thus Count the length of the Rafter from the Center and take the lateral extent thereof being 15 between your Compasses the Compasses being so set set one point in 10 For the Perpendicular the half width of the frame and turn the other point parallelly to the common Line and there it shall shew 11 Foot 2 1 7 Inches the true height of the perpendicular at true pitch required I F 11 Foot 2 Inches 1 7. 3. For the Length of the Hips Count the half breadth of the house on one Leg and the Rafters length on the other Leg and take the parallel distance between and it shall give the true length of the Hips required As here the parallel For the Lips Length distance between 15 and 10 shall be lateral 18 the true length of the Hips required in a square frame 20 foot wide true pitch C G 4. For the Diagonal line from the Corner to the King-post Count the half width of the house viz. 10 Foot on both Half the Diagonal Line Legs and take the parallel extent between and it shall reach from the Center to 14 foot 1 Inch â… being the line C E in the figure 5. Otherwise for the Hips length Count the half Diagonal on one Leg 14 1 â… and the perpendicular 11 foot 2 In. Hip. Inches 2 2 7 on the other Leg and the parallel distance between measured from the Center will give the true Hips length 18 viz. the line C G. Thus much for the Rafters and Hips in square frames at true pitch the measure of whose Angles and Lengths are as followeth    F. Inc. 100 parts F. 100 parts  deg min. The breadth of the house being 20 f. and the two ends square and the Rafters at true pitch the Lengths and angles of the frames are C F Rafter 15 00 00 15 000 Rafter Top 41 50 C G Hip 18 00 00 18 000 Foot 48 10 I F Perpendicular 11 02 17 11 180 Hips Top 51 38 B E ½ Diagonal 14 01 68 14 140 Foot 38 22 B I Half Breadth 10 00 00 10 000    B C Whole Breadt 20 00 00 20 000 Difference 13 16 K G Neerest Dâ—st 16 07 50 16 624     B D Whole Diag 28 03 36 28 281 Outsid Ang. 116 12 The Rule to find the Angles of the Rafters and Hips 1. For the Angles that the Rafters make at Foot and Head with the Raising-piece and King-post do thus When the Rule stands square in the 30 Scale then lay any streight piece to the Compass points when one stands in the half breadth and the other point in the perpendicular then if you apply a Bevel severally to that streight edge and the 30 Scale at each end the one is the Angle at Foot 48 10 and the other the Angle at Head viz. 41 50. 2. For the Angle at Head or King-post and foot of the Hips do thus the 30 Scale being set square set one point of the Compasses in the half Diagonal-line on one 30 Scale and the other point in the perpendicular on the other 30 Scale and to those points lay any streight piece then set a Bevel to that Streight piece and each 30 Scale and the one shall be the Angle at Foot 38 22 and the other the Angle at the head viz. 51 38 then this last Angle doubled and the difference between the Angle at Head and Foot viz. 13 16 added makes 116 12 the Angle of the out-side of the Hip-rafter in a square frame at true pitch required 3. A General way to find this outward Angle of the Hip or mould is thus in any pitch First you must find the neerest distance from one corner to the opposit Hip set up or from a point in the raising piece as far from one corner as the house end is broad as the point K or L in the Figure may be done thus Count the length of the Hip on one Leg and also take it between your Compasses and count the breadth of the house on the other Leg and set one point of the Compasses in the Hips length on one Leg and the other point in the breadth of the house on the other Leg altering the 30 Scales but not your Compasses then when the 30 Scales are so set one represents the Raising piece and the other the Hip set up Then the neerest distance from the breadth of the house on one 30 Scale to the Common-line on the other 30 Scale is the neerest distance required being measured from the Center 16 7 ½ Then take the whole Diagonal Line viz. C L or B K from the Center Laterally and make it a parallel in the neerest distance last found and that shall set the 30 Scales to the Angle of the out-side of the Hip required which you may measure in degrees thus take parallel 15 as the Rule stands and lay it from the Center and it shall reach to 116 Degrees on the Chords next one 30 Scale the Angle of the outside of the Hips required Example and more briefly in a House 20 Foot broad The house end is 20 Foot broad the whole Diagonal Line is 28 foot 3 inches â…œ the Hip Rafter 18 Foot Take 18 Foot between your compasses and set one point in 20 and open or shut the Rule till the other fits 18 then the neerest distance from 20 to the Common Line will be 16 foot 7 inches 4 8. Then take 28 foot 3 inches â… the whole Diagonal Line and make it a parallel in 16 foot 7 inches ½ the neerest distance and the 30 Scales are set to the Angles required For If you take out Parallel 15 the Chord of 60 and measure it Laterally from the Center it shall reach to 116 the Angle in Degrees and Minutes required Note If the whole breadth and whole Diagonal Line is too large for your Compasses then the half breadth and half Diagonal will do as well taking the half length of the Hip also between your Compasses and on the Scale also and that shall set the Scales to the same Angle as before Use XV. To find the Lengths and Angles of the Rafters and Hips or Sleepers in Bevelling Frames at any Pitch 1. For the length of the Rafter set the 30 Scales square then count the half length of the bevel-end on one Leg being always more there than the half breadth and the Perpendicular resolved on on the other Leg Then the Parallel distance between measured laterally shall be the length of Rafter the Rafter required and a Ruler laid to the two Points of the Compasses so set on the 30 Scales and a Bevel set as before in Square Frames is shewed shall give the Angles at head and foot required 2. For the Hips length count the Rafters length on one Leg and the half breadth of the bevel-bevel-end of the House more by half the number of inches bevelling on the other Leg and take the
parallel distance between and Measure it Laterally Hips from the Center and it shall be the length of the longest Hip-Rafter And for the shortest Hip count less than the bevel-Bevel-end by half the number of Inches bevelling and that shall be the short Hip required 3. The Inches or Feet and Inches of Bevelling being given to find how much one corner is under and the other over 90 Degrees or just Square open the 30 Scales and take the breadth of the House over at the neerest distance between Angle of the Frame your Compasses from the 30 Scale from the Center laterally and make it a parallel in 15 and 15 for 60 of the Chords Then take the Feet and Inches Bevelling from the same 30 Scale laterally and carry it parallelly till it stay in like parts then just against it on the Degrees or Tangents are the Degrees and Minutes required that one corner is more and the other less than 90 Degrees 4. To find the Diagonal Line Take the distance in the Chords to the Degrees above or under 90 last found from the Center Laterally and make it a Parallel in 15 and 15 and then the 30 Scales are set to the Angle the end is over or under 90 Degrees Then count Diagonal Lines the whole or half Bevel end on both Legs and the Parallel distance between shall shew the length of the whole or half Diagonal Line measured from the Center Note That when the Rule stands at the Blunt Angle it gives the longest Diagonal Line and when it stands at the Sharp Angle it gives the shortest Diagonal Line 5. By the Diagonal Line and Perpendicular to find the Hips Length and the Angles at Head and Foot of the Hip or Sleeper Count the half Diagonal on one Leg and the Perpendicular Hips height on the other Leg the 30 Scales being square then the Parallel distance between shall be the length of the Hip required being longer or shorter as the Diagonal Line is Also a Rule laid to the two Points of the Compasses measuring the Parallel Extent and a Bevel laid to the Rule and Angles of the Hips the two 30 Scales at each end gives the Angles at head and foot of Hips required To find the Neerest Distance from the Corner of the Rombus to the Opposite Hip set up in his true place 6. Count the length of the Hip on one 30 Scale and take that distance also between your Compasses Laterally count also on the other 30 Scale the length of the Bevel end and there set one point of the Compasses and open or shut the Neerest Distance Rule till the other point falls in the length of the Hip-Rafter first counted then one 30 Scale Represents the Raising Piece and the other the Hip set up then the neerest distance from the bredth of the Frame over at the Bevel end and more or less by half the Feet and Inches Bevelling to the Common Line of the other 30 Scale being measured from the Center shall be the neerest distance required To find the Angle on the Outside of the Hip. 7. To find the Out-side Angle of the longest Hip Take the shortest Diagonal Line between your Compasses and make it a Parallel in the neerest distance belonging to that Hip and the 30 Scales will be set to the Angle required and to measure it take Parallel 15 and 15 and measure it Laterally from the Center in the Chords and you shall have the measure Outside Angle of the Angle required Example In a House of 20 foot over at nearest distance and 4 foot or 48 inches Bevelling out of Square See fig. II. Let A B C D represent a Frame 20 foot over and 4 foot Bevelling the bevel-Bevel-end B C is longer than right over by 5 inches for if you set the Rule square and take the parallel extent from 20 the measure over to 4 foot the measure of Bevelling and measure it Laterally you shall find it reach Laterally to 20 foot and 5 inches the true length of the Bevel end The bevel-Bevel-end being 20. 5 inches the Perpendicular resolved on which at true Pitch ought to be about 11 foot 5 inches Fere. Then first for the Rafters length 1. Set the 30 Scales square and set one Point in 11. 5 the Perpendicular and the other point in 10 foot 2½ the half bevel-Bevel-end and to the Compass-Points lay a Rule and to the Rule and the 30 Scales at both ends set a Bevel and one shall Rafters Length and Angles be the Angle at foot the other the Angle at the top of the Rafter And the one Angle will be 42 Degrees for the foot and the other 48 for the top of the Rafters and the measure between the Compasses measured from the Center shall be 15 foot 4 inches the Rafters length required As by Inspection on the 40 and 30 Scales you may see 2. For the longest Hip set one Point in 12 foot 2 inches and ½ more by two foot then 10 foot 2 in ½ the half bevel-Bevel-end and the other Point in 15. 4 the Rafters length and measure it from the Center it gives 19 foot 6 inches the longest Hip. Again Set one Point in 8 foot 2 inches ½ 2 foot shorter than the half bevel-Bevel-end by 2 foot the half of 4 foot the Bevelling Hips Length and the other Point in 15 foot the Rafters length and measure it from the Center it gives 17 foot 4 inches the length of the shortest Hip. 3. For the length of both Diagonal Lines set the Scales of 30 to the Angle of the Frame at each corner and the measure from the half or the whole bevel-Bevel-end taken Parallelly shall be the length required of the half or whole Diagonal Line according as you take the whole or half Bevel-end As here in our Example the Blunt end is 101 gr 30 min. Diagonal Lines or 11 30 more then 90 gr therefore take the distance from the Center to 101. 30. on the Chords and make it a Parallel in 15 and 15 the Chord of 60 then is the 30 Scales set to the Angle of the Blunt-end of the frame and the Parallel distance between 20-5 the whole Bevel-end gives 31 foot 6 inches the whole Diagonal Line B L or the Parallel between 10-2½ give 15-09 inches B E the half Again The sharp end is 78. 30 11 deg 30 less then 90 then the Lateral Chord of 78. 30. made a Parallel Chord of 60 at 15 then is the Rule set to the sharp end of the Frame for the shorter Diagonal Line And the Parallel distance between 10-2½ gives 12 foot 11. the half or 25 foot 10 inches the whole Diagonal Line C F whose half is C E the shortest whole and half Diagonal Lines 4. For the Hips Length and Angles at Foot and Head Set the 30 Scales Square and count the shortest half Diagonal 12 foot 11 on one Leg and the Perpendicular 11 foot 5 on the other Leg then the
Compass Points so set lay a Rule Hips length and take the Bevel at both ends and it shall give the two Angles at head and foot of the shortest Hip and the same distance of the Compass Points shall be 17 foot 4 inches fere the Hip Hips length and Angles length as before and the Angle at the top 48. 30 and at foot 41 30 his Complement Again Set one point in 15 foot 10 the longest half Diagonal and the other point in 11 foot 5 inches the Perpendicular and lay a Rule to them and set the Bevel to both ends and you shall find 54 gr the Angle at the top and 36 the Angle at foot and the distance between the Compasses laid from the Center gives 19 foot 6 inches the longest Hip. 5. For the Out-side Angles of both Hips the longest first Take 19 6 between your Compasses the Hips length from the 30 Scale Set one point in 20 foot 5 the bevel-Bevel-end and close the Rule till the other Point touches 19. 6 the Hips length Then take the neerest distance from 18 foot 5 2 foot less Outside Angle of long Hip. than 20 foot 5 the breadth of the bevel-Bevel-end of the frame to the other 30 Scale and it is the neerest distance from the Point of the Rombus A to the Hip B G setup 15 foot 9 inches Then take out 25 foot 10 the shortest Diagonal and make it a Parallel in 15. 9 the neerest distance and then the 30 Scales are set to the Angle required for the Outside of the Long Hip being 110 degrees for Parallel 15 measured laterally on the Chord gives 110-0 6. For the Out-side Angle of the Shortest Hip. Take 17 foot 4 inches between your Compasses and set one Point in 20 foot 5 the Bevel end and open or shut the Rule till the other Point reaches 17 4 on the other 30 Scale Then Outside Angle of short Hip. the neerest distance from 22 foot 2 foot more than the breadth of the Bevel end of the Frame to the Common Line on the other 30 Scale and that shall be the neerest distance from L to C G the shortest Hip set up which is 18 foot 9 inches Then take out 15 foot 10 inches the half greater Diagonal Line because 31 foot 8 inches is more than the Scale of 30 and make it a Parallel in 9. 00 the half of 18 foot and the 30 Scales are set to the Angle required viz. 122 degrees for if you take out parallel 15 and measure it in the Chords Laterally it shall be 122 the Angle required Fig. III Use XVI To find the Rafters Hips and Angles in Bevel and Taper Frames being broader at one End that the other See Fig. III. First when the Frame is broader at one end than the other then the middle breadth is to be the guide for the Rafters length and the Perpendicular to be equal to the middle Rafters perpendicular on both ends though one pair of Rafters is longer than another and the Roof in winding thereby Which winding may many times be remedied by some convenient artifice or other a the ingenious workman will soon perceive As thus in brief Let A B C D represent the Frame of a House Bevelling at both ends and broader by 2 foot at one end than the other as here in the Bevelling figure being 20 foot on one side and 24 foot on the other side at one bevel-Bevel-end 10 foot 4 inches and at the other end 8 foot 1 inch ½ but at the nearest distance over only 10 foot and 8 foot First for your more apparent fatisfaction draw the true form of the Frame by as large a Scale as you conveniently can with the Sides and Angles as exact as you can as A B C D then draw the middle Line E F quite through the length and G H through the breadth of the Frame Perpendicular one to the other then measure G H as suppose 9 foot then lay off the half of G H from H to I and K then take out ¾ of G H and lay it from K and I to L 6 foot 9 inches for a pair of middling Rafters for this Taper House L G being the common Perpendicular at the middle and both ends of the Roof 5 foot o inch â… of an inch And for the Principal Rafters or other Rafters open the To find the length of the very Principal or single Rafter in a Taper Frame 30 Scale square and then count the Perpendicular 5. 0. â… on one Leg and half the distance between the Rafter-feet or half breadth of the Frame at that place on the other Leg and the Parallel distance between shall be the true length of the Rafter required Then for the Hips length first Make A M and B M equal to A E or B E at one end and make C N and D N equal to D F or C F and draw the lines M M and N N at both ends also make A b and B b equal to A B and C a and D a equal to C D and draw the Diagonal Lines A S and B S and C O and D O at each end extended and set down the measures of them These Lines being drawn you have the half Diagonal Lines A S and B S and C O and D O and may measure them by your Scale to find their Lengths or set the Rule to the Angles A E S and B E S find them by the 30 Scale also you have the Bevel ends of the frame and the neerest distance over from side to side of the frame Then for the Hips Length and Angles thus Set the 30 Scales square and count the perpendicular height L G on one Leg 5 0 â… and each half Diagonal Line c. one after another on the other Leg and the Parallel distance between shall be the true length of the several Hips required Example The Parallel distance between 5 foot 0 Inches â… the common perpendicular counted on one 30 Scale and 8 foot 1 Inch the longest Diagonal line A S counted on the other 30 Scale shall give 9 foot 6 Inches for the Length of one Hip A P Laterally And the Parallel distance between 5 0 â… the common perpendicular and 6 foot 6 Inches the other Diagonal line B S shall give the Lateral measure of 8 foot 3 Inches the length of B P the other Hip-rafter for the broadest end of the frame A B. Again the parallel distance between 5 0 â… the common perpendicular and 6 foot 4 Inches the greater Diagonal Line C O at the narrowest end shall give 8 foot 1 Inch for the Hip-Rafter C P. And the parallel distance between 5 0 â… the prependicular and 5 foot 0 Inch. 0 the Lesser Diagonal Line D O shall give 7 foot 2 Inches for the Hips Length D P to stand over the Diagonal Line D O. Note also that if to the Compasse points standing parallelly you lay a Rule and to the Rule so laid
and the 30 Scales at each end a Bevel and set it according to the Rule and 30 Scales it shall give the true Angle of the Hips at the Raising-piece King-Post which 8 Angles in these 4 Hips are exprest by the Lines and Letters in the Scheam thus P A S and P B S P D O and P C O the 4 Angles at the Raising Piece And S P A and S P B O P C and O P D the 4 Angles at the King-Post Whose length you may prove by Mr. Pope's excellent way making S P and O P equal to L G on the extended Diagonal Line and drawing the Lines A P B P C P D P for the 4 Hip-Rafters length required Or for more proof thus also by the Rule as before in Square Roofs Set the 30 Scales Square and take the Parallel Extent from 7 foot 2 ¾ the Rafters length at the broader end on one 30 Scale to 6 foot 2 inches more then ½ the bevel-Bevel-end by 1 foot the half quantity of Bevelling and it shall give 9 foot 6 inches for the Hip A P and from 7 2 ¾ to 4 2 one foot less than the half bevel-Bevel-end to 8 foot ¾ the Hip B P. Also the Par. Extent from 6 foot 5 ½ the Rafters length at the lesser end to 5. foot 0 ¾ one foot more than 4. 0 ¾ the half little bevel-Bevel-end shall give 8 1 ½ for the Hip P C. And the Extent from 6 5 4 8 the Rafters length to 3 0 ¾ 1 foot less than the half bevel-Bevel-end gives 7 1 ½ the Hip D P as before Note That by the working these 3 ways you may be sure to prevent any mistakes that may happen in working one way only Lastly For the Angles on the out-side or backs of the Hips Take every several Hips length Laterally between your Compasses and set one Point in the length of the Bevel-end or rather in a mean between A B the Bevel-end and M M at the greater end or C D and N N at the lesser end and open or shut the 30 Scales till the other Point falls on the Hips length that you work for then for the longer Hip count more and for the shorter Hip less by half the inches Bevelling then the whole breadth over at the end and take the Parallel neerest distance from thence to the 30 Scale for a neerest distance which neerest distance you must keep Then take the whole Diagonal Line Perpendicular to the Hip wrought for viz. the shortest Diagonal Line for the longest Hip at each end between your Compasses and make it a Parallel in the neerest distance last found and then the 30 Scales are set to the Angle of the back of the Hip required Example in this Figure for the longest Hip. Take 9 foot 6 inches the Hips length A P between your Compasses and set one Point in 10 foot one inch a mean between A B 10 foot 4 and M M 9-10½ and open or shut the Rule till the other point stands in 9 foot 6 the Hips length first taken Then the neerest distance from 9 foot 4 inches one foot less than 10 f. 4 inc the bevel-Bevel-end distance over to the other 30 Scale shall give 7-10 a neerest distance from the Point b to A P the Hip raised over A S the Diagonal Then the Lateral Extent B b of the whole Diagonal Line 12 f. 7 inc being made a Parallel in 7-10 the neerest distance last found shall set the 30 Scales to the Angle at R the back of the Hip required measured by taking Parallel 30 and measuring it Laterally on the Chords it is about 108 Degrees Secondly Take 8 foot 4 inches the other Hips length between your Compasses and setting one Point in 10 foot one inch as before open or shat the Rule till the other Point falls in 8. 4. the Hips length Then the neerest distance from 11 foot 4 inches one foot more than 10 foot 4 inches the distance of the bevel-Bevel-end to the Common Line of the other 30 Scale shall be when measured Laterally from the Center 9 foot fere for a neerest distance Then 15 foot 8 inches the longer Diagonal Line being made a Parallel in 9 foot fere the neerest distance last found sets the 30 Scales to 121 Degrees the Angle required the back of the shorter Hip required The same work serves for the other end being neer the same Angles Which you may prove by Mr. Will. Pope's excellent way thus find the middle between S and A or S and B at Q then the neerest distance from Q to B P or A P neer lay to R and draw the Lines R E R M for the Angles at R the back of the Hips required Moreover if you raise 4 Perpendiculars cutting the Points O and S the 2 places of the King-posts being Perpendicular to the raising Pieces A C and B D as the 4 Prick-lines ♉ S ♠S ♌ O and ♊ O do shew and lay the length of each Hip from his proper corner A B C D as A P from A to ♈ and ♠B P from B to ♈ and ♉ D P from D to ♊ and ♋ C P from C to ♌ and ♋ then draw lines from point to point as in the Figure Then ♠♌ and ♉ ♊ are the two Ridges when turned right over O S and C ♋ D is the least Hip and A ♉ B is the greater Hip as Mr. Pope hath well shewed Thus much for Hipt Roofs Use XVII To find the Length and Angles of every Principal Particular Rafter in Frames broader at one end than the other The Perpendicular as before was hinted is to be the same all over the Roof Therefore open the Rule square and take from the Perpendicular on one Leg to the half breadth of the Frame on the other Leg measure it from the Center and that is the length required For the Angles lay a Rule to the compass Points and set a Bevel as before is shewed and you have the Angles at the Raising-Piece and Ridge of the House to cut the Rafters feet by The same Rule serves to draw out a pair of well Stairs to give Hypothenusaes or strings at any particular height and breadth for the 30 Scales set Square and the Perpendicular height counted on one Leg and the breadth on the other Leg the measure between is always the Hypothenusa or string in flying Stairs as may plainly appear Use XVIII To find the Length and Angles of Collar Beams in any Roof Take the whole breadth of the Frame between your Compasses and set one point in the length of the Rafter on one Leg and the other point in the same place on the other Leg then the two Legs represent the two Principal Rafters and a Rule laid to the Compass Points represents the Raising piece then at any height that you please above the Raising-piece apply a Rule parallel to it and the measure between laid from the center gives the length remembring to add wood
for the tenons and a Bevel laid to the 30 Scale and Rule gives the true Angle to cut it by where the Timbers be square Use XIX To find the Lengths and Angles of Rafters and Purloyns in Bevel Frames The Length of the Rafters is shewed before to find it by the half breadth of the Square or Bevel-end and the perpendicular answerable to that Roof as afterward in the Example And the Angle of the Foot and outsides or backs of the bevel-Bevel-end Rafters and the upright of the Gable end must be to an Angle less and more than 90 degrees by the Angle at the Corner of the frame where that Rafter is to stand being more at the sharp Angle and less at the blunt Angle as in figure IIII you may see the true quantity of which Angle is thus found by the Rule Take the Length of the Rafter for the Bevel-end in Feet and Inches and make it a parallel in 15 then half the quantity of Feet and Inches Bevelling taken from the same Scale and carried parallelly till it stay in like parts shall shew right against it in the Tangents the Degrees and minutes required And this is the Angle that the blunt corner is to be laid in Legement more than a square and the sharp Angle less than a square or 90 degrees both out of Level and out of square also when you tumble or stripe in the tenons of the purloins the thing desired Fig. III Let A B represent the Bevel-end of a frame being out of square from the Line A C 6 Foot as the line C B sheweth then if A C be 20 foot A B will be 20 foot 10 Inches Then draw E L the middle Line of the frame and G M and H K the Lines at three quarter of the breadth of the frame then take E D the half Bevel and lay it from G to F and from H to I and draw the Line A F and B I for the outside Lines of the two bevel-Bevel-end Rafters and the two other Lines parallel to them according to the breadth or scantling of your bevel-Bevel-end Rafters as here in the figure 8 Inches broad Thus the Lines A F and B I represent the two end Rafters laid in Legement to fit in the purloins as for their lying out of square from the Raising pieces And to the same Angle they are to be laid out at Level that the cutting of the purloin ends may fit the Rafter sides when erected in their places according as the ends F and I of the bevel-Bevel-end Rafters are according to the Angles A F G and A F M the one being 11 degrees and 20 minutes under and the other 11 20 above 90 degrees Also Note that if P K and O M do represent a pair of square Rafters at any intended distance from A then T S and R Q will give the true Length of the purloins fit for those places R Q being the shortest purloin and T S the longest purloin To find by the Rule only how long the purloin must be on the outside more or less than the distance on the Raising pieces where you intend the two square Rafter feet shall stand do thus set the 30 Scales to the same Angle that the bevel-Bevel-end Rafters lie out of square when they lie in Legement to frame which here is to 11 degrees and 20 minutes Then count from the Center the quantity of Feet and Inches you intend to make the mortise holes from the Rafter foot in the Rafters for the tenons of your purloins and take from thence to the neerest distance to the other Line and that shall shew the quantity that the one purloin is to be longer and the other shorter than the distance between the Rafter-feet on the Raising pieces Example thus Set the 30 Scales to the Angle G A F then count A S the place for the mortise hole from A on the Rafter and take the neerest distance from thence to the other 30 Scale that shall give S V the quantity how much T S is shorter than A O and how much R Q is longer thâ—n P B. Note that if you count 3 quarters of 20 foot 10 Inches being the length of the bevel-Bevel-end that then you will make the Rafters too long by 4 Inches and a half as in the figure you may see therefore the surest way to find the Rafters length is to set the Scales to a square and then to take the parallel extent from the common perpendicular to the ½ breadth between the place for the Rafter feet on the Raising piece So that the length of the Rafters for the bevel-Bevel-end at true pitch is A F 15 foot 3 Inches and not A W 15-7-½ which is just 3 quarters of A B the bevel-Bevel-end over which the two Rafters are to stand For then the top of the bevel-Bevel-end Rafters would be too high for the top of the square Rafters being cut Just 15 foot three quarters of 20 foot the breadth of the frame Use XX. The use of the Scales to lay down or measure out on Paper or Board the Members and parts of the fine Columns and their Ornaments with their names and measures digested into a Table for the more ease and use of work-men For the drawing of the fine Columns and their Ornaments is largely before treated of in the former part by models minutes and quarters and considering the harmony between it and our Scales to 30 that are parted into 12 parts so that when every figure or foot on the Scales to 30 represent a model then every Inch is 5 Minutes and for small paper draughts that way of counting is neer enough but when you use it for a bigger draught that the whole scale of 30 may represent 3 models only then every figure will represent 6 minutes and the 12 Inches between are half minutes but in very great work as in Temples or Castles then the whole foot or scale to 30 may represent half of one model only and then every figure or foot is a minute and every Inch or small division is the 12th part of a minute for I count it to be a large Column whose Diameter at the base which is a model is above 3 foot and by this manner of computation which is natural to most scales the scales to 30 may be rendred convenient for small or moderate or large uses as in the following Examples may somewhat appear In Reading of which you must have recourse to the former figures whose members are marked with 1 2 3 4 5 6 7 8 9 10 and which figures direct you to the names in every of the columns which I found to be wanting in the former part so that the method here used is thus 1. The first Column in the Table is 1 2 3 4 5 6 7 8 10 11 12 c. answerable to the figures on the 6th figure in our present Example which will streight way guide you to the meaning of the names or termes in
THE Description and Use Of an Ordinary JOYNT-RULE Fitted with LINES For the ready finding The Lengths and Angles of Rafters and Hips and Collar-Beams in any Square or Bevilling Roofes at any Pitch and the Ready Drawing the Architrave Freize and Cornice in any Order WITH Other Useful Conclusions by the said Rule By John Browne LONDON Printed for William Fisher at the Postern Gate near Tower-Hill 1669. The Description and Use of a Joynt-Rule fitted with Lines for the ready finding the Lengths and Angles of Rafters Hips and Collar Beams in any Square or Bevelling Roofs at any Pitch FIrst The Rule is an ordinary Joynted Rule of a foot long when shut together or two foot being opened to a streight Line And the Lines delineated thereon for this purpose are 1. First a Line of Lines drawn Sector-wise from a center on both legs of the Rule but continued to 30 at the end in stead of 10 the old usual manner and every fingle Integer of the 30 is divided into 12 parts to represent every particular inch of the 30 foot according to the common reckoning by feet and inches 2. There is another Scale of equal parts also of the same length lying as neer to the other of 30 as may be on one leg only which is divided into 40 parts to represent 40 feet and each of those 40 feet parted into 6 parts to represent every two inches only because the room for one foot will not admit of more parts 3. In the same place on the other leg is divided a Line of natural Sines and Tangents to 45 but numbred as a Line of Chords to 180 Degrees to set the Rule to or to find the quantity of any Angle in the proper terms of expression all the world over degrees and minutes 4 On the innermost Line of the 30 Scale that runs to the Center is set 20 pricks beginning at 2 at the Center-pin at 30 and so proceeding with 3 4 5 and 6 at the Center-pin at 15 and then 7 8 9 10 and so forwards to 20 towards the Center which serves to divide a Circle into any number of parts very useful and ready in the practice of Drawing or Architecture Thus much for Description the Uses follow The uses of this Line of Lines or Scale of equal Parts to 30 drawn from the Center is of a general and manifold use as Mr. Gunter in his Book of the Sector hath shewed A brief touch whereof take in the first place as by the way and then the use of the Rule to the business intended And for the better doing hereof it is needful to explain three or four terms for the avoiding of many words and needless repetitions in this brief yet plain Discourse 1. First by the word Lateral is meant any distance taken either in feet and inches on the 30 Scale or degrees and minutes on the Chords taken and counted from the Center in the midst of the head of the Joynt-Rule a long any one leg as thus Suppose I would take out 15 foot Laterally set on point of the Compasses in the Center at the head and open the other to 15 on any one Leg on the 30 Scale this extent I call a Lateral Extent of 15 foot Also if you takethe Lateral Chord of 60 Degrees you shall find the extent of the Compasses from the Center to 60 to be the same as from the Center to 15 foot on the 30 Scale of Feet and Inches 2. By the word Parallel I mean any distance taken by setting one point of the Compasses in any number of feet and inches on one Leg and the other point in the same or any other number on the other Leg across from one Leg to the other as thus the Rule being opened then the extent of the Compasses from 20 on one Leg to 20 on the other Leg is a Parallel extent 3. In all Parallel Extents you must set one point of the Compasses in the Common Line on one Leg to the Common Line on the other Leg which Common Line is that only of the 30 Scale which runs to the Center in which the Center-pins at 15 and 30 are 4. The neerest distance from a point to a Line is only thus Set one point of the Compasses in the point given and open or shut the other being turned about till the other will but just touch or cleave the Line that I call the neerest distance Use I. To lay down a line that shall represent any Number of Feet and Inches given or required Take the Number given laterally from the 30 or 40 Scale from the Center and that is the Line required But if these Scales are too great or too small then take your Number of parts and the length thereof Laterally As for Example suppose I would have 3 Inches to represent 30 Foot take out 3 Inches between your Compasses and make it a parallel in 30 and 30 and the 30 Scale is set to your desire Use II. To Increase or Diminish a Line to any Proportion Take the given Line between your Compasses and make it a parallel in the parts thereof then the parallel extent of the parts you would have it Increased or Diminished to is the Augmentation or Diminution which was required Example Let 3 Inches represent 8 foot and to the same proportion I would have 10 Foot or 5 foot viz. more or less Take 3 Inches between your Compasses and make it a parallel in 8 and 8 on the 30 Scale then the parallel distance between 5 and 5 doth diminish the Line and the parallel between 10 10 doth Increase the Line to the proportion required Use III. To divide a Line into any Number of parts or models under 30. Take the given Line and make it a parallel in the parts on the 30 Scale into which you would have it divided then the parallel extent between 1 and 1 shall divide the Line accordingly Example Let 4 Inches be a Line to be divided into 9 parts take 4 Inches or any distance whatsoever and make it a parallel in 9 and 9 on the 30 Scale then the parallel distance between 1 and 1 shall divide 4 Inches into 9 parts required Note that for more exactness and conveniency you may Double 9 or Triple 9 viz. 18 or 27. and then if you make the Line to be divided a parallel in Triple the Number you must take out 3 in stead of one and that shall divide the Line into the parts required Example I would have 5 Inches put into 10 parts take 5 Inches between your compasses and make it a parallel in 30 and 30 the Triple of 10 then take out parallel 3 and 3 the triple of one and that shall divide the Line given being 5 Inches into 10 parts or models exactly the like for any other And note as the Rule stands you may take out any number of parts or models whatsoever to that Scale Use IV. Any two Lines given to find their
Side of the Square in Inches make it a Parallel in 12 then take out the Parallel Side of the Square and it shall give a Lateral 4th Number Then take out Lateral 12 and make it a parallel in the 4th number and then take out Parallel 12 again and it shall shew the answer in Inches laid Laterally from the Center Example At 9 Inches Square what makes a Foot As lat 9 to par 12 so is par 9 to lat 6 ¾ Agair As lat 12 to par 4th viz. 6 ¾ so is par 12 to 21 inches ¼ the length to make one foot required Use XII The Inches Square and Length given in Feet to find the Content in Feet and Inches As the Lat. Side of the Square to parallel 12 so is the parlength to a lat 4th Again As the la◠4th to parallel 12 so is the par side of the Square to Lateral Content Example at 9 Inches Square and 20 Foot long As lat 9 to par 12 so is par 20 to lateral 15 a 4th number Again As lat 15 the 4th to par 12 So is parallel 9 to lat 11 ¼ the Content required Thus much for the General Use being too long a Digression from the matter mainly intended Use XIII The Bredth of any Frame being given to find the Length of the Rafter and Perpendicular by Inspection only It being a General received Rule that the length of the Rafters should be three quarters the bredth of the House or frame for true pitch and 40 the feet in one Scale being equal in length to 30 the Number of feet in another Scale and 30 being 3 quarters of 40. Therefore If you seek for the breadth of the house on the 40 Scale then right against it on the 30 Scale is the length of the Rafter required Also if you seek the length of the Rafter on the 40 Scale on the 30 Scale right against it is the height of the perpendicular required viz. from the Raising-piece to the top of the Gable end or Rafter required Example If a house be 30 foot broad the Rafter ought to be 22 foot 6 Inches and the perpendicular 16 foot 9 Inches and a ½ for right against 30 counted on the 40 Scale on the 30 Scale is 22 6 the Rafter and right against 22 6 on the 40 Scale on the 30 is 16 9 ½ the perpendicular 1. Also by the Line of Numbers the extent of the Compasses from 20 to 15 will reach the same way from the breadth of any house to his proportionable Rafter at true pitch 2. And the extent from 20 to 18 turned the same way from any breadth of a house to his proportionable Hip Rafter in square frames 3. Also the extent from 20 to 11 18 100 will reach the same way from the width of any house to his proportionable perpendicular at square and true pitch 4. And the extent from 20 to 28 28 will reach from any other house breadth to his proportional whole Diagonal Line required at square and true pitch 5. And the extent from 20 the breadth to 16 63 the neerest distance at that breadth shall reach from any other breadth to his proportional neerest distance required if it were needful But the Angles in all Roofs great or small if true pitch and square are the same in all Frames Use XIV The breadth of the house and the height of the Perpendicular being given to find the Rafters length the Hips Length the Diagonal Line from Corner to the King-post and any Angle required in Square Frames What the Perpendiculer height of the Gable end ought to be at true pitch by the last Rule you may readily see and the better make estimate of the quantity of alteration Which being once Resolved on then thus proceed First open the two 30. Scales to a Right Angle by making Lateral 21 Foot 2 Inches ½ a parallel in 15 15 in the Brass Center pins 1. Then count half the breadth of the house on one Leg and the Length of the Perpendicular resolved on on the other leg then the parallel distance between them measured Laterally from the Center shall give the true length of the Rafter required Rafter 2. For the Hips Length count the Length of the Rafter last found on one Leg and the half breadth of the house on Hips the other Leg and take the parallel distance between and measure it from the Center Laterally and it shall be the true Length of the Hips required 3. For the Diagonal Line count the half breadth of the Diagonal house on both Legs and take the parallel distance between and measure it from the Center and it shall be the Length of the Diagonal Line from the Corner to the King-Post 4. Again for the Hips count the Diagonal Line last found Hips Length on one leg and the perpendicular height on the other Leg and the parallel distance between shall be the Hips true Length measured us before from the Center Example in a house of 20 Foot wide at true pitch See fig. 1. let A B C D represent a frame of a house 20 Foot wide B I the half width B A and C D equal to B C the whole width being 2 points to draw the Diagonal Lines by I E being equal to I B and the half E I laid from E to G gives I G the true length of the Rafters and G B or G C the Length of the Hips Or thus Three quarters of C B viz. C H gives C F and B F the Rafters Length the same extent also laid upon the middle Line from I to G gives CG and BG the Hips Length IF is the perpendicular height E is the point of the Diagonal Line or King-post perpendicularly opposite to or right under the meeting point of the 2 Hips and the two Rafters when raised and set in their places Thus much for Illustration what to do now for Application how to do The width of the house is always given the Length of the Rafter or the height of the perpendicular is next resolved on which in our Example being true pitch is also resolved on either by inspection as before or else by operation in this manner 1. The Breadth of the house being 20 Foot and the perpendicular resolved on to be 11 Foot 2 Inches ½ To find the Rafters Length work thus Open the 30 Scale to a Right Angle by use the 7th count 10 the half breadth of the house on one Leg and the For the Rafters length perpendicular height 11 2 ½ on the other Leg. viz. both on the 30 Scale from the Center and take the parallel distance between them on the common Line and measure it from the Center and it shall give just 15 Foot the length of the Rafter required C F. 2. But if according to the width of any frame you resolve on the Rafters length and would have the perpendicular height of the Gable end then