Right Honourable Sir WILLIAM BOLTON Knight, Lord Major of the City of London. And the Right Worshipful the Aldermen of the same City. As also to JOHN AUSTEN and THOMAS NEVILE Esquires. Commoners, Appoined by a Committee of Common-Council to direct the Admeasurement of the Ruins by the late Fire there. WILLIAM LEYBOURN, One of those Employed by Order in the Survey of those Ruins, Humbly presents (with the best of his Services) this Manual. Necessary for all Builders, and those that shall Employ them. TO THE READER. THE Line of Proportion or Numbers, commonly called (by Artificers) Gunter's Line, hath been discoursed of by several persons, and variously applied to divers uses; for when Mr. Gunter had brought it from the Tables to a Line, and written some Uses thereof, Mr. Wingate added divers Lines of several lengths, thereby to Extract the Square or Cube Roots, without doubling or trebling the distaence of the Compasses: After him Mr. Milbourn a Yorkshire Gentleman disposed it in a Serpentine or Spiral Line, thereby enlarging the divisions of the Line. Again, Mr. Seth Partrîdge contrived two Rulers, to slide one by the side of the other, having upon them two Lines of one length, which exactly and readily performeth all Operations wrought thereby, very exactly and speedily, without the help of Compasses. Now whatsoever all the forementioned Contrivances will perform, I have here showed in this Manual, and so ordered the Line that it will perform the work without Compasses, by Inspection, looking only upon the Ruler. And thereby may be measured (let the Line be of what length soever) not only Board, Glass, Timber, and Stone, but also all manner of Hang, Pavements, Wainscots', Plastering, Tyling, Brickwork, etc. To all which Uses I have particularly applied it, as will appear by several Instances in all the forementioned particulars; and the rather, because this Treatise may be beneficial and useful as well to Gentlemen and others (who at this time may have more than ordinary occasion to make use thereof, in the Re-building of the Renowned City of London,) as to Artificers themselves, for whose sakes chief it was intended. Vale. ADVERTISEMENT. IF any Gentleman studious in the Mathematics, have or shall have occasion for Instruments thereunto belonging, or Books to show the use of them, they may be furnished with all sorts, useful both for Sea or Land. either in Silver, Brass, or Wood, by Walter Hayes at the Cross-Daggers in moorfield's, next door to the Popes-head Tavern; where they may have all sorts of Maps, Globes, Sea-plats, and Mathematical Paper, Carpenter's Rules; Post and Pocket-dials' for any Latitude, Steel Letters, Figures, Signs, Planets, or Aspects, at reasonable Rates. How to Measure Board and Timber BY THE Carpenters PLAIN RULE. ALL manner of Superficial and Solid Measures may be measured the most absolute and artificial ways that are yet known, by the Precepts and Examples in this Book delivered: But although, every Capacity may not attain to the knowledge and understanding thereof, I thought good here to insert the Use of that Rule which is commonly made and sold, and which every Artificer continually carries about him. It's Description. I. Of the FORE-SIDE. It consisteth of two flat sides, one of which towards either edge thereof, is divided into 24 equal parts called Inches, and numbered by 1, 2, 3, 4, and so forth to 24 at the end thereof: Every one of the parts or Inches is again divided into two equal parts by Lines about half the length of the other, representing half Inches; and every of these half Inches is divided into two other equal parts, called quarters of Inches; and each of those again into two other equal parts, called half quarters of Inches: So that each Inch is divided into eight equal parts, representing Inches, Halves, Quarters, and Half-quarters. Both the edges on the one side of the Rule are thus divided and numbered, only where 24 stands at one end of the Line on one edge, there 1 stands on the other edge; so that which end of the Rule soever you measure with, you may count your number of Inches and parts right, without turning of the Rule. II. Of the BACKSIDE. On the other side of the Rule you have two other Lines or Scales drawn near to the edges of the same side; one is called the Line of Board-Measure, the other the Line of Timber-Measure. At the beginning of either of these Lines you have a little Table of Figures, the one for Board, the other for Timber or Stone. The line or Scale of Board-Measure gins at 6 towards your left hand, and so goes on to 36, ending just 4 Inches short of the other end of the Rule; but sometimes this Line is continued up to 100, but nor often, and then it goes nearer to the end of the Rule, namely, to within an Inch and an half of the end thereof. At the beginning of this Line there is a small Table from 1, to 6 Inches, which shows (in Figures) the quantity of the length of a Foot of any Board from one Inch broad to 6 Inches broad, and then the divisions supply the greater breadths. On the other edge, on the same side you have the Line or Scale of Timber-measure. This Scale gins at 8 and an half, and so goes on, (by divisions) to 36, towards the other end of the Ruler, namely, 36 ending within almost an Inch and half of the Rules end. To this Scale also there belongeth a Table, which standeth at the beginning of the Line, and goes from 1 Inch to 8 Inches, and gives the quantity of the length of a Foot of any Timber under 8 Inches square in Figures, as the other did for Board from 1 to 6: And these are called the Tables of Under-measure. The Table for UNDER-BOARD-MEASURE. 1 2 3 4 5 6 12 6 4 3 2 2 0 0 0 0 4 0 The Table for UNDER-TIMBER-MEASURE. 1 2 3 4 5 6 7 8 144 36 16 9 5 4 2 2 0 0 0 0 9 0 11 3 Thus much for the Description of the Lines upon the Carpenters plain Rule: Now for Their Use. I. Of the Fore-side, or Side of Inches. This side is only to measure the length and breadth of any thing to be measured, in Inches and parts; the manner of doing whereof is natural to every man; for, taking the Rule in the left hand, apply it to the thing to be measured, so have you the length, breadth, or thickness of the thing desired. But, II. Of the Backside. and, I. Of the Line of Board-measure. PROB. 1. The breadth of any Board being given, to find how much thereof in length will make a Foot square. Look for the number of Inches that your Board (or Glass) is broad, in the Line of Board-measure; and the number of Inches and parts of an Inch which stand against that on the other side of your Rule, is the quantity of Inches that will make a Foot square of that Board, or Glass, or what other thing soever it be to be measured. Example 1. There is a Board or Plank that is 9 Inches broad, how much of that in length will make a Foot square? Look for 9 Inches upon the Line of Board-measure (which you shall find at the Figure 9 upon the same Line), and just against that, on the other side of your Rule you shall find 16 Inches, which shows that every 16 Inches of that piece in length, will make a Foot square. Example 2. A Pain of Glass is 22 Inches broad, how much thereof in length will make a Foot square? Look for 22 Inches in the Line of Board-measure, and right against it (on the other side of your Rule) you shall find 6 Inches and almost an half; and so much in length of that breadth will make a Foot square. Example 3. If any plain Superficies be 30 Inches broad, how much thereof in length will make a Foot square? Seek for 30 Inches in the Line of Board-measure, and right against it, on the other side of the Rule, you shall find 4 Inches and ⅘, that is, 4 Inches and 4 fifth parts of an Inch. Example 4. If a Board be 9 Inches and an half broad, how much thereof in length will make a Poot square? Seek 9 Inches and an half in the Line of Board-measure, and against that on the other side of the Rule you shall find 15 Inches and about one sixth part of an Inch, to make a Foot square. ¶ NOTE. All these Examples might be performed otherwise by the Line; for if you take the Rule in your left hand, and apply the end thereof noted with 36 to the end of the Superficies, the other edge of the Superficies will show how many Inches, Halves, and Quarters will make a Foot square. This needs no Example. PROB. 2. The length and breadth of a Superficies being given, to find how many Square Feet are therein contained? By any of the ways before taught) find how much of the breadth given will make a Foot square; then run that length from one of the ends of the Superficies as often as you can, and so many square feet is there in that Superficies Example. A Board is 9 Inches ●road and 15 Poot long, how many square Feet are therein contained? By the first Example you find that 〈◊〉 9 Inches broad, 16 Inches in length to make a Foot; wherefore take 16 Inches of your Rule, and run that length along the Board from one end thereof, and you shall find that length to be contained in the Board of 15 Foot long, 11 times and 4 Inches over, which is ¼ of a Foot; so that the Board of 15 Foot long and 9 Inches broad, contains 11 Foot and one quarter: The like of any other. II. Of the Line of Timber-measure. PROBL. 1. The Square of any piece of Timber at the end thereof being given, to find how much of that piece in length shall make a Foot solid? The Use of the Line of Timber-measure is in all respects the same at that of Board-measure; for knowing the square of your piece of Timber at the end thereof, you have no more to do than to look for the quantity of the Square thereof in the Line of Timber-measure, and right against it, on the other side of the Rule, you have the quantity of Inches that will make a Foot solid of that piece. Example 1. A piece of Timber is 10 Inches square, how much thereof in length will make a Foot solid? Look for 10 Inches in the Line of Timber-measure, and right against it, on the other side of the Rule, you shall find 17 Inches and somewhat above a quarter of an Inch; and so much of that piece in length will make a Foot solid. Example 2. If the Square of a piece of Timber be 21 Inches, how much thereof in length will make a Foot solid? Seek 21 Inches in the Line of Timber-measure, and against it you shall find, on the other side of the Rule, almost 4 Inches; and so much in length will make a solid Foot of Timber. Note 1. If Timber be broader at one end then at the other, the usual way is to add both ends together, and take half thereof for the true square; but if the difference be very much, this way is erroneous, though for the most part practised. Note 2. Also for Round Timber, the usual way is to gird it about the middle with a String, and take a fourth part thereof for the Square; this also is erroneous: Therefore for such as desire curiosity and exactness, let them repair to the Rules in this Book delivered for that purpose, where they may receive ample satisfaction. Concerning the Tables at the beginning of the Lines of Board and Timber-Measure. The Table of Board-Measure gives the length of a Foot square of any Board under 6 Inches broad; therefore by the Table there set you may find that Foot. In. parts. If a Board be 1 Inches broad 12 0 0 will make a Foot square. 2 6 0 0 3 4 0 0 4 3 0 0 5 2 4 5 6 2 0 0 By this small Table you may see, that a Board of 4 Inches broad will require 3 Foot thereof in length to make a Foot square.— Also, a Board of 5 Inches broad will require 2 Foot 4 Inches and 4 fifth parts of an Inch. The Table of Timber-measure gives the length of a Foot folid of any piece of Timber or Stone whose square is under 8 Inches: Wherefore by the Table at the beginning of the Line of Timber-measure you may find that If a piece of Timber be 1 Inch quare 256 0 0 will make a Foot solid. 2 36 0 0 3 16 0 0 4 9 0 0 5 5 9 0 6 4 0 0 7 2 11 0 8 2 3 0 By this Table (which is the same in effect with that which standeth at the end of the Line of Timber-measure) you may see that a piece of Timber that is 4 Inches square requires 9 Foot in length to make a solid Foot: Also, a pitce of 5 Inches square, require 5 Foot 9 Inches and 1/16 parts of an Inch to make a solid Foot: And so of the rest. But because these Tables go only to whole Inches, I have here added two Tables, one for Board, the other for Timber; the Table for Board, from one quarter of an Inch to 6 Inches in breadth; and the Table for Timber, from 2. Inches square to 8 Inches, by Inches, Halves, and Quarters. The Tables follow. The Table for Board-measure. Inches & qu. feet. in. 10 par. 1 48 0 0 2 24 0 0 3 16 0 0 I. 0 12 0 0 1 9 7 2 2 8 0 0 3 6 10 2 II. 0 6 0 0 1 5 4 0 2 4 9 6 3 4 4 4 III. 0 4 0 0 Inches & qu. feet. in. 10 par. III. 0 4 0 0 1 3 8 3 2 3 5 1 3 3 2 4 IU. 0 3 0 0 1 2 9 9 2 2 8 0 3 2 6 3 V. 0 2 4 8 1 2 3 4 2 2 2 2 3 2 1 0 The Table for Timber-measure. II. 0 36 0 0 1 28 4 3 2 23 0 4 3 19 0 3 III. 0 16 0 0 1 13 7 6 2 11 9 1 3 10 1 8 IU. 0 9 0 0 1 7 11 6 2 7 1 3 3 6 4 6 V 0 5 9 1 1 5 2 7 2 4 9 1 3 4 4 2 VI 0 4 0 0 1 3 4 2 2 3 4 9 3 3 1 9 VII. 0 2 11 2 1 2 8 1 2 2 6 7 3 2 4 7 Place here the FIGURE Of the RULE THE Line of Proportion (or Numbers,) Commonly called Gunters Line; Made Easie. WHat this Line is, and how to make it, is best known to those who make Mathematical Instruments; but the uses of it are so general, that all sorts of men, of what faculty soever, may apply it to his particular use, though it more immediately, and particularly, concerns such Artificers whose employment consists in Menstruation: as Carpenters, Joiner's, Masons, Bricklayers, Painters, Glasiers, and such like; for that all kind of mensurations, either SUPERFICIAL; as Board, Glass, Pavement, Tyleing, etc. Or SOLID, as Timber, Stone, Pillars, Pyramids, etc. are by this Line, most easily, speedily, and exactly performed: For, whatsoever thing, concerning measure, that may be performed by Arithmetic, this Line will do exactly, and much sooner; as by the working of the several Rules in Arithmetic, by this Line shall be plainly made appear. CHAP. I. NUMERATION upon the Line. BEfore I show you how to number upon the Line, it will be necessary to let you understand how the Line is divided and numbered, as also, what those divisions and numbers set to them upon the Ruler, do sigdifie. Know therefore, that the Line of numbers gins at the Figure One, and so proceeds successively from I to 2.3.4.5.6.7.8.9. and then on farther, by 1.2.3.4.5.6.7.8.9.10. at the end of the line. The first 1. which standeth at the beginning of the line, representeth the One tenth part of any Unite or Intiger, as One tenth part of a Foot, One tenth part of a Yard, el, Perch, Mile, etc. Or it may signify One tenth of a Year, Month, Hour, etc. Or the one tenth of a Pound, Shilling, or Penny, etc. Or the one tenth part of any thing either in Number, Weight, Measure, Tine, or the like. The Figure 2. signifies two tenth parts of any thing. The figure 3. three tenth parts. The figure 4. four tenth parts, etc. till you come to the second 1. which standeth in the middle of the line, which 1. signifieth One whole Unite or Intiger, as One whole Foot, Yard, Perch, etc. Now the other intermediate divisions, those which stand between the figures 1 and 2 (which are in number ten) do represent (each of them) one hundred part of one Unite or Intiger; so the first division beyond the figure 1, represents 11 hundred parts of the Intiget, the second division, 12 hundred parts of the Intiger, and so on; the figure 2 representing 20 hundred parts of the Intiger, and the next division beyond 2 is 21 hundred parts, and so on, till you come to the figure 1 in the middle of the line, which representeth one whole Intiger. The figure 2 signifieth two whole Intigers; the figure 3, three whole Intigers, and so on, till you come to 10 at the end of the line, which signifieth ten whole Intigers: and the intermediate divisions, which stand between 1 and 2 in the middle of the line are (every of them) tenth parts of the Intiger. So the Rule contains 10 whole Intigers, every of which is divided into 10 parts. But if upon the line you would count numbers of more places than two, (which are all numbers above 10) then the 1 which is at the beginning of the line, must be accounted one Intiger; and the 1 in the middle of the line, ten Intigers; and the 10 at the end, will be 100 Intigers. But yet farther, if upon the Line you would express numbers of more places than three (which are all numbers above 100). Then the 1 at the beginning of the line, is to be accounted ten Intigers, the 1 in the middle a hundred Intigers, and the 10 at the end of the line 1000 Intigers. And if you proceed yet farther; then the 1 at the beginning must be accounted for a hundred Intigers, that in the middle a thousand, and the 10 at the end of the line for 10000, ten thousand Intigers. In this manner you might proceed farther, by counting the first 1 for 1000, 10000, etc. Intigers, but to four places is sufficient, which by a Rule of a competent length (as of two Foot) any question concerning measuring, may be by one exactly enough performed. The Divisions and Numbers on the line being thus explained, it resteth now to show you how to find that point upon the line, which shall represent any number proposed, and that I shall show you in these Propositions following, which may fitly be called. NUMERATION. PROP. I. A whole number consisting of two, three, or four places, being given; to find the point upon the Line, which representeth the same: NOTE, Let your number given be of how many places soever; for the first figure of your number, you must take the same figure upon the Line: For the second figure in your number, take the number thereof on the grand (or larger) intermediate divisions on the Line: For the third figure in your number, take the number thereof on the smaller intermediate divisions on the line. And for your fourth figure, you must find its place by estimation. Example I. Let it be required to find the place of 15 upon the Line. For your first figure 1, count the 1 in the middle of the Line, then for the 5 which is your second figure, count five of the grand (or larger) intermediate divisions upon the line, and that point is the very place upon the line representing 15. Note, that every fifth of the grand intermediate divisions, is drawn forth with a longer line than the rest for case in counting. Again, To find the place upon the Line representing 37. For your first figure 3, count the figure 3 upon the Line, then for the 7, count 7 of the intermediate divisions, and that point is the place upon the Rule representing 37. Example 2. Let it be required to find the place of 134 upon the line. For your first figure 1, count 1 upon the line; for your second figure 3, count three of the grand divisions; and for the third figure 4, count 4 of the smallest intermediate divisions, and that very point is the place upon the Line representing 134. Again, To find the place representing 308. For your first figure 3, count the 3 upon the Line; for your second figure o (which is a Cipher) count none of the grand divisions; but for your last figure 8, count 8 of the intermediate divisions, and that point shall be the place upon the line representing 308. Example 3. Let it be required to find the place of 1350. For your first figure 1, take 1 on the middle of the Line: For your second figure 3, take the figure 3 upon the line upwards; for the 5, count five of the grand intermediate divisions, and that is the place of 1350. Again, To find the place of 1626. For your first figure 1, count the 1 on the middle of the Line; for your second figure 6, count the figure 6 upon the line upwards; then for your third figure 2, count two of the grand divisions; and for your last figure 6, estimate six tenth parts of the next grand division (which is something more than half the distance, (because 6 more than half 10), and that is the point upon the line representing 1626. Note, By these Examples last mentioned, you may perceive, that the figures 1.2.3.4.5.6.7.8.9. do sometimes signify themselves alone, sometimes 10.20.30, etc. sometimes 100.200.300, etc. as the work performed thereby shall require. The first figure of every number is always that which is here set down, and the rest of the figures are to be supplied according as the nature of the Question shall require. And by this variation and change of the powers of these numbers, from 1 to 10, or 100 or 1000, any proportion either Arithmetial or Geometrical may be wrought. One whereof I will insert for your better exercise of numbering on the Rule; by the often practice whereof, you will find the work facile and delightful, which shall be this following. PROP. 2. Having two numbers given, to find as many more as, you please, which shall be in continual proportion one to the other, as the two numbers given were. FOr the working of this proposition, this is THE RULE. Place one Foot of the Compasses in the first given number on the line, and extend the other Foot to the second given number, then may you turn the Compasses from that second number to a third, from that third to a fourth, from that fourth to a fifth, a sixth, a seventh, etc. towhat number of places you please. Example 1. Let the two given numbers be 2 and 4. Place one Foot of your Compasses in 2, and extend the other Foot to 4, than that Foot which now standeth in 2, being turned about, will reach from 4 to 8, and from 8 to 16, from 16 to 32, from 32 to 64, from 64 to 128. But when your Compasses stand in 64, if you turn them about yet farther, they will fall beyond the end of the line, wherefore you must place one Foot in some other, 64 nearer the beginning of the line, and then the other Foot will reach to 128, and from 128 to 256, and from 256 to 512, and from 512 to 1024, but here it will go off of your line again, whereof (as before) you must choose another 512 nearer the beginning of the line, and there placing you Compasses, they will reach to 1024. from 1024 to 2048, from 2048 to 4096, etc. Example 2. But if the given numbers were 10 and 9 decreasing, then place one Foot in 10 at the end of line, and extend the other downwards to 9, the same extent will reach still backwards to 8.1 (or 8 1/10) and from 8.1 to 7.29, and still backwards from 7.29 to 6.56. Likewise, if the two first numbers had been as 1 to 9, the third proportional would have been 81, the fourth 729, and the fifth 656, with the same extent of the Compasses. Again: Let the two numbers be 10 and 12, place one Foot in 10, and extend the other to 12, that extent will reach from 12 to 14.4, and from thence to 17.28. But if the numbers were 1 and 12, than the third proportional would be 144, and the fourth 1728, and all with the same extent of the Compasses. CHAP. II. MULTIPLICATION by the Line. IN Multiplication, the Proportion is this: As 1 upon the Line, Is to one of the numbers to be multiplied: So is the other of the numbers to be multiplied, To the Product of them. Which is the number sought. Example 1. Let it be required to multiply 5 by 7. The proportion is; As 1 ∶ to 5 ∷ so is 7 ∶ to 35. Therefore Set one Foot of your Compasses in 1, and extend the other Foot to 5; with that extent of the Compasses, place one Foot in 7, and the other Foot will fall upon 35, which is the Product. Example 2. Let it be required to multiply 32 by 9 The Proportion is; As 1 ∶ to 9 ∷ so 32 ∶ to 288. Set one Foot in one, and extend the other Foot to 9, that same extent will reach from 32 to 288, which is the product or sum of 32, being multiplied by 9 Otherwise Set one Foot in 1, and extend the other to 32, the same extent will reach from 9, to 288, as before. Example 3. Let it be required to multiply 8 75/100 by 5 45/100. The Analogy or Proportion is; As 1 ∶ to 8.75 ∷ so 6.45 ∶ to 56.44. fere. Set one Foot in 1, and extend the other to 8.75, the same extent applied forward upon the line, will reach from 6.45 to 56.44. fere. Or if you set one Foot in 1, and extend the other to 6.45: The same extent will reach from 8.75 to 56.44, almost (namely to 43 ¾) as before. CHAP. III. DIVISION by the Line. IN Division three things are to be minded, viz. The Dividend, or number to be divided. The Divisor, the number by which the Dividend is divided. The Quotient, which is the number sought. And so often as the Divisor is contained in the Dividend, so often doth the Quotient contain Unity. For the working of Division, this is the Analogy. As the Divisor is to Unity, or 1. So is the Dividend to the Quotient. Example 1. Let it be required to divide 35 by 7. The Proportion is; As 7 ∶ to 1 ∷ so 35 ∶ to 5. Set one Foot of the Compasses in 7, and extend the other Foot downwards to one; that same extent will reach from 35 downwards to 5, which is the Quotient, and so many times is 7 contained in 35. Otherwise Extend the Compasses upwards from 7 to 35, that same extent will reach upwards from 1 to 5, as before. Example 2. Let it be required to divide 288 by 32. The Proportion is; As 32 ∶ to 1 ∷ so 288 ∶ to 9 Extend t●e Compasses downwards from 32 to 1, that same extent will reach downwards from 288 to 9, which is the Quotient. Or extend the Compasses upwards from 32 to 288; the same extent will reach upwards from 1 to 9 as before. Example 3. Let it be required to divide 56.44 by 8.75. The Proportion is; As 8.75 ∶ to 1 ∷ so 56.44 ∶ to 6.45. Extend the Compasses downwards from 8.75 to 1, the same extent will reach downwards from 56.44 to 6.45. Or extend them upwards from 8.75, to 56.44, the same will reach upwards from 1 to 6.45. as before. Note this in Division, That so many times as the Divisor may be orderly set under the Dividend in Arithmetical Work, so many places of figures shall be in the Quotient of your Division: As if 34785 were to be divided by 75, the Quotient shall consist of three figures only, namely of 463, because 75 can be but three times set orderly under 34785 in Arithmetical operation. CHAP. IU. The GOLDEN-RULE Direct by the Line. THis Rule may well be termed the Golden-Rule, it being the most useful of all others; for having three numbers given you, may, by it, find a fourth in proportion to them, as by divers Examples following shall be made plain. And this Rule is performed upon the Line, with the like ease and exactness, as any of those before mentioned: And for the working of it upon the Line, this is the general Analogy or Proportion. As the first number given, Is to the second number given; So is the third number given, To the fourth number required. Or As the first number given, Is to the third number given; So is the second number given, To the fourth number sought. Wherefore Always, Extend the Compasses from the first number to the second, and that distance or extent applied, the same way, upon the Line, shall reach from the third, to the fourth number required. Or otherwise, Extend the Compasses from the first number to the third, and that extent applied the same way, shall also reach from the second to the fourth. Either of these ways will effect the same thing, as by Examples following shall be made appear. And it is necessary thus to vary the proportion, sometimes, to avoid the opening of the Compasses too wide; for when the Compasses are opened to a very large extent, you can neither take off any distance exactly, nor give so a good estimate of any parts required, as you may do when they are opened to a lesser distance: But this you will find out best by practice; and therefore I will now proceed to Examples. Example 1. If 45 yards of Cloth cost 30 pound, what will 84 yards cost at the same rate? As 45 ∶ to 38 ∷ so 84 ∶ to 56. Extend the Compasses downward from 45 to 30, that extent will reach downward from 84 to 56 l. the price of 84 yards. Or extend the Compasses upwards from 45 to 84, the same will reach from 30 to 56, as before. Example 2. If 26 Acres of Land be worth 64 l. a year; what is 36 Acres of the like Land worth by the year. As 26 ∶ to 64 ∷ so 36 ∶ to 88.615. Extend the Compasses from 26 to 64, the same extent will reach from 36 to 88 615/1000 parts (which is about 12 s. 3 d. 2 q.) and so much is 36 Acres of the like Land worth by the year. Example 3. If 100 l. yield 6 l. Interest for one year, or 12 months, what shall 75 l. yield? As 100 ∶ to 6 ∷ 75 ∶ to 4.50. Extend the Compasses from 100 to 6, the same extent will reach from 75. to 4.50. (or 4 ½) which is 4 l. 10 s, and so much will 75 l. yield interest in the year. Example 4. If 75 l. yield 4 l. 10 s. interest for one year, or 12 months, what will 100 l. yield? As 75 ∶ to 4.50 ∷ so 160 ∶ to 67. Extend the Compasses downwards from 75 to 4.50, the same extent will reach from 100 to 6, and such interest will 100 l. yield. Many other Questions might be added, but the Rule (and manner of working it) is so plain, that it needs them not, and so general, that he which can resolve one, may as well resolve any other, and therefore I shall say no more of it in this place. CHAP. V The GOLDEN-RULE Reverse by the Line. IN this reverse or backward Rule of Three, this note is specially to be observed, That if the third number be greater than the first, then will the fourth number be less than the second. And on the contrary, if the third number be less than the first, than the fourth number will be greater than the second. As by Examples will appear, Example 1. If 12 Workmen, do any piece of Work in 8 days, how many Workmen shall do the same piece of Work in 2 days? It is here to be noted, That in this Question 12 is not the first number (though it be first named) but 2; for the middlemost term of the three must be of the same kind with the fourth number which is to be sought; as in this Example it is Men, therefore 12 (which are men) must stand in the middle, or second place, because the fourth number which is to be sought is also Men: And therefore the numbers will stand thus, days men days 2, 12, 8. For if 8 days, require 12 men, than 2 days (which is but a fourth part of 8 days) shall require four time 12 men, that is, 48 men. For here, Less requires More; that is, less Time, more Hands, and hence the Work is contrary to the direct Rule: Wherefore to effect it, Extend the Compasses, from 2 to 8, the same extent will reach from 12 (the contrary way on the Line) to 48, which is the number of men that will effect the same piece of work in two days. Example 2. If one Close will graze 21 Horses for 6 weeks, how many Horses will the same Close graze for 7 weeks? Extend the Compasses from 6 to 7, (for you must always extend your Compasses to numbers of one kind, or denomination, as here 6 and 7 are both Horses) the same extent will reach from 21 backward to 18, and so many Horses will the same Close graze for 7 weeks. CHAP. VI Of DUPLICATE PROPORTION by the Line. DUplicate proportion is such a proportion as is between Lines and Superficies, or between Superficies and Lines. 1. Of the proportion of LINES to SUPERFICIES. In this Case, extend the Compasses from the first to the second number of the same denomination, that same extent (being doubled) shall give the distance from the third number unto the fourth. Example 1. If the Diameter of a Circle be 14 Inches, and the Area, or content thereof be 154 Inches, what will be the content of another Circle, whose Diameter is 28 Inches? Extend the Compasses from 14 to 28, that extent doubled, will reach from 154, to 616; for first it will reach from 154 to 308, and from thence to 616, and that is the Area or Content of a Circle whose Diameter is 28. Example 2. If a piece of Land that is 20 Pole square, be worth 30 pounds; what is a piece of Land of the same goodness worth that is 35 Pole square. Extend the Compasses from 20 to 35, that extent doubled, will reach from 30 to 91.8, that is, 91 pound 8/10 of a pound, which is 16 shillings, and so much is such a piece of Land worth. II. Of the Proportion of SUPERFICIES to LINES. In this Case, Extend the Compasses unto the half of the distance between the two numbers of the same denomination; that same extent shall reach from the third number, to the fourth required. Example 1. Let there be two Circles given, the Area or Content of the one being 154, and its Diameter 14 The Area of the other Circle is 616, what is the length of its Diameter? Upon your Line, divide the distance between 154 and 616, into two equal parts, then with that distance set one Foot in 14, and the other shall fall upon 28, which is the length of the Diameter of the other Circle, whose Area is 616. Example 2. There is a piece of Land containing 20 Pole square worth 30 pound, there is another piece worth 91 pound 16 s. how many Pole square aught that piece to contain? Take with your Compasses half the distance between 30 l. and 91 l. 16 s. Then set one foot in 20 pole, and the other foot will reach to 35 Pole, and so many Pole square must the Land be that is worth 91 l. 16 s. CHAP. VII. OF TRIPLICATE PROPOTION by the Line. TRiplicate proportion is such a proportion as is between Lines and Solids, or between Solids and Lines. 1. Of the Proportion between LINES and SOLIDS In this Case, Extend the Compasses from the first number to the second of the same denomination, that extent (being tripled) shall reach from the third number to the f●rth. Example. There is a Bullet whose Diameter is 4 Inches, weighing 9 l. What shall another bullet of the same mettle weigh, whose Diameter shall be 8 inches. Extend the Compasses from 4 to 8 (the two Diameters) the same extent (being tripled) will reach from 9 to 72. which is the weight of a bullet whose diameter is 8 inches. II. Of the proportion of SOLIDS to LINES. In this Case Extend the Compasses unto the third part of the distance between the two numbers of like denomination, that same extent shall reach from the third to the fourth number required. Example. The weight of a Cube being 72 pound, the Side whereof was 8 inches; and the weight of another Cube of the same matter weighting 9 pound, what must the side be? Upon your line divide the distance between 9 and 72 into three equal parts, than set one foot of that distance in 8, and the other foot shall rest in 4, the length of the side of the Cube required. CHAP. VIII. The Extraction of the SQARE-ROOT By the Line. TO Extract the Square-Root, is to find a mean proportional Number between 1 and the number given, and therefore is to be found by dividing the space between them into two equal parts. Example. Let it be required to find the Square-Root of 36. Extend the Compasses from 1 to 36, the middle way upon the Line between these two numbers is 6, which is the Square-Root of 36. In like manner may you find the Square-Root of 81 to be 9, of 144 to be 12, of 256 to be 16; and of other numbers as in this Table. Root. Square. Root. Square. 1 1 11 121 2 4 12 144 3 9 13 169 4 16 14 196 5 25 15 225 6 36 16 256 7 49 17 289 8 64 18 324 9 81 19 361 10 100 20 400 If you suppose the number to have pricks over every second Figure, as is usual in the Arithmetical Operation, then if the last prick towards the left hand fall over the last Figure (which will always be when the number of Figures are odd) than it will be best to place Unity at the 1 in the middle of the line, so that the Root and the Square may both fall forward towards 10 at the end of the Line. But if the number of Figures be even, it will then be best to place Unity at 10 at the end of the Line; so the Root and the Square both will fall backwards towards the middle of the Line. CHAP. IX. The Extraction of the CUBE-ROOT By the Line. TO Extract the Cube-Root, is to find the first of two mean proportionals between 1 and the number whose Cube-Root you require, and is therefore to be found upon the Line, by dividing the space between them into three equal parts. Example. Let it be required to find the Cube-Root of 216. Extend the Compasses from 1 to 216, one third part of that distance shall reach from one to 6, which is the Cube-Root of 216. In like manner may you find the Cube-Root of 729 to be 9, of 1728 to be 12, of 110592 to be 48, of 493039 to be 79, etc. as in this Table. Root. Cube. Root. Cube. 1 1 11 1331 2 8 12 1728 3 27 13 2197 4 64 14 2744 5 125 15 3375 6 216 16 4096 7 343 17 4913 8 512 18 5832 9 729 19 6859 10 1000 20 8000 CHAP. X. The Use of the Line, applied to SUPERFICIAL-MEASURE, Such as Board, Glass, Wainscot, Pavement, Hang, Paintings, etc. of what kind soever. THe Measures by which Board, Glass, Timber, Stone, and such like are measured, is by the Foot, a Foot containing 12 Inches, and each Inch into eight parts called halves, quarters, and half-quarters; but this kind of division not being consentaneous or agreeable to the divisions upon your Line of Proportion, where between 1 and 2 is divided (not into 8, but) into 10 parts, the like between 2 and 3 into 10 parts, and so between 3 and 4,4 and 5, etc. Therefore, I hold it requisite both for ease and exactness, to have every Inch on your two-foot Rule divided not into 8, but into 10 equal parts, which hereafter (throughout this Book) we will call Inch-Measure. Again, Whereas your Foot is divided into 12 equal parts called Inches, I would have your Foot divided into 10 equal parts, and each of those parts subdivided into 10 other equal parts, so will you whole Foot contain 100 equal parts, which will be agreeable to the divisions of your Line, and facilitate the work, as by the Examples in this kind given will be made to appear; and this we shall hereafter call Foot-Measure. But if any person be so wedded to Inches halves and quarters, that he will not be beaten out of his opinion, but persist therein, and yet is desirous to have knowledge in the use of this Line, I say, such person may have added to the side of his Inches halves and quarters (by way of faceing, as I term it) a line of Foot-measure, and also his Inches into 10 as well as 8, so that he may measure by one, and work upon his Line by the other. And this indeed will be necessary to be done upon the Rules of those ingenious Artificers who need them not, for that they many times meet with wilful persons that will have them to measure their way, how disconsentaneous to Reason soever it be. In this nature would I have the Rule divided, and in this manner have I caused them to be made both for myself and others, a figure whereof I have inserted at the beginning of the Book. And here note, That what is here said concerning dividing the Inch and Foot into 10 parts, the like is to be understood of the Yard, el, Pole or Perch, or any other measure whatsoever. These things being premised, we will now proceed to Examples. I. Examples in Inch-Measure only. Example 1. Let a Board or Plank be 37 Inches broad, and 263 Inches long, how many Square Inches is there in such a Plank? The Proportion is, As 1, is to 27, the breadth in Inches; So is 263, the length in Inches, To 7101, the number of Square Inches in the whole Plank. Extend the Compasses from 1 to 27, the same extent forwards, will reach from 263, to 7101 the content. Or, you may extend the Compasses from 1 to 263, the same will reach from 27 to 7101, as before. Example 2. Let a Pain of Glass be 53.4 Inches broad, and 126.8 Inches long; how many Foot is there in that Pain? The Proportion is, As 144 (because 144 Inches make 1 Foot) is to 53.4, the breadth in Inches; So is 126.8, the length in Inches, to 47.06 the content in Feet. Extend the Compasses from 144 to 53.4, the same will reach from 126.8, to 47.06, which is 47 Foot and ●● 6/9 parts of a Foot, the content of the whole Pain. Example 3. If a Marble Footpace or Walk be 20 Inches broad, how much in length of that will make a Foot, square 3● The proportion is, As 20, the breadth in Inches, is to 144, the Inches in one Foot; So is 1 Foot, unto the length of one Foot in Inch-measure. Extend the Compasses from 20 to 144, that extent will reach from 1 to 7.2; so that 7 Inches and 2/10 of that breadth will make a Foot square. II. Examples in Foot-Measure only. Example 1. Let a Floor or Stone-pavement be 52 Foot broad, and 1105 Foot long, how many Foot square in that Floor or Pavement? The Proportion is. As 1 Foot, to 52 Foot the breadth; So 100.5 Foot the length, to 5746 the content in square Feet. Extend the Compasses from 1 to 52, the same will reach from 110.5, to 5746, the content of the Pavement or Floor in square Feet. Example 2. There is a Plank of Cedar 2 Foot 25 parts broad; how much in length thereof will make a Foot square? The Proportion is, As 2.25 the breadth, is to 1; So is 1, or any number of Feet, to the length of a Foot square in Foot-measure. Extend the Compasses from 2.25 to 1, that extent will reach back from 100, which is one Foot, to 44 parts; and so many parts in length of that Plank will make a Foot. In like manner 88 parts will make 2 Foot, 1 Foot 32 parts will make 3 Foot, etc. For, As 2.25, is to 1 Foot; So is parts. 100 200 300 to parts. 44 88 132, etc. III. Examples in Inch-Measure and Foot-Measure together. Example 1. Let a Board be 30 Inches broad, and 15 Foot and ¼ or 25 parts long; how many Foot square d●th such a Board or Plank contain? The Analogy is, As 12 Inches, to 30 the breadth in Inches; So 15.25 the length in Feet, to 38.125 the content in Feet. Extend the Compasses from 12 to 30, the same will reach from 15.25 to 38.125; and so many Foot square is contained in such a Plank. I will conclude this Chapter with this useful and necessary Problem, Namely, By having the length and breadth of any Long Square, or Parallelogram, to find the length of a side of a Geometrical Square equal thereunto. Note, By a Long Square or Parallelogram is meant any Square whose sides are longer one than another, as any lo●g Table, etc. But a Geometrical Square is that whose 4 sides are all of one length. This by the Line is easily effected; for if you take the half-distance upon your Line between the length and the breadth, the Number upon which the Compass point resteth, shall be, the length of the side of the Geometrical Square, equal to the Long Square or Parallelogram. Example. Let the longer side of a Parallelogram be 183 Inches, and the breadth 30 Inches: If you divide the distance upon your Line between these two Numbers into two equal parts, the Compass point shall-rest upon 74 Inches 10 parts: So that a Geometrical Square whose side is 74.10, shall be equal in Area to a Long Square whose sides are 30 and 183. CHAP. XI. OF YARD-MEASURE By the Line. MAny Artificers, as Joiner's, Painters, Plasterers, Paviers, Upholsters, measure and sell their Work, not by the Foot but by the Yard, it will be necessary to give Examples in this kind of Measure also. And here also it is requisite that your Yard be divided into 100 parts, and not into Halves, Quarters, and Nails; which supposed, take these Examples following. Example 1. A joiner hath Wainscoted a Gallery containing 130 Yards 25 parts about, and in height 15 Yards 50 parts; how many square Yards is in that Gallery? The proportion is, As 1 yard, to 15.50 yards the height; So 130.25 the compass in yards, 10 2018.87 the content in yards, Extend the Compasses from 1 to 15.50 the breadth, the same extent will reach from 130.25 the length, to 2018.87; and so many Square Yards of Wainscoting is in that Gallery. Example 2. A Painter hath painted Landscape, or other Work, over the Wainscot of a Room, which is 1 Yard 75 parts of a Yard deep; how much in length thereof will make a Yard Square? As the breadth 1.75, is to 1 yard or 100 parts; So is 1, or any other numb. of yards, to the length of a yard square. Extend the Compasses from 1 to 1.75, the same extent will reach from 100 (or one yard) to 75.14; and so much in length of that painting will make a yard square. Example 3. A Plasterer hath laid and beautified a Ceiling containing 13 yards broad, and 63 yards 30 parts long; how many square yards is there in that Ceiling? As 1 yard, to the breadth 13 yards; So the length 63.30, to the content. Extend the Compasses from 1 to 13, the same extent will reach from 63.30, to 823 almost; and so many square yards are there in such a Ceiling. Note, It may so fall out sometimes, that it will be required to measure some piece of Work, and to give an estimate of the quantity of the yards therein contained, when you have not a yard thus divided by you, but only your two-foot Rule; for the supplying whereof I will add this following Problem. PROBLEM. The length and breadth of any Superficies being given in Feet, in find the content thereof in Yards. Let the breadth of a piece of any Work, to be paid for by the Yard, be 4 Foot, and the Length thereof 12 Foot, how many square Yards are contained therein? The Analogy or Proportion is, As 9, is to 4, the breadth in Feet; So is 12, the length. to 5 36/100 the content in yards. Extend the Compasses from 9 to 4, the same extent will reach (the same way) from 12 to 5.35, that is, to 5 yards, and 35 hundred parts of a yard, which is 35 yards, one quarter, and almost half a quarter of a yard. CHAP. XII. OF LAND-MEASURE By the Line. THe usual Measures for Land are Chains, of which there are divers sorts; but the denominations of the quantity of Land is given in in Acres and Perches. The Chains now most in use are principally two; One containing 1 Perch in length, The other 4 Perches in length, each of them divided into 100 links. For the practice of them take these Examples. I. By the 1 Pole Chain. Example 1. There is a plat of ground 30 Perches broad, and 183 Perches long; how many Perches doth it contain? As 1, to 30 the breadth; So 183 the length, to 5490 the content. Extend the Compasses from 1 to 30, that shall reach from 183 to 5490 the content. Example 2. But the length and breadth of the same piece of ground being given as before in Perches, if it were required to find the content in Acres, then, As 160, to 30 the breadth; So 183, to 34 Acres 31/100 parts of an Acre. Extend the Compasses from 160 to 30, the same shall reach (being extended the same way) from 183 to 34.31, that is, 34 Acres, 31 hundred parts of an Acre, which is something above a Rood. II. By the Four Pole Chain. Example 1. A piece of Land containing 16 Chains 25 Links in breadth, and 57 Chains 30 Links in length, how many Acres doth it contain? The Analogy is, As 10, to 16.25 the breadth in Chains and Links; So is 57.30 the length in Chains and Links, to 93 Acres 925/10060 parts of an Acre. Extend the Compasses from 10 to 16.25, the same extent will reach from 57.30, to 93.0925. Example 2. The Base and Perpendicular of a Triangle being given in Chains and Links, to find the content in Acres. This is a right useful and necessary Proposition; for by it all manner of irregular plaits of Land are cast up: But my intent here is not to teach Surveying, but to show the use of the Line of Proportion. Wherefore, Let the Perpendicular of the Triangle be 7 Chains 50 Links, and the Base 45 Chains 75 Links, the proportion will be, As 20, to 7.50 the Perpendicular; So is 45.75 the Base, to 17.15 the content. Extend the Compasses from 20 to 7.50, that extent shall reach from 45.75 to 17.15, which is 17. Acres, and 15/100 parts. Example 3. Having the length of any Furlong given, to find what breadth it must have to make an Acre. Let the length of the Furlong be 12 Chains 50 Links, then to find the breadth for one Acre this is the Analogy. As 15.20 the length in Chains, is unto 10; So is 1 Acre, to 80 Links, which must be the breadth of the Furlong. Wherefore, Extend the Compasses from 10 to 12.50, the same will reach from 1 Acre to 80 Links the breadth of the Furlong. CHAP. XIII. Of the Mensuration of divers Regular SUPERFICIAL-FIGURES by the Line. HAving sufficiently shown the manner of measuring of such Superficial-Figures as are measured by length and breadth, I will now show you how by the Line to measure some other Regular Figures, as ahe Circle, etc. I. Of the Circle. Example 1. The length of the diameter of any Circle being given, to find the Circumference thereof. The proportion between the diameter and the circumference of any Circle is as 7 to 22; or, in exacter terms, as 1000 to 3140. Wherefore, If the diameter of a Circle given be 23 Inches, the circumference therefore may be found by this following Analogy. As 1.000, is to 3.140; So is 12 the diameter, to 37.68 the Circumference. Wherefore extend the Compasses from 1.000, to 3.140, the same extent shall reach from 12, to 37 Inches 68 parts, which is the Circumference. Example 2. The circumference of any Circle being given, to find the length of the diameter. This is the converse of the former Example, and the Analogy is the converse also. Let the circumference of a Circle be 37 Inches 68 parts, what is the length of the diameter? As 3.140, to 1.000; So is 37 Inches 68 parts the circumference, to 12 Inches the diameter. Extend the Compasses from 3.140, to 1.000, the same extent will reach from 37.68, to 12, the diameter required. Example 3. Having the diameter of a Circle, to find the length of the side of a Square, which shall be equal in content to the same Circle. If the diameter of a Circle be 12 Inches, the proportion is, As 1.0000, is to 12 Inches the diameter, So is 8862, to 10.63 the side of the Square. Extend the Compasses from 10000 to 12, the same extent will reach from 8862, to 10 Inches 63 hundred parts, the side of a Square equal in Area to the Circle whose diameter is 12 Inches. Eample 4. Having the Circumference of a Circle given, to find the side of a Square equal to that Circle. Let the circumference of the given Circle be 37 Inches 68 parts; The proportion is, As 10000, to 37.68 the circumference; So is 2832, to 10.63 the side of the Square. Extend the Compasses from 10000 to 37.68, the same will reach from 2821, to 10 Inches 63 parts, the side of the Square required. Example 5. The diameter of a Circle being given, to find the superficial content thereof. Let the diameter of a Circle be 15 Inches. Extend the Compasses from 1 to 15 the diameter, then apply one foot of that distance (always) to 7854; then turn that distance twice from this number the same way, and the Compass point will fall upon 176 Inches 74 parts, which is the Area of that Circle whose diameter is 15 Inches. Example 6. The circumference of a Circle being given, to find the Area thereof. Let the circumference of a Circle given be 47 Inches 13 parts. Exetnd the Compasses from 1 to 47.13 the Circumference; this distance being applied (always) to this number 7958, and from thence twice repeated, the point of the Compasses at the second remove will fall upon 176 Inches 74 parts, equal to the Area of the Circle as before. Note. Here note, That your Compasses being opened from 1 to 47.13 the Circumference, when you come to set one foot upon 7958, the other will reach at your first turning over to 37.8; and when you turn them over again it will fall out of the line; wherefore you must set one foot in 37.8 in the lower part of the Line, and then the other will fall upon 176.74. And thus you must do in other cases, whenever your Compass point goes beyond your Line. CHAP. XIV. II. Of the Triangle. A Triangle is a Figure consisting of three sides and three angles, the longest side whereof we call the base; and a line drawn from the Angle opposite to the base we call the perpendicular. To measure Triangles there are several ways; I will only show you one or two to be done by the Line. Example 1. There is a Triangle whose base is 14 foot, and his perpendicular 6 foot, I would know how many square Foot is contained in this Triangle. The proportion is, As 2, is to 6 the perpendicular; So is 14 the base, to 42 the Area. Or, As 1, is to 3, half the base; So is 14, to 42 the Area. Or, As 2, is to 6 the perpendicular; So is 7 half the base, to 42 the Area. Or, As 1, is to 6 the perpendicular; So is 14 the base, to 84 the double Area. All these ways produce the same effect, but the first is the best: Wherefore, The base of your Triangle being 14, and the perpendicular 6; Extend the Compasses from 2 to 6, the same extent will reach from 14, to 42 the Area. III. Of the Trapezia. A Trapezia is any right lined figure consisting of four unequal sides and as many angles: For the measuring of it, you must first reduce it into two Triangles, by drawing a line or diameter from one opposite angle to another the longest way, then from the two angles opposite to this line, let fall two perpendiculars; so is the Trapezia divided into two Triangles. The manner how to measure it is this. Example. There is a Trapezia whose diameter is 12.34, and one perpendicular is 4.20, the other 5.07; I would know the Content or Area thereof. The two perpendiculars added together make 9.27. Then the analogy is, As 2, is to 9.27, the sum of the perdendiculars; So is 12.34, the base, to 57.19, the Area. There are as many ways to measure Trapezias, as in the last Example I gave you for Triangles; but this is the best. And here note, That if you are to measure any irregular piece, of what nature soever, whether Land, Board, Glass, Pavement, or the like, your best and exactest way is to reduce them to Trapezias, and measure them as before is taught. FOUR Of Regular Figures of 5, 6, 8, 10, or 20 equal sides. These Figures by Geometricians are called Regular Polygons, and the way to measure them is by adding all the cider together. Then measure the length from the centre of the figure to the middle of one of the sides; by the help of these two you may find the Area of the Figure. Example. Let there be a Regular Polygon of 11 equal sides, each side being 7 Inches; and let the length of the line from the centre to the middle of one of those sides be 12 Inches. Add all the sides together they make 77; then, As 2, is to 77 the sum of the sides; So is 12, the length of the line from the middle of the figure, to 462, the content of the figure. CHAP. XV. The Use of the Line applied to SOLID-MEASURE Such as Timber, Stone, etc. TImber and Stone are usually measured by the same Rule or Measure as Board and Glass are, namely, by Feet and Inches: Therefore such a Rule as was mentioned in the beginning of the Tenth Chapter is fit for this business also. Before we come to show the way of measuring of Stone or Timber, it will be necessary to premise thus much, That the base or end of every piece of Timber or Stone is (or must be supposed) either exactly square, that is, every side alike, or else one of the sides longer than the other: wherefore the first thing to be done is to find the Area, or superficial content of the base or end of any piece of Timber or Stone to be measured, which may be done several ways, either in Inch-Measure, as by the first Example of the first part of the tenth Chapter; or in Foot-Measure, by the first Example in the second part of the same Chapter; or both in Foot-measure and Inch-measure, as in the first Example of the third part of the same tenth Chapter; and therefore need not be here repeated again: Wherefore, we will proceed to our intended purpose of Measuring, first, by Inch-measure only; secondly, by Foot-measure only; and thirdly, by both together; as we did before in the measuring of Board, etc. 1. In Inch-Measure only. Example 1. There is a piece of Timber 30 Inches broad, 21 Inches 6 parts deep, and 183 Inches long; how many square Inches is there in this solid piece of Timber? The proportion is, 1. As 1, unto 30 Inches the breadth; So is 21.6 Inches the depth, to 648 Inches, the content of the base of the piece. 2. As 1, unto 648, the content of the base; So is 183 Inches, the length of the piece, to 118584, the solid content in Inches. Wherefore, Extend the Compases from 1, to 30 the breadth; the same will reach from 21.6 the depth, to 648 the content of the base.— Again, Extend the Compasses from 1, to 648 the content of the base; that extent will reach from 183 the length, to 118584 Inches the solid content. But so many places of Figures cannot well be estimated upon your Line, except. it be very large; but by following Examples you shall have your desire accomplished exactly and easily. Example 2. To find the content of the same piece of Timber in Foot-measure, the dimensions being given in Inches and parts. The proportion is, 1. As 1, to 30 the breadth; So 21.6 the depth, to 648 the content of the base, as before. 2. As 1728, the number of solid Inches in a Foot of Timber, is to 648, the content of the base; So is 183, the length in Inches, to 68 Foot, and 62/100 parts of a Foot, as before. Wherefore, as before, Extend the Compasses from 1, to 30 the breadth; the same will reach from 21.6 the depth, to 648 the content of the base, as before.— Again, Extend the Compasses from 1728, to 648 the base, the same extent will reach the same way, from 183 the length, to 68.62 the content of the piece of Timber in Feet and parts, that is, 68 Foot, and above half a Foot. Example 3. Let a squared Stone or piece of Timber be 30 Inches broad, and 21 Inches six parts deep; how much in length shall make a Foot square of that piece of Timber or Stone? You may find the content of the base, as in the last Example, to be 648 Inches; then the proportion is, As 648, the content of the base, is to 1728, the inches in a Foot; So is 1, to 2 inches 67 parts, the length of a Foot solid. Therefore, Extend the Compasses from 648 the base, to 1728; the same will reach from 1, to 2.67: So that 2 inches 67/100 parts will make a Foot solid of that piece of Timber or Stone. This may be done another way by this Analogy or Proportion. 1. As. 12, to 30 the breadth in inches; So 21.6, the depth in inches, to a fourth number (which here will be about 54.) 2. As this fourth number 54, is to 144; So is 1, to 2.67, the length of a Foot solid. Wherefore, Extend the Compasses from 12, to 30 the breadth, that extent will reach from 21.6 the depth, to a certain place upon the Line (about 54), where keep the point of the Compass fast, and open the other to 144, then will this extent of the Compasses reach from 1, to 2 inches 67 parts, the length of a Foot solid, as before. II. In Foot-Measure only. Example 1. Let a Stone, or a piece of Timber, be 2 Foot 50 parts broad, 1 Foot 80 parts deep, and 15 Foot 25 parts long, how many solid or cubical Feet doth such a piece contain? The proportion is. 1. As 1, is to 2.50 Foot the breadth; So is 1.80 Foot the depth, to 4.50 Foot, the base in such measure. 2. As 1, unto 4.50 the base; So 15.25. the length, to 68.62, the content in Feet. Extend the Compasses from 1 to 2.50 the breadth; the same will reach from 1.80 the depth, to 4.50 the base.— Again, Extend the Compasses from 1, to 4.50 the base; that extent will reach from 15.25 the length, to 68.62, the content in Feet. Example 2. In the forementioned piece of squared Stone or Timber, being 2 Foot 50 parts broad, and 1 Foot 80 parts deep, Let it be required to find how much thereof in length will make a Foot. The proportion is, 1. As 1, is to 2.50, the breadth; So is 1.80, the depth, to 4.50, the content of the base in Foot-measure. 2. As 4.50, the base, is to 1; So is 1 Foot, to 222 parts, the length of a Foot solid. Wherefore, Extend the Compasses from 1, to 2.50 the breadth; the same extent will reach from 1.80 the depth, to 4.50, the content of the base.— Again, Extend the Compasses from 4.50 the base, to 1; the same will reach from 10, to 222 parts, the length of a cubical or solid Foot of that Stone or piece of Timber. III. In FOOT-MEASURE and INCH-MEASURE together. Example. Let a squared Stone or piece of Timber be 30 Inches broad, 27.6 Inches deep, and 15 Foot 25 parts long, how many cubical or solid Foot of Stone or Timber is shear in that piece? The proportion is, 1. As 1, is to 30 inches, the breadth; So is 21.6 inches, the depth, to 640, the content of the base in inches. 2. As 144, the inches in a Foot superficial, is to 648, the content of the base in inches; So is 15.25, the length of the piece in Foot-measure, to 68 Foot 62 part. Wherefore, Extend the Compasses from 1, to 30 the breadth, the same will reach from 21.6 the depth, to 648, the content of the base.— Again, Extend the Compasses from 144, to 648 the content of the base, the same extent will reach from 15.25, the length of the piece, to 68.62, the solid content of the Stone or Timber in Feet and 100 parts of a Foot By having the same things given in the same piece of Stone or Timber (or in any other) the work may be varied several ways: The Analogies or Proportions I will only give you, leaving the practice thereof to yourself. Breadth of the piece 30 inches. Depth of the piece 21.6 inches. Length of the piece 15.25 foot. The Proportions. 1. As 144, to 30, the breadth; So 21.6, the depth, to a fourth Number. From which fourth Number, if you extend your Compasses to 1, and place one foot in 15.25, the length of the piece, the other Foot shall fall upon 68.62, the content of the Stone. Or, 2. As 12, unto 30, the breadth; So 21.6, the depth, to some fourth Number. From this fourth Number, Extend the Compasses to 12, that distance will reach from 15.25, the length of the piece, to 68.62, the content of the piece. CHAP. XVI. How to measure Stone or Timber by the Line, by having the Square of the Base, and the Length of the Piece given, both in Foot and Inch Measure. HOw to find the length of a Side of a Geometrical Square, that shall be equal to any Parallelogram or Long Square, is taught at the latter end of the Tenth Chapter of this Book, by which Rule it may at any time be found: That being done there, I shall only here begin with Examples. Example 1. There is a Squared piece of Timber whose length is 183 Inches, and the Side of the Square equal to the base or end thereof is 25 Inches 45 parts; how many Foot doth that piece contain? 1. As 41.57, to 25.45, the side of the Square; So is 183, the length in Inches, to a fourth Number. 2. And that fourth Number, to 68.62, the content in Feet. Extend the Compasses from 41.57, to 25.45, the side of the Square; the same will reach from 183, the length, to some other part of the Line; from whence if you again extend the same distance, the point will rest upon 68 Foot 62/100 parts of a Foot; and so many Foot is in the piece. Example 2. Let the side of a Square, equal to the Base of a piece of Stone or Timber, be 2 Foot 12 parts, and the length of the same piece 15 Foot 25 parts; how many solid Foot is there in that piece? 1. As 1. to 2 Foot 12 parts the side of the Square; So 15 Foot 25 parts, the length, to a fourth Number. 2. And that fourth Number, to 68.62, the content in Feet. Extend the Compasses from 1, to 2.12, the side of the Square; that will reach from 15.25, the length, to some other number on the Line, from whence the Compasses being extended, the movable point will fall upon 68.62, the content, as before. Example 3. The side of a Square, equal to the Base of a Stone, being 25 Inches 45 parts, and the length of that Stone 15 Foot 25 parts, how many Foot doth it contain? 1. As 12, to 25.45, the Square in Inches; So is 15.25 Foot, the length, to a fourth Number. 2. And that fourth Number, to 68.62, the content. Extend the Compasses from 12, to 25.45 the side of the Square; the same will reach from 15.25, to some other point upon the Line, from whence the Compasses being extended, the movable point will fall upon 68 Foot 62 parts, the content of the Stone. Example 4. There is a piece of Timber whose side of the Square of the Base is 25 Inches 45 parts; how much in length of that piece will make a Foot solid? 1. As 25.45, the side of the Square, is to 41.57; So is 1 Foot, to a fourth Number. 2. And that fourth Number, to 6 Inches 67 parts. Wherefore, Extend the Compasses from 25.45 the side, to 41.57; the same will reach from 1, to some other point, from whence the Compasses being extended, will reach to 6.67, the length of a Foot solid of that piece of Timber. Example 5. The length of the side of a Square, equal to the Base of a piece of Timber being 2 Foot 12 parts, to find how much in length of that piece will make a Foot solid in Foot-measure. As 2.120, the side of the Square, is to 1.000; So is 1.000, to a fourth Number: And that fourth Number, to 0.471 parts of a Foot, to make a Foot square. Extend the Compasses from 2.120 the side of the Square, to 1000; the same extent will reach from 1000, downwards, to some other point upon the Line, and from thence downwards, to 222 parts of a Foot; and so much in length will make a Foot solid. CHAP. XVII. Concerning Timber that is bigger at one end than at the other, either Round or Square; and how to measure it. I. For SQUARED-TIMBER. IN large Timber-trees, when they are squared, there is a great disproportion between the Squares of both ends; wherefore some do use to take the Square of the middle of the piece for the mean or true Square; but this is not exact, though much used; but the best way is this: Find by the Problem at the end of the tenth Chapter of this Book, the length of the side of a Spuare equal to both the ends of the piece, add these two sides together, and take the half thereof for the true Square; and with that Square you may by the Rules of the last Chapter measure it as if it were perfectly Square. II. For ROUND-TIMBER. The ordinary way used for the measuring of Round Timber, is to gird it about the middle with a Line, and to take one fourth part thereof for the side of a Square equal thereto: but this is false, though most men use it, Custom having made it bear the face of Truth; for it is more in measure than in reality it should be. But the exact way of measuring of Round Timber (especially if it be growing) is this: About the middle thereof, in some smooth place, gird the same about with a string: Then have you this proportion. As 1000, is to the number of Inches about; So is 2821, to the length of the side of a Square equal thereunto. So if a Tree, being girt about, as abovesaid, should contain in circumference 47 Inches 13 parts: If you extend the Compasses from 1000 to 47 Inches 13 parts, the same extent will reach from 2821, to 13 Inches 29 parts, which is equal to the side of a Square equal to that Tree, which being obtained, the Tree may be measured divers ways, according to the Examples in the last Chapter. CHAP. XVIII. Concerning the measuring of Regular Solids, as Cylinders, Globes, Cones, and such like. I. Of the CYLINDER. A Cylinder is a round Figure, of equal circumference in all parts thereof, as a standing Pillar, a Rowling-stone for Garden-walks, etc. To measure such a Figure there are several ways, both by having the circumference given when it is standing, or by having the diameter at the end thereof when it is lying, or by having the side of a Square equal to the base thereof. I. By having the Diameter given. Example 1. The Diameter being 15 Inches, how much in length makes a Foot? As 15 the diameter, to 46.90; So is 1, to a fourth. And that fourth, to 9.78, the length of a Foot. Extend the Compasses from 15 the diameter, to 46.90; that extent will reach from 1, to another point upon the Line, and from thence to 9 Inches 78 parts, the length of a Foot solid. Example 2. The diameter being 1 Foot 25 parts, how much in length makes a Foot in Foot-measure? As 1.25, the diameter in Feet, unto 1.128; So 1, to a fourth number, And that, to 8.14, the length of a Foot solid in Foot-measure. Extend the Compasses from 1.25 the diameter, to 1.128; the same will reach from 1, to some other number, and from thence to 1 Foot 128 parts of a Foot. Example 3. Having the diameter 15 Inches, and the length 105 Inches, how many solid Inches doth the Cylinder Contain? As 1.128, to 15 Inches the diameter; So is 105 Inches, the length, to a fourth number, And that, to 18555.34 Inches, the content. Extend the Compasses from 1.128, to 15 the length; the fame extent will reach from 105 the length, to some other number, and from thence to 18555.34 Inches, the content of the Cylinder in Inches. Example 4. Having the diameter 1 Foot 25 parts, and the length 8 Foot 75 parts, to find the content in Feet. As 1.128, to 1.25, the diameter; So 8.75, the length, to a fourth, And that fourth, to 10.74 Foot, the content. Extend the Compasses from 1.128, to 1.25 the diameter; that extent will reach from 8.75 the length, to some other number, and from that to 10 Foot 74 parts, the content. Example 5. Having the diameter 15 Inches, and the length 105 Inches, how many Foot doth it contain? As 46.90, to 15 Inches, the diameter; So is 105 Inches, the length, to a fourth, And that fourth, to 10 Foot 74 parts, the content. Extend the Compasses from 46.90, to 15 the diameter; that extent will reach from 105, the length, to another number, and from that to 10 Foot 74 parts, the content. Example 6. The diameter being 15 Inches, and the length 8 Foot 75 parts, how many Foot doth it contain? As 13.54, to 15 Inches, the diameter; So 8.75 Foot the length, to a fourth, And that fourth, to 10.74, the length in Feet. Extend the Compesses from 3.54, to 15 the length, that extent will reach from 8.75 the length, to another number, and from thence to 10.74 Foot, the content in Feet. II. By having the Circumference given. Example 1. The Circumference of a Cylinder is 47 Inches 13 parts, how much thereof in length shall make a Foot solid. As 47.13 Inches, the circumference, to 147.36; So 1, to a fourth number, And that, to 9.78 Inches, the length of a Foot. Extend the Compasses from 47.13, the circumference, to 147.36; that extent will reach from 1, to a fourth number, and from thence to 9 Inches 78 parts, the length of a Foot solid. Example 2. Having the circumference of a Cylinder 3 Foot 927 parts, to find the length of a Foot solid thereof in Foot-measure. As 3.927 Foot, to 3.545; So 1, to a fourth number, And that, to 815 parts of a Foot, the length. Extend the Compasses from 3.927, the circumference, to 3.545; that extent will reach from 1, to some other number, and from thence to 815 parts of a Foot, for the length of a solid Foot of that Cylinder. Example 3. The circumference of a Cylinder being 47 Inches 13 parts, and the length thereof 105 Inches, how many Inches is there in such a Cylinder? As 3.545, to 47.13, the circumference; So 105 Inches, the length, to a fourth number, And that, to 18555, the content in Inches. Extend the Compasses from 3.545, to 47.13 the circumference; that extent will reach from 105 the length, to another number, and from thence to 18555, the number of solid Inches in the Cylinder. Example 4. The circumference being 47 Inches 13 parts, and the length 105 Inches (as before); how many solid Foot in that Cylinder? As 147.36, to 47.13 Inches, the circumfer. So 105 Inches, the length, to a fourth number, And that, to 10 Foot 74 parts, the content. Extend the Compasses from 147.36, to 47.13 the circumference, that extent will reach from 105 the length, to another number, and from that to 10 Foot 74 parts of a Foot, the solid content. Example 5. Let the length of the Cylinder be 8 Foot 75 parts, and the circumference 3 Foot 927 parts; how many Foot doth it contain? As 3.545, to 3.927 Foot, the circumference; So 8.75 Foot, the length, to a fourth number, And that, to 10 Foot 74 parts, the content. Extend the Compasses from 3.545, to 3.927; the same extent will reach from 8.75 the length, to 10.74, the content in Feet. Example 6. Let the circumference given be 47 Inches 13 parts, and the length 8 Foot 75 parts, how many solid Foot doth the Cylinder contain? As 42.54, to 47.13 Inches the circumference; So is 8.75 Foot, the length, to a fourth, And that fourth, to 10.74 Foot, the content. Extend the Compasses from 42.54, to 47.13 the circumference; that extent will reach from 8.75 the length, to another number, and from thence to 10 Foot 74 parts, the content of the Cylinder in solid Feet. III. By having the Side of a Square, equal to the Base or End of a Cylinder. Example. Let the Side of a Square, equal to the Base or End of the Cylinder, be 13 Inches 29 parts, and the length thereof 105 Inches; how many square Feet are contained in that Cylinder? As 41.57, to 13.29 Inches, the side of the Square; So is 105, the length in Inches, to a fourth number, And that, to 10 Foot 47 parts, the content of the Cylinder in Feet and parts. Extend the Compasses from 41.57. to 13.29 Inches, the side of a Square equal to the base of the Cylinder; that extent will reach from 105 Inches, the length, to another number, and from thence to 10 Foot 47 parts, the content of the Cylinder in Feet. II. Of the CONE. A Cone is a round Figure, having for the base thereof a Circle, the side whereof riseth from the circumference of the Circle round about the same equally, till it meet in a point just over the centre of the Circle, and is in the form of a Spire-Steeple: And it is thus measured. Example 1. Let there be a Cone the diameter of whose Base is 10 Inches, and whose height is 12 Inches; I would know how many solid or cubical Inches are contained therein. The diameter being 10 the content of the Circle or Base will be found to the 78 Inches 54 parts; as by the fifth Example in Chap. 13. of this Book. The Area of the Base being thus found, the proportion is, As 3, to 78.54 Inches, the content of the base; So is 14 Inches; the height, to 314 Inches 16 parts of an Inch, for the content of the Cone in Inches. Extend the Compasses from 3, to 78.54 the base; that extent will reach from 12 the height, to 314 Inches 16 parts, the content of the Cone in solid Inches. Example 2. Let the diameter of the Base be 12 Inches, as before, and the length of the side he 13 Inches; kow many solid Inches is there in this Cone? 1. Extend the Compasses from 1, to 5 Inches, half the diameter of the Base, that extent will reach from 5 to 25. 2. Extend the Compasses from 1, to 13 the length of the side; that extent will reach from 13, to 169. 3. From this 169, take the 25 before found, and there remains 144. 4. Upon your Line take half the distance between 1 and 144, and you shall find it to be 12, which 12 is the height of the Cone: So the height being had, you may find the content, as in the last Example. III. Of SPHERICAL BODIES. A Spherical Body, is such a Body, whose Superficies in all the parts of it are equally distant from the centre of the body; as Globes, Bullets, etc. Example 1. The Circumference of a Globe or Bullet being 28 Inches 28 parts, to find the length of the diameter. As 22, to 7; So is 28.28, the circumference, to 9 Inches, the diameter. Extend the Compasses from 22, to 7; the same extent will reach from 28.28 the circumference, to 9 Inches, the length of the diameter of that Bullet. Example 2. The diameter of a Spherical Body being given is 9 Inches, and its circumference is 28 Inches 28 parts, how many square Inches is there in the superficies of that Spherical Body? As 1, is to 9 Inches, the diameter; So is 28.28 Inches, the circumference, to 254.5 Inches, the superficial content. Extend the Compasses from 1, to 9 the diameter; the same extent will reach from 28.28, the circumference, to 254 Inches 5 parts, the superficial Inches in this Spherical Body. Example 3. The diameter of a Spherical Body being 9 Inches, how many solid Inches are therein contained? 1. As 1, is to 9 the diameter; See is 9, to a fourth number, And that fourth number, to 729, the cube of the diameter. 2. As 9, the diameter, to 729, its cube; So is 11, to 891 Inches, the solid content of the Spherical Body. Extend the Compasses from 1 to 9, that extent will reach to 81, and from 81, to 729, the cube of the diameter.— Then, Extend the Compasses from 9, the diameter, to 729 its Cube; that extent will reach from 11, to 891 Inches, the solid content of the Spherical Body. I might here add the manner how to measure other kind of Bodies, both Regular and Irregular; as Ellipses, Parabolas, etc. Also of Prismes, Scalene Cones, Spheroiades, etc. But these being out of the reach of ordinary Artificers, for whose sakes this Treatise was chief composed, I shall here conclude this Treatise of the Use of the Line of Proportion, with a short Supplement of Gauging of Vessels. CHAP. XIX. Concerning the Gauging of Vessels By the Line. BEfore you can measure your Vessel, to find the content thereof in Gallons or parts, you must find the content thereof in solid Inches; and to effect this, you must find the content of two third parts of a Circle, agreeable to the diameter at the Bung; and one third part of another Circle, agreeable to that of the diameter at the Head; these two added together, and multiplied by the length of the Vessel, that product will be the content of that Vessel in Inches. Example. Let there be a Vessel whose diameter at Head, 18 diameter at Bung, 32 length is 40 And let the content thereof, first in Inches, and then in Gallons, be required. I. For the two third parts of the Circle at the Bungler As 1, to this universal number [5236]; So 1024, the square of the diameter at the Bung 32, to 536.166 Inches, which is two third parts of the content of the Circle at the Bungler Wherefore, Extend the Compasses from 1, to 5236; the same extent will reach from 1024, (the square of 32 the diameter at the Bung) to 536.166 inches, the content of two third parts of the circle at the Bungler II. For one third part of the Circle at the Head. As 1, to this general number [2618]; So is 324, the square of the diameter at head 18, to 84.823 Inches, which is one third part of the content of the circle at the head. Wherefore, Extend the Compasses from 1, to 2618; the same extent will reach from 324 (the square of 18, the diameter at the Head) to 84.823 Inches, the content of one third part of the diameter at the head. III. For the number of Square Inches in the Vessel. Add these two numbers 536.166 and 84.823 They make 620.989 40 Which multiplied by 40, the length of the Vessel, produceth 24839.560 And so many Square Inches are contained in such a Vessel, whose diameter at the Head is 18 Inches, at the Bung 32 Inches, and is 40 Inches long. iv For the Content in Wine or Ale Gallons. Divide this number 24839.56. by— 231 for Wine, 282 for Ale, and the Quotients shall tell you the number of Gallons and parts of a Gallon. Wine. gall. parts. 231) 24839.56 (107.52. 231 1739 1617. 1225 1155 706 693 13 Ale. gall. parts. 282) 24839.56 (88.08 2256 2279 2256 2356 2256 100 By this work you may perceive that this Vessel containeth 107 Gallons 53 parts of Wine measure. 88 Gallons 08 parts of Ale measure. How to multiply and divide by the Line, is taught in the Second and Third Chapters of this Book, and therefore were needless here to repeat it again: But I chose rather to do it Arithmetically, for the better illustration, and for the satisfaction of such as have a delight in Numbers. How to Measure Board, Glass, Timber, Stone, etc. By a Line of Equal Parts, drawn from the Centre of a Two-foot joint-rule. ALL Proportions that may be wrought upon a strait Ruler by the Line of Proportion or Numbers, the same may be wrought by a Line of Equal Parts, drawn from the Centre of an opening Joint. And whereas this Line of Equal Parts is numbered from the Centre of the Rule towards the end thereof, by 1, 2, 3, 4, etc. to 10; that these Figures (as in the other Line) do sometimes signify themselves only; sometimes 1, 2, 3, etc. do signify 10, 20, 30, etc. sometimes 100, 200, 300, etc. according to the quality of the Question propounded. By this Line you may also Multiply, divide, work the Rule of Proportion, and perform divers things which the Line of Numbers performeth, and some others which that will not; but I shall here only show you how Board, Glass, Timber, Stone, etc. may be thereby measured; which I shall do in these following Propositions. And, I. For SUPERFICIAL MEASURE, as Board, Glass, etc. I. In INCH-MEASURE. PROP. I. A Plank being 27 Inches broad, and 263 Inches long, how many Square Inches are contained therein? As I, is to 27: So is 263, to 7101. Take in your Compasses the distance from the Centre, to 27 (the breadth) upon your Line of Equal parts; with this distance set one foot in 10 at the end of the Line, and open the Rule till the other foot fall in 10 on the other Leg of the Rule. The Rule thus standing, take with your Compasses the distance between 263 on one Leg of the Rule, to 263 on the other Leg, this distance will reach from the Centre of the Rule to 7101; and so many square Inches are in that piece. PROP. 2. If a Board, or Plank, or piece of Pavement, or of Glass, be 20 Inches broad, how much thereof in length shall make a Foot square? As 20, is to 144; So 1, to 7.2. Take 144 out of your Line of Equal parts, from the Centre, and setting one foot in 20, open the other Leg till the other Compass point fall in 20 also. The Rule thus standing, take the distance between 10 and 10, and that distance will reach from the centre of the Rule to 7 Inches 2/10 parts of an Inch; and so much in length will make a Foot Square. II. In FOOT-MEASURE. PROP. 3. A Room is 52 Foot broad, and 110.5 Foot long; how many Square Foot are there in that Room? As 52, is to 1; So is 110.5, to 5746. Take in your Compasses 52 the breadth; with this distance open the Ruler in 10, and 10; it so resting, take the distance between 110.5 and 110.5 on either side, that distance applied to the Centre of the Rule, will reach to 5746; and so many Square Foot is in that Room. PROP. 4. A Plank being 2 Foot 25 parts broad, how much in length thereof shall make a Foot Square? As 2.25, the breadth, is to 1 or 10; So is 10, to 44, the length of a Foot. Take in your Compasses the distance from the Centre of your Rule to 1; then set one foot in 2.25, and open the other Leg till the other Compass point fall in 2.25 on the other side; the Rule thus standing, take the distance between 10 and 10; that distance applied from the Centre of the Rule, will reach to 44 parts of a Foot; and so much in length will make a Foot. III. In YARD-MEASURE. PROP. 5. A Room is hung with Tapestry, containing 130 Yards 25 parts in compass, and in depth 5 Yards 20 parts; how many Yards of Tapestry is in that Room? As 1, to 5.20; So 130.25, to 677.4. Take 5.20 in your Compasses, and that distance put over in 10 and 10; the Rule thus standing, take the distance between 130.25 and 130.25 on each Leg of the Rule, that distance will reach from the Centre of the Rule, to 677 Yards 4 tenths of a Yard. II. For SOLID-MEASURE, as Timber, Stone, etc. By the Line of Equal parts. I. In INCH-MEASURE. PROP. I. A piece of Timber being 30 Inches broad, 21 Inches 6 parts deep, and 183 Inches long; how many Foot is contained in that piece of Timber? 1. As 1, is to 30; So is 21.6, to 648. Take the distance from the Centre to 30; then set one foot in 10, and open the Rule till the other Compass point fall in 10 on the other Leg of the Rule: Then take the distance between 21.6 and 21.6, that distance will reach from the Centre of the Rule, to 648, the content of the base or end of the piece of Timber in Inches: Then, 2. As 1728, the number of Inches in a Foot solid, is to 648, the content of the base; So is 183 Inches, the length, to 68 Foot 62 parts, the content in Feet. Take in your Compasses the distance from the Centre to 1728, with this distance set one foot in 648, and open the other Leg of the Rule till the other point of the Compasses fall in 648 on the other Leg; then take in your Compasses the distance from the Centre to 183; with this distance move both points of the Compasses gently along on both the Lines on either side the Rule, till the Compass points rest upon one and the same number on either Leg, which you shall here find them to do at 68: 62 parts; so the piece containeth 68 Foot and 62/100 parts of a Foot. This kind of work may seem troublesome at first; but little practice will render it easy. Note. If you take the first number of your proportion from the centre of your Rule, you must take your third number thence also, and then will your number sought be found as this here in this Example. But if you take your first number cross the Rule, than your third number must be so taken also, and your number sought must be taken from the centre, as those before were. PROP. 2. If a Stone be 30 Inches broad, and 21 Inches 6 parts deep; how much in length of that Stone will make a Foot square? You must first find the content of the base, as in the first Proposition, and it will be 648 Inches: Then, As 648, the content of the base, is to 1728, the Inches in a solid Foot; So is 1, to 2.67 parts. Take 1728 in your Compasses from the Centre; with that extent open the Rule from 648, to 648: The Rule so resting, take the distance between 10, and 10; that distance applied to the Line from the Centre, shall reach to 2 Inches 67 parts; and so much in length will make a Foot solid of that Stone or piece of Timber. II. In FOOT-MEASURE. PROP. 3. If a Stone or piece of Timber be 2 Foot 50 parts broad, 1 Foot 80 parts deep, and 15 Foot 25 parts long; how many solid Feet doth that piece contain? 1. As I, is to 2.50, the breadth; So is 1.80, the depth, to 4.50, the content of the base in Feet. Take 2.50 in your Compasses from the Centre; with that extent open the Rule in 10 and 10, then take the distance between 1.80 and 1.80, that extent will reach from the Centre of the Rule, to 4 Foot 50 parts, the content of the base. 2. As I, to 4.50, the base; So 15.25, the length, to 68.62, the content in Feet. Take 4.50 in your Compasses, and thereto open the Rule from 10 to 10; then take the distance between 15.25 and 15.25, that distance will reach from the Centre of the Rule, to 68 Foot 62 parts, the content of the Stone. PROP. 4. The breadth being 2 Foot 50 parts, the depth 1 Foot 80 parts; how much in length thereof will make a solid Foot? You may find the quantity or content of the base by the first of the last Proposition to be 4 Foot 50 parts: Then, As 4.50, the base, is to 1; So is 10, or one Foot, to 222 parts. Open the Compasses from the Centre to 1; then setting one foot in 4.50, open the other Leg till the Compass point falleth in 4.50 on the other Leg; then take the distance between 10 and 10, and that will reach from the Centre to 222: and so many parts of a Foot will make a solid Foot of that piece of Stone or Timber. PROP. 5. To divide a Right Line into any number of Equal Parts, at the first opening of the Compass. Let a Line be given you to be divided into 6 equal parts: Take the length of the Line given in your Compasses: then because it is to be divided into 6 parts, put one Foot in 6 on one Leg, and open the other Leg till the other point fall on 6 on the other Leg: The Rule thus standing, take the distance between 1 and 1, that distance shall divide your given Line into 6 equal parts: The like for any other number of parts whatsoever. Many other Conclusions may be done by this Line; but I shall reserve them, and divers other Conclusions of the like nature, to a more convenient place. The Use of the LINE of PROPORTION IMPROVED. By which Board, Glass, Land, Wainscot, Hang, Pavement, Brickwork, Tyling, Plastering, and any other Superficial; As also, Stone, Timber, and other Solid Measure; may be Measured thereby, without the Use of Pen, Ink, Paper, Compasses, or other Motion (as sliding, or the like) whatsoever, by Inspection, only by looking upon the Line. The ARGUMENT. I Am not ignorant how many have written of the Use of this Line of Proportion since the invention of Logarithms, from which Tables this Line is constitnted and made; as namely, After Mr. Gunter's first contrivance, Mr. Wingate seconded him in making divers Lines to several Radiuses, thereby to bring it to Extract the Square and Cube Roots without doubling or trebling, or dividing the distance into two or three parts; but this made a great number of lines to small purpose; for nothing here could be done without the help of the Compasses. Again, One T. Browne, a Maker of Mathematical Instruments, made it in a Serpentine or Spiral Line, composed of divers Consentrick Circles, thereby to enlarge the divisions; which was the contrivance of one Mr.— Milburn a Yorkshire Gentleman, who writ thereof, and communicated his Uses to the aforesaid Brown, who (since his death) attributed it to himself: But whoever was the contriver of it, it is not without inconvemence; for it can in no wise be made portable; and besides (instead of Compasses) an opening Joint with thirds must be placed to move upon the centre of the Instrument, without which no proportion can be wrought. There is yet a third way contrived, by which this Line is made very serviceable, and convenient both for use and carriage, and is to be used without Compasses, and is composed of two Lines of one length upon either side of two Rulers, to slide one by the side of the other; the uses whereof in the measuring of Board, Glass, Timber, Stone, etc. and in other parts of Geometry, Astronomy, Fortification, Trigonometry, Geography, Navigation, Gauging, Dialling, etc. together with the uses of the Lines of Artificial Sins and Tangents, in the same manner contrived, all upon one Ruler, are largely written upon by Mr Seth Partridge, in a Book of his lately published, entitled, The Description and Use of the Double Scale of Proportion; which Book and Instrument are both sold by Mr. Walter Hayes, at the Sign of the Cross-Daggers in moorfield's, near the Popes-head Tavern. There is yet another way of disposing of this Line of Proportion, by having one Line of the full length of the Ruler, and another Line of the same Radius broken in two parts between 3 and 4; so that in working your Compasses never go off of the Line: This is one of the best contrivances, but here Compasses must be used. These are all the Contrivances that I have hitherto seen of these Lines: That which I here speak of, and will show how to use, is only two. Lines of one and the same Radius, being set upon a plain Ruler of any length (the larger the better) having the beginning of one Line, at the end of the other, the divisions of each Line being set so close together, that if you find any number upon one of the Lines, you may easily see what number stands against it in the other Line. This is all the Variation: and what this easy contrivance will effect, will appear by the uses following. The Lines are the same with the Line of Proportion or Numbers, mentioned and treated of in the former part of this Book; and therefore how to number upon them is showed in the first Chapter of this Book, and therefore needs not here again be repeated. Also Multiplication, Division, the Golden Rule, Duplicated and Triplicated Proportion, the Extraction of Roots, etc. delivered in the second, third, fourth, fifth Chapters, etc. as also in Measuring of Superficies and Solids, and the mensuration of other Figures treated of through the whole Book, these Lines thus disposed will effect with Compasses: But some of those Uses which they will effect in measuring without the help of Compasses, I will here show. CAUTION. What measure soever you measure by, let the Integer or grand Measure be divided into 10 or 100 parts (it matters not of what length your Lines of Proportion be, for to them all measures are alike). Thus, If you measure any thing by the Foot, let your Foot be divided into 100 parts: If by the Yard, divide your Yard into 100 parts: If by the Ell, divide that into 100 parts. So likewise, if by the Perch, Square, etc. or by what Measure soever, let the grand Measure (as I said before) be divided into 100 parts. CHAP. I. OF SUPERFICIAL MEASURE, BY Superficial Measure is meant all kind of flat Measure, such as is Board, Glass, Pavement, Hang, Plastering, Tyling, Land-measure, etc. And these several things are measured by distinct Measures, as some by the Foot, others by the Yard, others again by the Ell, some by the Rod, and some by the Square: of all which I shall give Examples: and, I. Of FOOT-MEASURE. Example 1. If a Board be 2 Foot 64 parts broad, how much in length of that Board will make a Foot Square? Look upon one of your Lines (it matters not which) for 1 Foot 64 parts; and right against it on the other Line you shall find 61: and so many parts of a Foot will make a Foot square of that Board. Example 2, A Plank is 3 Foot 50 parts broad, how much thereof in length will make a Foot? Find 3 Foot 50 parts upon one Line, and right against it on the other Line you shall find 28 parts and 4/7 or something more that half a part. Example 3. If a Board be 75 parts of a Foot broad, how much thereof in length shall make a Foot square? Look upon one of your Lines for 75, and right against it you shall find 1 Foot, and 33 parts; and so much in length makes a square Foot. Note. If the breadth of any thing given be more than one Foot, than the length of a Foot square must be less than a Foot, as in the two first Examples it was: But if the breadth given be less than a Foot, (as in this last Example) than the length of a Foot square must be more than a Foot. Example 4. A Pain of Glass is 35 parts broad, how much in length makes a Foot? Find 35 in one Line, against it you shall find 2 Foot 854/7 parts; and so much in length makes a square Foot. Example 5. A Pain of Glass is 3 Foot broad, how much in length makes a Foot? Find 3 Foot in one Line, against it in the other you shall find 33 ⅓ parts, and so much in length makes a Foot square. Example 6. If a piece of Glass be 1 Foot 98 parts broad, how much in length will make a Foot? Look 1 Foot 98 parts in one Line, and against it in the other you will find 5 Foot and half a part, and so much in length makes a Foot. II. Of YARD-MEASURE. Example 1. A Gallery is Wainscoted 2 Yards 56 parts deep, how much of that length will make a Yard square? Seek 2 Yards 56 parts in one Line, and against it in the other you shall find 39 parts and somewhat more; and so many parts of a Yard will make a Yard square. Example 2. A Room is Wainscoted 1 Yard 13 parts high; how much in length thereof will make a Yard square? Look one Yard 13 parts in one Line, against it in the other you will find 88 parts, and above half a part; and so much in length makes a Yard square. Example 3. If the Frieze about a Room be 62 parts of a Yard broad, how much in length thereof will make a Yard square? Find 62 parts in one of your Lines, and against it in the other you shall find 1 Yard 61 parts, and somewhat more; and so much in length makes a Foot square. Example 4. There is a Gallery paved with Marble, being 5 Yards 70 parts broad; how much of that in length will make a Yard square? Seek 5 Yards 70 parts in one Line, and against it in the other you shall find 17 parts and an half; and so much in length of that Pavement will make a Yard square. Example 5. A Parlour being 7 Yards 29 parts broad, hath a Ceiling of Fretwork plastered, how much of that breadth will make a Yard Square? Find 7 Yards 29 parts in one of your Lines, and right against it in the other Line you shall find 13 parts and 7/10, which is above half a part: So that 13 parts and a little more than half a part, will make a Yard square of that Ceiling. Example 6. A Plasterer hath Rendered the inside of a Wall containing 2 Yards 36 parts in height; how much of that will make a Yard square? Find 2 Yards 36 parts in one of your Lines, and right against it on the other you shall find 42 parts 3/10 of a part, that is something more than one third part of a part; and so much in length makes a Yard Square. III. Of MEASURE by the ELL. Example 1. There is a Room hung with Tapestry, which is 4 els 25 parts high; how much Tapestry in length will make an Ell square? Note. Here by els we understand Flemish els (for by that Measure are Hang sold); which Ell contains three quarters of our Yard; that is, 75 parts of our Yard: So that if an Upholster have his Flemish Ell divided into 100 parts, he may measure his Hang as in the Examples following is showed. Here because the Hang are 4 els 25 parts deep, Look for 4 els 25 parts in one of your Lines, right against which in the other you shall find 23 parts and a half; and so many parts of his Ell will make a Flemish Ell square. Example 2. T●● Embroidery of a pair of Vallens about a Bed is 28 parts of a Flemish Ell deep; how much of that Embroidery in length will make a Flemish Ell square? Look for 28 parts in one of your Lines, and against it in the other Line you shall find 3 els and 57 parts of an Ell; and so much in length will make an Ell square. Example 3. A Gallery being 3 els 98 parts deep, is hung with Arras; how much of that depth will make an Ell square? Seek 3 els 98 parts in one Line against which in the other you shall find 25 yarts and 1/10 of a part; and so much in length will make an Ell square. iv Of MEASURE by the ROD. Example 1. There is a Brickwall which is 75 parts of a Rod high, how much in length of that Wall will make a Rod square? Note, That all Wall-work is by the Bricklayers measured by the Rod, which contains 16 Foot and a half in length: Wherefore, let his Rod, being 16 foot and an half in length, be divided into 100 equal parts, and then let him work as followeth. The Wall being 75 parts of a Rod high, Look for 75 parts in one Line, and in the other Line right against 75 you shall find 1 Rod 33 parts of a Rod; and so much of that Wall in length is contained in a Square Rod. Example 2. A Carpenter hath Railed and Paled in a Garden with Pales 52 parts of a Rod high, how much of that Paling shall make a Rod square? Seek 52 parts in one Line, against it in the other Line you shall find 1 Rod 92 parts; and so much in length will make a Rod square of that Paling. Example 3. A Bricklayer hath made a Shower to carry Water, the Bottom, Sides, and Arch together contains 1 Rod 64 parts; how much of that Drain or Shewer makes a Square Rod? Find 1 Rod 64 parts in one of your Lines, and right against that number you shall find in the other Line almost 61 parts; and so many parts of a Rod in length will make a Rod square. And here note, That though I have here put these two last Examples, that Paling is not measured by the Square Rod, but (let the height thereof be what it will) it is measured by the Rod in length. In like manner is Hedging, Ditching, and many other things that are measured by the Rod. Example 4. If a piece of Land be 2 Rod 31 parts broad, how much in length thereof shall make a Rod square? Seek 2 Rods 31 parts upon one of your Lines, and over against it upon the other Line you shall find 42 parts and about ⅓ of a part; and so much in length makes a square Rod. Example 5. A piece of Land being 80 parts of a Rod broad, how much thereof in length shall make a Rod square? Look for 80 parts in one Line, and in the other Line opposite thereunto you shall find 1 Rod 22 parts, and so much in length makes a Rod square. V Of MEASURING by the SQUARE. There are two things principally which are measured by the Square, and they are Tyling of Houses, and Flooring of Rooms; and in this reckoning they account a Square to be 10 Foot every way: So that for this kind of Measure divide a Line or Rod of Ten Foot long into 100 parts, and it is fit for the purpose. Exampler 1. A Barn, the breadth of the Tyling whereof on both sides is 1 Square 30 parts, how much in length of that Tyling will make a Square Square? Find 1 Square 30 parts upon one of your Lines, and right against it on the other Line you shall find 77 parts almost, and so much in length of that Tyling will make a Square Square. Example 2. The Tyling of a house is 76 parts of a Square broad, how much in length thereof will make a Square Square? Seek 76 parts in one Line, and against it in the other you shall find 1 Square 31 parts and an half almost; and so much in length will make a Square Square, that is, 10 Foot every way, in all 100 Foot. CHAP. II. OF SOLID MEASURE. BY Solid Measure is meant such Measure as hath Length, Breadth, and Thickness, such as Timber, Stone, or the like. But before Timber or Stone can be measured, you must find the content of the square of the base thereof, which is taught by the Problem at the end of the Tenth Chapter: but that being performed by Compasses, I will here show how it may be (by these Lines thus disposed) performed without; and that shall be my first Proposition or Example. Example 1. Let a piece of Timber or a Stone be 80 parts of a Foot deep, and 3 Foot 75 parts broad; how much in length of that piece will make a Foot square? Here (by any of the former Rules of Superficial Measure) find at 80 parts broad how much in length will make a Foot, which you shall find to be 1 Foot 25 parts: For, If you find 80 parts, the depth of the piece, in one Line, against it in the other you shall find 1 Foot 25 parts. Take 1 Foot 25 parts of your Foot Rule, and measure it along the breadth of the piece, which is 3 Foot 75 parts, and see how often it is contained therein, which you shall find to be three times; wherefore, you may conclude, that the square of the base of that piece of Timber whose depth is 80 parts, and whose breadth is 3 Foot 75 parts, is just 3 Foot. Now the square of the base of the piece being thus obtained, you may find the length of a Foot square thereof in this manner. Example 2. Let the Square of the base of a piece of Timber or Stone be 3 Foot, how much in length of that piece will make a Foot solid? Look for 3 Foot in one of your Lines, and in the other right against it you shall find 33 parts and ⅓ part of a part; and so much in length will make a Foot solid. Example 3. Let a piece of Stone or Timber be 2 Foot 50 parts broad, and 50 parts deep; how much of that Stone in length shall make a solid Foot? By any of the ways before prescribed you shall find that the depth of your Stone being 50 parts, it will require 2 Foot in length thereof to make a Foot square: Wherefore, measure how often you can find 2 Foot in the breadth of your Solid, which you may find only once, and 50 parts more, which is one quarter of two Foot: Wherefore, the Square of this Solid contains 1 Foot 25 parts. Wherefore, Look in one of your Lines for 1 Foot 25 parts, and right against it you shall find 80 parts; and so much in length will make a Foot solid. Example 4. The Square of the Base of any regular Solid being given, together with the length of the same Solid, to find how many solid Feet are contained in the same. Let the forementioned Solid serve for this Example also, whose length was 35 Foot: We found that the Square of the Base was 1 Foot 25 parts, and that 80 parts in length would make one solid Foot: Wherefore, take 80 parts of your Rule, and run it along the piece so often as you can, which you shall find to be 43 and 60 parts, which is just three Quarters; so that in this piece of Timber there is 43 Foot and three Quarters. I might add many more Examples of this kind, and some to other purposes; but these are sufficient for the purpose intended; and so I shall conclude this Treatise, leaving the farther practice thereof to yourself: For, Usus optimus Magister. CHAP. III. OF CIRCULAR MEASURE. By having either the Circumference or Diameter of any Circle given, thereby to find the Side of a Square Equal to the same Circle; or the Side of a Square that may be inscribed within the same Circle. IN the Thirteenth Chapter of this Book you have six Examples by having the Circumference or Diameter of any Circle given, thereby to find the Side of a Square equal, the Superficial Content, etc. But I have seen upon some Two-Foot-Rules Lines to effect this thing by only opening the Compasses to the distance given upon one Line, and applying the same to some of the other Scales. One of those Scales is noted at the end thereof with C, signifying the Circumference of any Circle; the other with D, signifying the Diameter; the other with S, E, signifying Square Equal to the Circie; the other with S. W, signifying Square within. Example. So that if you should have given you the Diameter of a Circle being 15 Inches, Out of the Line noted with D take 15 Inches, apply that distance to the Line noted with C, it will reach to 47 Inches and 13/100 parts of an Inch; and so much is the Circumference of that Circle. Again, The Diameter being 15 Inches, as before, take that distance out of the Line D, and it will reach upon the Line S E, to 13 Inches 29/100 parts; and that shall be the Side of a Square equal to the Circle whose Diameter is 15 Inches. Again, The Diameter being 15 Inches, if you take that distance out of the line noted with D, it will reach upon the Line S W, to 10 Inches 60/1 parts of an Inch; and that is the length of the Side of the greatest Square that can be drawn within that Circle whose Diameter is 15 Inches. The like may be done if the Circumference were given, by taking the Circumference thereof out of the Line noted with C, and applying it to the other Scales. This I thought convenient to add here, because sometimes these Lines are put upon Two-foot Rules. FINIS.