64 0 0 0  3 47 0 2 4 2 2 36 0 0 0  1 28 4 3 3  2 23 0 4 1  3 19 0 3 1 3 3 16 0 0 0  1 13 7 5 9  2 11 9 0 6  3 10 1 8 8 4 4 9 0 0 0  1 7 11 6 6  2 7 1 3 3  3 6 4 5 9 5 5 5 9 1 2  1 5 2 6 9  2 4 9 1 2  3 4 4 2 6 6 6 4 0 0 0  1 3 8 2 3  2 3 4 9 0  3 3 1 9 3 7 7 2 11 2 8  1 2 8 8 6  2 2 6 7 2  3 2 4 7 7 8 8 2 3 0 0 8¼  2 1 3 9 A Table for the under Board-measure to Inches and quarters  fe in 10. 100  48 0 0 0  24 0 0 0  16 0 0 0 1 12 0 0 0  9 7 2 0  8 0 0 0  6 10 2 9 2 6 0 0 0  5 4 0 0  4 9 6 0  4 4 3 6 3 4 0 0 0  3 8 3 0  3 5 1 4  3 2 4 0 4 3 0 0 0  2 9 8 8  2 8 0 0  2 6 3 1 5 2 4 8 0  2 3 4 2  2 2 1 8  2 1 0 4 6 2 0 0 0  1 11 0 5  1 10 1 5  1 9 3 3 7 1 8 5 8  1 7 8 6  1 7 2 0  1 6 5 8 8 1 6 0 0 8¼ 1 5 4 5 Note also that this Table or any smaller part of under-measure may be supplyed by the divisions of the board and timber-measure only as thus Double the inches and parts of breadth for board-measure or of squares for timber-measure and seek it in the Lines of board or timber-measure and count twice from thence to the rules end for board or 4 times for timber and that shall be the true length that makes a foot of board or timber Ex. 4. At 4 inches and ½ square or broad 4½ doubled is 9. then look for 9 on the board-measure and two times from thence to the end shall make a foot of board Or look for 9 on the Line of timber-measure and 4 times from thence to the end of the Rule shall be the true length to make a foot of timber at 4 inches ½ square But if it be so small a piece that when it is doubled the number is not on the divided part of the rule then double it again and count 4 times for board-measure and 16 times for timber Ex. 5. At 2 inches and a half a quarter broad or square that doubled is 4¼ which is not on the rule therefore I double it again saying 4 ¼ and 4¼ is 8½ which is on the rule then for board count 4 times from 8½ on the board-measure to the upper end by 36 to make a foot of board at 2â…› broad And for timber count 16 times from 8½ near the beginning of timber-measure which will be near 32 foot to make a foot of timber at 2â— square But if twice doubling will not do then double again and count 8 times for board and 64 times for timber as in the Table you may see which will be very slender timber Also between the two lines of Inches is set four scales of equal parts called Circumference Diameter Square-equal and Square-within Whose Use may be thus The Diameter of any circle being given to find the circumference or the side of a Square-equal oâ— the side of the square within Example suppose the Diameter of a circle be 15 inches Take 15 from the scale called Diameter and measure it in the scale called circumference and iâ— gives 47. 10. Also the same extent measured in the line called square-within and it gives 10. 55. For the side of a square-within in that circle of 51 inches Diameter Again the same extent being measur'd in the scale call'd square-equal and it gives 13. 27 for the side of â— square equal to a circle of 15 inches Diameter Lastly this 13. 27 the side of the square-equall multiplyed by it selâ— gives 176 the Area of that Circle in Inches whose Diameter is 15 Inches The same may be done if the Circumference be first given then that taken first from that Line and measured in the other Lines you shall have the respective Answers as before But if the Area be first given then to find the Square-equal find the Square-root of the Area and that root shall be the side of the Square-equal Example Suppose the Area of a Circle be 288 what is the side of the Square-equal The middle between 1 and 288 is at near 17 the side of the Square-equal for 17 squared is 289. Then 17 taken from the Scale call'd square equal gives you any of the rest These Scales serve to draw any Platform of a House or Field very convenlently being of several bignesses The Description and Use of the Line of Numbers commonly called Gunter's Line CHAP. II. The definition and description of the Line of Numbers and Numeration thereon THE Line of Numbers is only the Logarithmes contrived on a Ruler and the several ranks of figures in the Logarithms are here express'd by short and longer and longest division and they are so contrived in proportion one to another that as the Logarithmes by adding together and substracting one from another produce the quesita so here by turning a pair of Compasses forward or backward according to due order from one point to another doth also bring out the quesita in like manner For the length of this Line of Numbers know that the longer it is the better it is and for that purpose it hath been contrived several ways as first into a Rule of two Foot long and three Foot long by Mr. Gunter and I suppose it was therefore called Gunter's Line Then that Line doubled or laid so together that you might work either right on or cross from one to another by Mr. Wingate afterwards projected in a Circle by Mr. Oughtred and also to slide one by another by the same Author and â—ast of all projected and that best of all hitherto for largeness and conseqenly for exactness into a Serpentine or winding circular Line of 5 or 10 or 20 turns or more or less by Mr. Brown the uses being in all of them in a manner the same only some with Compasses as Mr. Gunter's and Mr. Wingat's and some with flat Compasses or an opening Index as Mr. Oughtred's and Mr. Browne's and one without either as the sliding Rules but the Rules or Precepts that serve for the use of one will indifferenly servâ— for any But the projection that â— shall chiefly confine my self to iâ— that of Mr. Gunter's being the most proper for to be inscribed oâ— a Carpenters Rule for whose sakeâ— I undertake this collection of the most useful convenient and
produceth 21 foot 69 cube inches the near content PROB. IX Having the Girt and length given in foot measure to find the content in feet The extent from 3.545 the feet and 100 parts about when one foot long makes one foot of Timber to the Girt in feet and parts The same extent laid twice from the length in feet gives the content in feet Examp. A Brewers Tun of 20 foot about and 4 foot and ½ deep how many solid feet is it The extent from 3.545 to 20 shall reach the same way at two turnings from 4 foot and ½ to 143 foot and 10 parts the solid content in feet and 6 foot being a full Beer-barrel it contains 24 barrels Beer-measure For 31 foot 8 parts the area of a Circle of 20 foot about being multiplied by 4 foot 5 parts the depth gives 143 foot and 10 pts as before PROB. X. Having the Girt in inches and the length of a Cillander given in inches to find the solid content in cube inches The extent from 3.545 to the Girt in inches being twice repeated the same way from the length in inches gives the content in inches Examp. At 48 inches about and 24 inches in length how many cube inches is it The extent from 3.545 to 48 the inches girt being twice repeated from 24 the length in inches gives 4398 the content in solid inches For 183.2 the area of a Circle of 48 inches about multiplied by 24 the inches deep produceth 4397 the near content as before To insure you the number of places the Print 11 times repeated doth certainly direct you PROB. XI It being an ordinary way in measuring of round Timber such as Oak Elme Beech Pear-Tree and the like which is sometimes very rugged uneven and knotty to take a line and girt about the middle of it and then to take the fourth part of that for the side of a Square-equal to that Circumference but this measure is not exact but more than it should be But either because of allowance for the faults abovesaid or for Ignorance the custome is still used and Men commonly think themselves wrong'd if they have not such measure Therefore I have fitted you with a Proportion for it both for Diameter and Circumference And first for Diameter The Diameter given in inches and the length in feet to find the content As 1.526 to the Diameter so is the length to a fourth and that fourth to the content in feet according to the rate abovesaid The extent from 1.526 to 9.53 being twice repeated from 8 shall reach to 3.12 the content PROB. XII Having the Circumference in inches to find the content in the abovesaid measure As 48 to the inches about so is the length to a fourth number and that fourth to the content The extent from 48 to 30 being twice repeated from 8 shall fall upon 3. 12 the content required PROB. XIII How to measure Taper Timber that is bigger at one end than at the other The usual way for doing of this is to take the Circumference of the middle or mean bigness but a more exact way is to find the content of the base of both ends and add them together and then to take the half for the mean which multiplied by the length shall give you the true content Examp. A round Pillar is to be measured whose Diameter at one end is 20 inches at the other end it is 32 inches Diameter and in length 16 foot or 192 inches the content of the little end is 314. 286 the area or content of the greater end is 773.142 which put together make 1087.428 whose half 543.714 multiplied by 192 the length gives 104393.143 Cubical inches which reduced into feet is 60 feet and 713 cubical inches for the solid content of the Pillar PROB. XIV To measure a Cone such as is a Spire of a Steeple or the like by having the height and Diameter of Base Examp. Let a Cone be to be measured whose Base is 10 foot and the height thereof 12 foot the content of the Base will be found by the 14th Problem of superficial measure to be 78. 54 then this 78. 54 multiplied by 4 a third part of 12 the perpendicular or height of the Cone will give 314. 4 for the content of the Cone required By the numbers work thus the extent from 1 to 4 will reach from 78. 54 to 314. 4. But because there may be some trouble in getting the true perpendicular of a Cone which is its height take this Rule First take half the Diameter and multiply it in its self which here is 25 then measure the side of the Cone 13 and multiply that by it self which here is 169 from which take the Square of half the Base which is 25 your first number found and the remain is 144 the Square-root of which is the height of the Cone or length of the perpendicular PROB. XV. To measure a Globe or Sphere Arithmetically Cube the Diameter then multiply that by 11 and divide by 21 gives you the true solid content let a Sphere be to be measured whose Axis or Diameter is 14 that multiplied by it self gives 196 and 196 again by 14 gives 2744 this multiplied by 11 gives 30184 and this last divided by 21 gives 1437. 67 for the content of the Sphere whose Diameter is 14. But more briefly by the Numbers thus the extent from 1 to the Axis being twice repeated from 3. 142 will reach to the superficial content that is the superficies round about But if the same extent from 1 to the Axis be thrice repeated from 5238 it will reach to the solid content as 1 to 14 so 3. 142 to 617 being twice repeated as 1 to 14 so 5278 to 1437 being thrice repeated As for many sided figures if they have length you have sufficient for them in the Chapter of superficial measure to find the base and then the base multiplied by the length giveth the content But as for figures of roundish form they coming very seldom in use I shall not in this place trouble you with them for they may be reduced to Spheres or Cones or Trirngles or Cubes and then measured by those Problems accordingly And so much for the mensuration of Solids CHAP. VII The Use of the Line of Numbers in Questions that eoncern Military Orders PROB. I. Any number of Souldiers being propounded to order them into a square Battle of Men. Find by the 12th Problem of the second Chapter the square-root of the number given for so much as that root shall be so many Souldiers ought you to place in Rank and so many likewise in File to make a square Battle of Men. Examp. Let it be required to order 625 Souldiers into a square Battle of men the square-root of that number is 25 wherefore you are to place 25 in rank and as many in file for Fractions in this practice are not considerable For had there been but 3 less there would have
inches 15. the Ale or Beer at 18. 95. Also the Gage-points for a Beer-barrel at 35 inches and 9525 parts and the gagepoint for a barrel of Ale at 33 inches and 89645 parts Fourthly the figures on the left side are not much unlike the right for 1 at the beginning is one inch and so it proceeds by quarters of inches to 1 foot then by figures at the feet and the divisions all whole inches to 10 foot then every whole foot and half and quarter or 10th to 100 or 140 or 150 foot and this I call the left side the other the right side so that from 1 inch at the lower end to one foot every inch hath a figure from 1 foot to 10 foot every foot hath a figure and from 10 foot to 100 every 10th foot only is figured I have been very plain in explaining this because I would avoid vain repetitions in the following uses wherein you shall have first the most ordinary and easie questions and then the more hard and critical and less useful The Uses follow PROB. I. A piece of Timber being not square to make it square or to find the Square-equal Set the breadth on the left side to the breadth on the right then right against the inch and quarters thick found on the left side on the right is the inches square required Examp. At 18 broad and 6 thick you shall find 10 inches â…œ the side of the square required For if you set 6 inches against 6 inches on the right and left side then right against 18 inches or 1 foot 6 inches on the left on the right you have 10 inches 1 quarter and half a quarter for the side of the square-equal to 18 one way and 6 the other way PROB. II. The side of the square given to find how much makes a foot For all pieces between 3 or 4 inches and 42 inches square which are the most useful this the best way set the inches or feet and inches square found out on the right side to one foot on the left then right against 1 foot on the right on the left is the inches or feet and inches required to make a foot of Timber But when the piece is small count 1 foot on the right for 1 inch and call 12 on the right for 1 foot Example At 8 inches square set 8 on the right to 1 foot on the left then right against 1 foot on the right on the left is 2 foot 3 inches the length required To find how much is in a foot long Just as the Rule stands even look for the inches the piece is square on the right and on the left is the inches or feet and inches required Example At 17 inches square there is 2 foot of timber in 1 foot long which if you multiply by the length you shall have the true content A very good way for large pieces and very exact PROB. III. The side of the square and length given to find the content For all pieces between 1 inch or â—â— part of a foot and 100 foot this is the easiest way Set the word square or 1 foot to the length on the left then right against the inches or feet and inches square on the right on the left you have the content Examp. At 9 inches square and 20 foot long set the long stroke by the word square to 20 foot on the left then right against 9 inches on the right side on the left side you have 11 foot and a quarter the content required But if it be a very great piece as above 100 foot then call 1 foot on the left side 10 foot and 2 foot 20 c. then 10 shall be 100 and 100 a 1000 that will supply to 1500 foot in a piece But for all small pieces under 3 inches square and above 1 quarter of an inch do thus Set 12 on the top or the small 12 when it is most convenient to use to the length on the left side then right against the inches or 12s of 1 inch squares found on the right side on the left is the true content required Example At 2 inches 3 twelves or 1 quarter square and 10 foot long you shall find 4 inches and a quarter ferè But note when you use the small 12 the answer is given in decimals of a foot therefore the top 12 is best ROB. IV. The square of a small piecè of Timber given to find how much makes a Foot. For all pieces from 12 inches to 1 inch square do thus set the inches and 12s or quarters square counting 1 foot on the right side for 1 inch and 2 foot for 2 inches c. found out on the right side to 100 on the left then right against the upper or small 12 on the right on the left is the length required to make a foot of Timber Examp. At 2 inches ¼ square you must have 28 foot 4 inches to make a foot PROB. V. Under 1 inch square to find the length of a foot Set 1 foot 9 inches 6 inches oâ— 3 inches found on the right side for 1 inch ¼ ½ or ¼ of an inch against 10 on the left side counted for 100 then right against the small 12 you have the feet in length required Examp. At 1 inch square you find 144 feet at ½ square 256 feet at ½ an inch square 576 feet at ¼ or an inch square you find 2034 feet in length to make 1 foot of Timber Or if you set the former numbers 12 9 6 1 against 1 inch on the left then right against the upper 12 is a number which multiplied by 12 is the number of feet required PROB. VI. A great piece above 3 foot ¼ square to find the length of a foot Set the feet and inches on thâ— right to 100 on the left theâ— right against small 12 is the incheâ— and 12s or 12s of a 12th thaâ— goes to make a foot Examp. At 4 foot square yoâ— have 9. 12ths or ¾ of an inch tâ— make a foot of Timber at 5 fooâ— square 5. 12ths and 10. 12ths â—â— a 12th to make a foot Thus you see the Rule as noâ— contrived resolves from 1 quarteâ—â— square to 12 foot square the content or quantity of a foot of Timber in length at any squarenesâ— without Pen or Compasses CHAP. V. For round Timber PROB. I. The number of inches that a piece â—â— Timber is about being given find how much makes a foot First for all ordinary pieces sâ— 1 foot on the left to the inches â—â— feet and inches above on the righâ— then right against TR for true measure or round for the usual measure is the feet or feet and inches required to make a foot of Timber at that circumference about Examp. At 4 inches about 113 foot 2 inches is for true measure but for the usual measure 142 foot goes to make a foot of Timber At 12 foot 3 inches about 1 inch is a true foot but for
the usual allowance as the fourth part of a fine girt about gives it must be 1 inch ¼ long to make a foot of Timber at that circumference But for very large pieces count 1 foot on the right for 12 foot 2. 24 c. and set 1 foot on the left as before then in the answer 1 foot on the left is 1. 12th of an inch and 1 inch 1. 14th of an inch Example At 144 foot about 1. 144th part of an inch is a foot of Timber PROB. II. For very small wood to find a foot in length But for very small pieces of under 4 inches about set 1 foot 2 foot c. on the right counted for 1 inch 2 inches 3 inches or 4 inches to 1 foot on the left then right against TR or round you have a number which multiplied by 12 is the number of feet required Example At 1 inch round true measure is 151 foot ferè but for the usual allowance 196 which numbers multiplied by 12 is the number of feet required viz. 1809 and 2352. But note you must read the 196 and 151 right as thus 1 foot on the left is 12 2 is 24 c. so that 12 foot is 144 and our number by the same account is 151 near To find how much is in a foot in length set round or TR to 1 foot on the left then right against the inches or feet and inches about found on the right on the left is the answer required PROB. III. The inches or feet and inches about and length given to find the content Set the word round or TR for the usual or true measure to the length on the left then right against the inches about on the right on the left is the content required Example At 2 foot 3 inches about and 20 foot long it is 6 foot 2 inches of the usual allowance or 8 foot of true measure But if it be a great Tree then set TR or round to 1 called 10 or to 10 called 100 then is the content augmented to 1000 foot as you did in the Rules for square Timber But if you would have it measure bigger still then set the 4 inches or a TR set close by the brass on the right side to the length on the left either as it is or augmented counting at last according the note 1 foot on the right is 12 foot and 12 at the top is 144 foot then right against the feet about on the right on the left is the content required Examp. A Brewers Tun 3 foot long or deep and 72 foot about set the TR by the brass to 36 inch which is thus counted on the left side 1 inch is 10 inches 2 is 20 3 is 30 3½ is 35 somewhat more is 36 so then 1 foot is 120 inches o◠10 foot then right against 6 times 12 foot on the right which is at ◠foot on the left you have 12 30 foot as near as the Rule will give it which counting 6 foot to a barrel is 205 barrels the content required PROB. IV. To find the content of a very smal◠piece Set the word round or TR to the length on the left as in the third Problem of this Chapter the◠right against the inches about o◠the right calling 1 foot 1 inch and 6 inches ½ an inch on the lef◠is the 13s of 1 inch or 12s of a 12th required Example At half an inch about and 10 foot long it is 2 12s and a half of 1 12th of an inch or 2 square inches and ½ true measure Again 2 inches ¼ about and 10 foot long is half an inch of true measure 12 inches to a foot solid or ½ a foot superficial of one inch thick CHAP. VI. To measure Timber having the Diameter and the length given PROB. I. The Diameter given in inches to find the length of a foot Set 1 foot on the left to the inches diameter on the right then right against TD for true diameter or the word diameter for the usual allowance of a string girt about and doubled 4 times for the side of the square you have the feet and inches required Example At 10 inches diameter 1 foot 10 inches makes a foot But for very great pieces set 1 foot as before but look for TD beyond the upper 12 right against it on the left you have the 12 of 1 inch or the 12s of a 12 that makes a foot But for very small sticks set 1 2 or 3 foot on the right for 1 2 or 3 inches to 1 foot on the left then right against TD or Diameter you have a number which multiplied by 12 is the number of feet required to make a foot of Timber Examp. At 1 inch Diameter you shall have 15 foot 3 inches and better which multiplied by 12 is 123 foot 3 inches Note that 1 on the left is reckoned 10 foot and 2. 20 foot as before in the same Rule for the circumference and then note 1 inch is 10 inches At any diameter to find how much is in 1 foot long do thus set Diam or TD to 1 foot then just against the inches or feet and inches diameter found on the right on the left is the answer Example At 2 foot diameter is 3 foot 2 inches in 1 foot of length which multiplied by the length gives the true content of any round piece and very exactly PROB. II. The diameter and length given to find the true content For all ordinary pieces set the word diameter for the usual measure or TD for true measure always to the length on the left then right against the inches or feet and inches diameter on the right on the left is the content required Examp. At 5 inches diameter and 30 foot long you shall find 4 foot and ½ an inch true measure for the content required But for very small pieces set TD or Diam to the length as before then counting 1 foot on the right for 1 inch and 6 inches for ⅓ an inch on the left you shall have the answer or content required But note as the right side is diminished so is the left for 1 foot on the left is a 12th of an inch of Timber whereof 12 makes a foot or 1 long inch a foot long and 1 inch square and every ineh on the left is 1 square inch thus at 2 foot long and half an inch diameter it is 4 □ inches ¾ in content But for a great piece under 1000 foot set TD or diameter to 1 2 or 3 foot called 10 20 or 30 foot then right against the feet and inches diameter you have a content augmented accordingly as at 30 foot long and 7 foot diameter you have 1140 foot for the true content using TD Note that in large Taper-timber whether square or round when i◠is measured by the usual way that is by the middle square or girt or the 2 squares or girts put together and the half counted for the equal square or girt I say a square of half the difference of the squares or girts and one part of the length is to be added to the former measure as is proved in the circles of proportion pag. 50. As thus for Example Suppose a Taper piece be at one end 16 inches square at the other 30 inches square and 30 foot long the square in the middle is like to be 23 inches the content then is 110 foot now half the difference of the two ends square is 7 inches and 1 third part of the length is 10 foot a piece 7 inches square and 10 foot long is 3 foot 5 inches which added is 113 foot 5 inches the true content of that taper piece abovesaid The general way of Gauging by this Rule is thus Set the W. or the A. for Wine or Ale-measure always to the length of the Vessel found out on the left Then right against the mean Diameter found out on the right side on the left is the answer required Examp. At 30 inches Diameter and 36 long you shall find about 90 gallons and a half Ale-measure The Gage-point for a Beer-barrel i◠near 3 foot and the Ale-barre◠near 34 inches which use thus Set 3 to the depth of the Tun then right against the mean Diam is the content in Barrels Example Set 3 to 36 inches and then right against 5 foot Diam is 10 barrel of Beer-measure a very good an◠spedy way FINIS 3. 3. 5. 2. 3. 4.
The Description and Use OF THE CARPENTERS-RULE Together with the vse of the LINE of NUMBERS Commonly called GUNTERS-LINE Applyed to the Measuring of all Superficies and Solids as Board Glass Plaistering Wainscoat Tyling Paving Flooring c. Timber Stone Square or Round Gauging of Vessels c. ALSO Military Orders Simple and Compound Interest and Tables of Reduction with the way of working by Arithmatick in most of them Together with the Use Of the Glasiers and Mr. White 's Sliding-Rules Rendred plain and easie for ordinary Capacities By Iohn Brown. London Printed for W. Fisher and R. Mount at the Postern on Tower-hill 1688. Courteous Reader THis Little Book was first written by me several years since and hath been accepted of among many that â—ave had the Perusal-thereof And several Impressions in that time beâ—â—g sold off and it being now out â—f Print and none to be had I have â—â—vised it and left out what might â—ell be spared and added that which â—ight make it more plain and easier 〈◊〉 be remembred As for Instance in the using of â—â—e Line of Numbers commonly â—â—lled Gunters Line for the meaâ—â—ring of Board or Timber or Stone â—â—e fixed Points or Centers is only 10 â—â—d 12 for square Timber or Stone And in measuring of round Timber or Stone as round Timber â—â—ere is used only 13. 54 for Inches â—â—d 1. 128 for Foot-measure being â—e Diameter in Inches and Footâ—easure of any round solid when one â—oot in length makes one solid Foot of 12 Inches every way or 1728 so lid Cube Inches which is a foot 〈◊〉 Timber or Stone And if the Circumference or Giâ—â— of the Piece about is given then tâ—â— fixed Point or Center used is at 4â—â— 54 the Inches and 100 part of â—â— Inch about when one Foot or 1â—â— Inches long makes one Foot solid Or else at 3.545 the Feet and 100â— parts about when one Foot long makâ— one solid Foot equal to 1728 Cube Iâ— Also after every Problem is tâ— brief way of working it by the Pen 〈◊〉 a proof of the truth of every Operâ— tion by the Rule being more thâ—â— was before in the former Impressionâ— Also the Line of Pence is aâ— ded to the Line of Numbers and plain way set forth of the use there by the Line on the Rule Or the mâ— plain description thereof in a Prâ— of Gunters Line 11 times repeateâ— which may be had with the Book Tower-Hill or the Minories The Use of the Gunters Line â—he Art of Gauging is here but briefâ—y hinted because there are several Books of Gauging purposely made for that Imployment more compleat â—han can be expected in this short Discourse And Lines of Area's of Circles in Ale-gallons at any Diameter given from 1 Inoh to 200 Inâ—hes which may be used for round â—r square Vessels to give the content â—f every Inch deep in any taper Vessel as fast as any one can write it down And Directions for the ready measuring the Drip or stooping Bottoms of round or square Tuns and the Liquor about the Crowns of Coppers Which Books are to be had at the Postern at Tower-Hill or at the Authors in the Minories To this Impression is added the Use of the Lines called Diameter Circumference Square-Equal and Square within a Circle and to find the Circles Area or Content by them or having the Area to find the Diameter or the Circumference or the Square-equal or Square-within Also to this is added the way bâ— the Pen to multiply Feet Incheâ— and 12 parts of an Inch together whereby any Superficies as Boarâ— Floor Wall Yard or Field way â—â— exactly measured by the Pen. Alsâ— by a second Operation or Multiplâ— cation may any Solid as Timber 〈◊〉 Stone or round Vessel be measured the Arithmetical way whereof 〈◊〉 worded as plain as in any Boâ—â— whatsoever Also the Use of the Glasier's Slâ— ding-Rule to measure Glass or aâ—â— Superficies And Mr. White 's Sliding-Ruâ—â— to measure Timber being as neâ—â— and ready a way as ever was used Thus you have a brief Account 〈◊〉 what is contained in this little Booâ— and I wish it may be helpful to maâ— a Learner for whom it is prepareâ— So I remain ready to serve you in thâ— or the like John Brown. From my House at the Sphere anâ— Sun-Dial in the Minories Londoâ— â—he Description and Use of the Carpenters Rule CHAP. I. IT is call'd a Carpenters Rule rather then a Ioyners Bricklayers Masons Glasiers or the like I suppose because â—â—ey find the most absolute necessiâ—â— of it in their way for they have 〈◊〉 much or more occasion to use 〈◊〉 than most other Trades though â—â—e same Rule may measure all kind â—f Superficies and Solids which â—wo Measures measure every visiâ—â—e substance which is to be meaâ—â—red And it is usually made of â—â—ox or Holly 24 Inches in length â—nd commonly an Inch and half or 〈◊〉 Inch and quarter in breadth â—nd of thickness at pleasure and â—n the one side it is divided into â—4 equal Inches according to the â—tandard at Londonâ—nd â—nd every one of those 24 Inches is divided into eight parts that iâ— Halfs Quarters and Half-quarterâ— or ten parts as you please and thâ— Half-inches are known from thâ— Quarters and Quarters from thâ— Half-quarters by short longer and longest strokes and at everâ— whole inch is set figures proceedinâ— from 1 to 24 from the right hand toward the left and these partâ— and figures are on both edges 〈◊〉 one side of the Rule both wayâ— numbred to the intent that howsoever you hold the rule you have thâ— right end to measure from provided you have the right side On the other side you have thâ— Lines of Timber and Board measure the Timber-measure is thaâ— which begins at 8 and a half thaâ— is when the figures of the Timber line stand upright to you then I saâ— it begins at the left end at 8 and 〈◊〉 and proceeds to 36 within an Incâ— and â…œ of an Inch of the end Also aâ— the beginning end of the Line oâ— Timber measure is a Table of figures which contains the quantity of the under-measure from one Inch square to 8 Inches square for the figure 9 comes upon the Rule as you may see near to 8 in the Table On or next the other edge and same side you have the Line of Board-measure and when those figures stand upright you have 6 at the left or beginning end and 36 at the other or right end just 4 Inches off the end unless it be divided up to a 100 then it is nigh an Inch and half off the end This Line hath also his Table of Under-measure at the beginning end and begins at 1 and goes to 6 and then the divisions on the Rule do supply all the rest to 100. Thus much for Description Now for Use. The Inches are to measure the length or breadth of any Superficies or Solid given and the manner of doing it were superfluous to speak of
or once to mention being not only easie but even natural to every man for holding the Rule in the left hand and applying it to the board or any thing to be measured you have your desire But now for the use of the other side â— shall shew it in two or three examples in each measure that is Superficial or Solid And first in Superficial or Board-measure Ex. 1. The breadth of any Superficies as Board or Glass or the like being given to find how much iâ— length makes a Square Foot or iâ— equal to 12 inches broad and 12 Inches long for so much is a â—rne Fooâ— Superficial To do this look for the number of Inches your Superficies is broad in the Line of Board-measure anâ— keep your finger there and right against it on the Inches side you have the number of Inches that goeâ— to make up a Foot of Board oâ— Glass or any Superficies Suppose â— have a peice 8 Inches broad How many Inches make a Foot I look for 8 on the Board-measure and just against my finger being set to 8 on the Inch side I find 18 and so many Inches long at that breadth goes to make a Foot Superficial Again suppose it had been 18 Inches broad then I find 8 Inches in length to make a Foot superficial but if 36 Inches broad then 4 Inches in length makes a Foot. Or you may do it more easie thus Take your Rule and hold it in your left hand and apply it to the breadth of your Board or Glass making the end that is next 36 even with one edge of the board or glass the other edge of the board shews how many Inches or Quarters of an Inch go to make a foot of the board or Glass This is but the converse of the former and needs no example for laying the Rule â—o it and looking on the Board-measure you have your desire Or else you may do thus in all narrow peices under 6 inches broad As suppose 3¼ double 3¼ it makes 6½ then I say that twice the length from 6½ to the end of the Rule shall make a Foot Superficial or so much in length makes a foot Ex. 2. Having A Superficies of any length and breadth given to find the Content that is how many Foot there is in it Having found the breadth and how much makes one Foot turn that over as many times as you can for so many Foot is there in that Superficies But if it be a great breadth then you may turn it over two or three times and then take that together and so say 2 4 6 8 10 c. or 3 6 9 12 15 18 21 and till you come to the end of the Superficies Note that the three short strokes between figure and figure are the Quarters as thus 8 and a quarter 8 and a half 8 and three quarters then 9 c. till you come to 30 and then 30 and a half 31 c. to 36. And if it be divided any further it is to whole Inches only to 100. The use of the Table at the beginning end of the Board-measure First you have five ranks of figures the first or uppermost is the number of Inches that any Superficies is broad and the other 4 are Feet and Inches and parts of an Inch that goes to make up a Foot of Superficial measure As for example at 5 Inches broad you must have 2 Foot 4 Inches and 4 Fifths of an Inch more that is 4 parts of 5 the Inch being divided into 5 parts but where you have but two figures beside the uppermost and Ciphers in the rest you must read it thus At two Inches broad you must have six Foot in length no Inches no parts Thus much for the use of the line of Superficial or Board measure The Use of the Line of Solid or Timber-measure The use of this Line is much like the former For first you must learn how much your piece iâ— square and then look for the same number on the Line of Timber-measure and the space from thence to the end of the Rule is the truâ— length at that squareness to make a Foot of Timber Ex. 1. I have a piece that is 9 Inches square I look for 9 on the Line of Timber-measure and therâ— I say the space from 9 to the end of the Rule is the true length to make a Foot of Timber and it iâ— near 21 Inches 3 eights of an Inch. Again suppose it were 24 Inches square then I find 3 Inches iâ— length makes a Foot for so I find 3 Inches on the other side just against 24 But if it were small Timber as under 9 Inches square then you must seek the square in the upper rank in the Table and right under you have the Feet Inches and parts that go to make a Foot square as was in the Table of Board-measure As suppose 7 Inches square â—hen you must seek the square Inches in the upper rank in the Table and right under you have the Feet Inches and parts that go to make a Foot square as was in the Table of Board-measure As suppose 7 Inches square right under 7 I find in the Table 2 Foot 11 Inches and 2 sevenths of an Inch divided into 7 parts and at 8 Inches square you find only 2 Foot 3 Inches 0 parts and so for the rest But if a piece be not just square but broader at one side than the other then the usual way is to add them both together and to take half for the square but if they differ much then this way will be very erroneous and therefore I refer you to the following Rules But if it be round Timber then take a string and girt it about and the fourth part of this is usually allow'd for the side of the square and then you deal with it as if it were just square Thus much for the Use of thâ— Carpenters Plain-rule I have also added a Table foâ— the Under-measure for Timber Board to Inches and Quarters anâ— the use is thus Look on the left side for the number of Inches anâ— Quarters your Timber is squareâ— or your Board is broad and right against it you have the Feet Inches tenth part of an Inch and tenth of tenth or hundredth part of an Inch that goeth to make a Foot oâ— Timber or Board Ex. 3. A piece of Timber 3 Inches 1 quarter F. Inch. 10. 100 square will have 13 7 5 9 parts to make a Foot. And a Board of 3 Inch and a quarter broad must F. Inc. 10. 100. have 3 8 3 0 in length to make a Foot and so of the rest as is plain by the Table and needs no further explication being common to most Artificers A Table for the under Timber-measure to Inches and quarters in qr feet in 10p 100  1 2304 0 0 0  2 576 0 0 0  3 256 0 0 0 1 1 144 0 0 0  1 92 1 9 7  2
part as here 5 for 3 times 5 is 15 open the Compasses from thence to the Principal and then turn the other two turns viz. 10 15 and this may avoid much errour or at the least much mitigate it for in these questions the larger the Line is the better PROB. IV. A yearly Rent or Annuity being propounded to find the worth in ready Money First find by the last what the Arrears come to at the term propounded and then what those Arrears are worth in ready Money and that shall be the value of it in ready money Examp. What may a Lease of 10 l. per annum having 15 years to come be worth in ready money I find by the last Problem that the Arrears of 10l per ann forborn 15 years is worth 231 l. 4 s. And likewise I find by the second Problem that 231 l. 4 s. is worth in ready money 96 l. 16 s. and so much may a man give for a Lease of 10 l. per ann for 15 years to come at the rate of 6 l. per cent But if it were not to begin presently but to stay a certain term longer then you must add that time to the time of forbearance as suppose that after 5 years it were to begin then you must say 231 l. 4 s. forborn 20 years is worth in ready money and it is 72 l. 8 s. and that shall be the value of the Lease required PROB. V. A sum of money being propounded to find what Annuity to continue any number of years at any rate propounded that summ of Money will purchase Take any known Annuity and find the value of it in ready money this being done the proportion will be thus as the value found out is to the Annuity taken so is the summ propounded to the Annuity required Examp. What Annuity to continue 15 years will 800 l. purchase after the rate of 6 l. per Cent. Here first I take 10l per ann for 15 years and find it to be worth in ready money 96 l. 16 s. by the last Problem then I say as 96 l. 8 s. is to 10 so is 800 to 82-7 which is 82l 14 s. and so much near do I conclude will an Annuity of 82 l. 14 s. per ann be worth for 15 years after the rate of 6 l. per Cent. viz. 800 l. Also in this Impression is added the use of the Line of Pence which is added to the Line of Numbers next to it when it shall be desired by any one being very convenient for casting up small summs of Money in any concern whatsoever The Line of Numbers and Pence together do give the Decimal Fraction of any summ under 20 s. very near as by the Print to 11 Radixes is most plainly seen Where 20 s. or 1 l. in Money is at 1 in the fourth line and 2 s. one tenth of a pound at 1 right over it in the third Line 9 farthings and 6 tenths of a farthing being one tenth of two shillings at 001 in the second line right over 01. Lastly 96 100 parts of a farthing at 0001 in the first Line of Numbers towards the right hand or the 96 1000 parts at the 00001 at the left end of the same Line Therefore note if 20 s. is 1 15 s. is 075 12 s. is 06 10 s. is 05 5 s. is at 025 and 2 s. or 24 d. at 01 which on two-Two-foot Rules is set at 10 at the end of the Rule though in this case called but one tenth when 1 is one pound Then if 1 or 10 at the end be 24 pence 18 pence is at 75 12 d. at 5 6 d. at 25 2 d. at 8.33 in the first part and 1 d. at 4166 and 1 farthing at 1042 a little beyond the first 1 on Two-foot Rules From hence you may see that the Print of the eleven Lines of Numbers sheweth the right Decimal Fraction of any summ under 1 pound Sterling by consequence for any summ above and was purposely made to explain this on two-Two-foot Rules The Use of the Line of Numbers and Pence laid together on two-Two-foot Rules may be in a brief manner thus 1. At any price 100 or 5 score what cost 1 counting 10 at the end of the Line 10 l. Examp. At 5 l. per 100 counted at 5 in the second part because 10 is called 10 l. just over or under it in the line of pence is 12 d. the exact answer for 100 s. is 5 l. 2. Examp. At 2 l. 10 s. per 100 right against 2.5 on Numbers representing 2 l. 10 s. right against it on the Line of Pence is 6 d. the Answer for 100 6 d. is 2 l. 10 s. Now observe if 1 in the middle is called 1 l. then in the first part 9 is 18 s. 9.5 is 19 s. 5 towards the beginning-end is 10 s. and 1 at the very beginning end is 2 s. Then to supply all under 2 s. to 1 farthing begin again at the upper end at 10 where is set 24 d. the same with 2 s. and back again 23. 22. 21. 20. 19. 18 pence at 75 then 17. 16. 15. 14. 13. 12 d. and 5 then 11 10. 9. 8. 7. 6 d. at 25 then 5. 4. 3. 2 d. at 0.834 then 1 penny at 0.4166 then 3 farthings 2 farthings 1 farthing at 0.1042 and every 10th of a farthing in pricks between the farthings 3. Examp. At 1 l. per 100 counted at the middle 1 is 2 d. 1 far 6 tenths of a far for 1. 4. Examp. At 10 s. per 100 counted at 5 in the first part just against it in the Line of pence is 1 penny or 4 farthings 8 tenths of a farthing the true answer 5. Examp. At 2 s. per 100 counted at 1 at the beginning of the Line of Numbers is no farthings but 96 parts of a farthing in 100 pts the answer for the half of 96 is 48 the farthings in 2 s. 6. Examp. But for any price under 2 s. per 100 count thus as 12 d. per 100 just against 12 d. on the Line of pence is 5 on numbers for 5 tenths of a farthing the near answer for 5 tenths of a farthing is half a farthing and 100 half farthings is 50 farthings being two farthings above 48 the farthings in one shilling 7. Examp. At 6 d. per 100 right against 6 in the Line of pence is 025 or one quarter of a tenth of a farthing on the Line of Numbers the answer near the truth For a hundred quarters of a farthing is 50 half farthings or 25 farthings one more than 6 d. But if you count 1 farthing less for every 6 d. you shall have it right Example at 18 d. per 100 count 17 d. and 1 far 3 far less than 18 d. and just against it on Numbers is 12 100 pts of 1 farthing the true price of 1. Again at 12 d. counted at 11 d. 2 far just against it on numbers is 48 100
agree together This being premised as to the using of it you may apply all the former precepts and examples to this Rule as well as the other CHAP. III. To measure Timber by the Sliding-Rule PROB. I. To measure Timber by this Rule is nothing else but to work the Double Rule of Three Examp. At 8 inches square 20 foot long I would know the content Set 12 if the side of the square be given in inches or 1 if in foot meâ—sure on the first side to 8 the inches square on the second then right against 12 on the second side on the first is 18 the fourth proportional part then for the second work set 18 the fourth proportional last found to 8 the inches square on the second then right against 20 the length is 9 the content required Or rather thus Set 12 against 8 then right against 20 on the same side 12 was is 13. 5 near on then look for 13. 5 fere on the first side and right against it on the second is 9 foot the content required PROB. II. To measure a piece that is not square Set 12 if you use the inches or 1 if you use foot-measure on the first side to the inches thick on the second then right against the inches broad on the first side on the second is a fourth proportional then in the second operation set 2 on the first side to the fourth proportional on the second then right against the length on the first side on the second is the content required Ex. At 8 inches thick and 16 broad and 20 foot long you shall find 18 foot fere PROB. III. The square given to find how much makes a foot Set the inches square on the first side to 12 on the second then right against 12 on the first on the second is a fourth proportional number then in the second work as the inches square to the fourth proportional so is 12 to the number of inches required to make a foot of Timber Ex. At 6 inches square set 6 to 12 then against the other 12 is 24 then set 6 to 24 then right against 12 you shall have 48 the length in inches required After the same manner are other questions wrought but the Compasses are easier and more ready therefore I shall say no more to this but only refer you to the former Rules in the third fourth and fifth Chapters Only note that in those Sliding-rules made for Glasiers use the one half of the Line of Numbers is on one side of the Rule and the other on the other side and whatsoever leg or piece of the rule is the first on the one side the same leg or piece is the first when the Rule is turned on the other side which must well be observed but note that for measuring of Timber those that use it may have one side fitted for that as I shall more plainly and fully shew in the next chapter being the easiest speediest and nearest way that ever yet was used by any man resolving any Contents by having the length and the diameter the circumference or square given A Table of the true size of Glasiers Quarries both long and square calculated by J. B. Square-Quarries 77. 19 gr Quarries Rang. Sides bread leng content in feet content in inc 8 in 100 i. 100 I. p i. pts F. p â—ts Inc. p. 8 4 20 4 30 5 36 6 70 0. 1250 1. 50 10 3 76 3 84 4 80 6 00 0. 1000 1. 20 12 3 43 3 51 4 38 5 47 0. 0833 1. 00 15 3 07 3 13 3 92 4 90 0. 0667 0. 80 18 2 80 2 86 3 57 4 47 0. 0555 0. 666 20 2 66 2 72 3 39 4 24 0 5000 0. 60 Long Quarries 67. 22 Quarries Rang. Sides bread leng content content  in pts I. pts â— 100 â— pts F. 100 â— pts 8 4 09 4 41. 4. 90 7. 34 0. 1250 1. 50 10 3 65 3. 95 4. 38 6. 57 0. 1000 1. 20 12 3 34 3 61 4. 00 6. 00 0. 0833 1. 00 15 2 98. 3 23 3. 58 5. 37 0. 0667 0. 80 18 2 58 2 79 3. 10 4. 90 0. 0555 0. 666 20 2 72 2 94 3. 27 4. 65 0. 0500 0. 60 Note that a prick after the 100 parts of an inch notes a quarter and a stroke a half of 100 parts of an inch to make this Table work thus by the Line of Numbers Divide the distance between the content of some known size as square 10s or long 12s and the content of the inquired size into two equal parts for that distance laid the right way increasing for a bigger or decreasing for a less from the sides of the known size shall give the reciprocal sides of the inquired size Example for square 12s The half distance on the Line of Numbers between 1000 the content of square 10s and 0.833 the content of square 12s shall reach from 6 the length of square 10s to 5.47 the length of square 12s and from 4. 80 the breadth of square 10s to 4 38 the breadth of square 12s and from 3 84 to 3. 51 and from 3 76 to 3 43 and so for all the rest CHAP. IV. The description of the Line of Numbers on a Sliding-rule to measure Solid measure only according to Mr. White 's first contrivance but much augmented by J. B. First when the figures on the Timber side stand right towards you fit to read then that half or piece next to your right hand I call the right side the other is of necessity the left Secondly the figures on the right side are first at the lower end where the brass is pin'd fast either 3 or 4 or 5 it matters not much which yet to have 3 there is best then upwards 4. 5. 6. 7. 8. 9. 10. 11. for so many inches then 1. 2 3. 4. 5. 6. 7. 8 9. 10. 11. 12. under the brass at the top for so many feet the divisions between to 1 foot are quarters of inches the next above 1 foot are only whole inches as you may plainly see Thirdly at 1 foot you have the word square at 1 foot 1 inch ½ is a mark and right against it is set TD noting the true diameter of a â—ound Cylinder at an inch further â—s 12 set which I call small 12 being in small figures Again at 1 foot 3 inches better is another mark and right against it the word diameter for the diameter of a piece of timber according to the usual english allowance Then again at 3 foot 6 inches ½ near is TR for the true circumference of a round cillander Lastly at 4 foot is the word round noting the circumference according to the usual allowance whose use followeth Also at 13 foot 7 inches is TD and at 3 inches and ½ is TR. Note also if you put on the Gage-points for Ale or Wine with the mean diameter and length you may gage any Wine or Beer-vessel the Wine at 17