' MEASURING made EASY, Or the Dcicription and Ufe of Coggefhall's Sliding Rule : Containing In flructions for Meafuring all Manner of TIMBER, Both by the Common Way, and the true Way : With Directions for taking the Dimenfions of Trees, and the Allowance for Bark, &c. Performed both by the Rule, and by Arithmetick ; By which may be meafured All MANNER of SUPERFICIES, As Board, Glafs, Plaiflering, Painting, Wainfcotting, Tyling, Paving, Land, &c, Both by the Rule and Arithmetic. To which is now added, The Defcription of Scamozzi's Lines, with the Ufe in finding the Length and Angles of Rafters, Hips, Collar-Beams, &c. By J. G o o d, Teacher of the Mathematics. Carefully Corrcded, and much Enlarg'd By J. ATKINSON. I O N D O N : Printed for MOUNT & PAGE, Tower-bill, 1786. 3 ) T O T H E READER. I Here prefent you with Jome Fruits of my /pare Hours, wherein I have endeavoured to render the Ufe of the Sliding-Rule more Plain and Eafy than ever it appeared before. I have here delivered in a few Rules, but yet in a Plain and Eafy Manner, Problems for the meafuring all Manner of Regular Figures, and Plain Superficies, fuch as Board, Glafs, Paint- ing, Plaiftering, Paving, Tyling, Joyners and Mafons Work, &c. and that both by the Sliding. Rule, and by Arithmetick. In the next Place, you have Directions, for the meafuring all Manner of 'Timber or Stone, by the Sliding-Rule, and Arithmetically, by a new Method, whereby the true Content of Round Tim- ber or Trees, may be found. I remain, a Well-Wifoer to the Mathematicks, JOHN GOOD. 2007357 ( 4 ) CHAP. I. The Defcription and UJe of Coggeihairj Rule. *i H I S Rule is framed three Ways ; Sliding by one another, like Glazier s Rule ; Sliding on one Side of a Two. Foot Joint Rule j and one Part Sliding on the other, in a Foot long, the back Part being flat, on which are fundry Lines or Scales. Upon the forefaid Sliding Side of the Rule, are four Lines of Numbers, three are Double Lines, and one a fmgle Line of Numbers ; marked in the Figure annexed, with A, B, C, and D. The three marked A, B, and C, are called Double Lines of Numbers •, being figured i, 2, 3, 4, 5, 6, 7, 8, 9. Then, 1, 2, 3, 4, 5, 6, y y 8, 9, and 10 at the End. That marked D, is the Single Line of Num- hers, and figured thus, 4, 5, 6, 7, 8, 9, 10, 20, 30, and 40 at the End, even with and under 10 in the Double Line next to it, this is called the Girt-line, and is fo marked in the Figure. * The ( 5 ) The Figures on the Three Dcuble Lines tf Numbers may be encreafed or decreafed at pieafure ; thus, i at the Beginning may be called 10, too, or iooo, then 2 is 20, 200, or 2000, fo that when 1 at the Beginning is 10, then 1 in the middle is 100, and then 10 at the End is 1000; but if 1 at the Beginning is counted for 1, then 1 in the Middle is 10, and 10 at the End is 100. And as the Figures are altered, fo mud the Strokes or Divifions between them be altered in their Value, according to the Num- ber of the Parts they are divided into : As thus, the Diftance from 1 to 2 is divided into 10 Parts, and each Tenth is divided into 5 Parts ; and that from 2 to 3 is divided into 10 Parts; and each Tenth into 2 Parts, and fo on from 3 to 5 •, then the Diftance from 5 to 6, is divided into 10 Parts only -, and fo on to 1 in the Middle of the Rule, or the Firft Part of the Double Line cf Numbers. The Se- cond Party or Radius, is divided like the Firfi Radius. The Girt Line marked D, is divided from 4 to 5 into 10 Parts, and each Tenth into 2 Parts, and fo on from 5 to ;c, then from 10 to 20, it's divided into 10 Parts, a,nd each Tenth is divided into 4 Parts ; and fo on all the way from 20 to 40 at A 3 the ( 6 ) the End, which is right againft 10 at the End of the Double Line of ' Numbers. The Lines on the Back-fide, of this Rule that flide on the one Side, are thefe, a Line of Inch Meafure from i to 12, each Inch divided into Halfs, Quarters, and Half- Quarters -, another Line of Inch Meafure from. 1 to 12, each Inch divided into 10 equal Parts ; and a Line of Foot Meafure, being one Foot divided into 100 equal Parts, and Figured 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100 even with 12 on Inch Meafure. And the Back-fide of the Sliding Piece is divided into Inches, Halves, Quarters, and Half Quarters, and figured from 12 to 24; fo that it may be (lid out to 2 Feet to mea- fure the Length of a Tree, or any Thing elfe you have occafion to meafure. Note, The Line of Foot-meafure turns Inches, Halves, Quarters, &c. into Decimal Parts of a Foot* which is moll ready for measuring by this Rule, as alfo by Arithmetick •, which is fully explained in the Table following. ( 7 ) STABLE Reducing Inches, Halves, Quar- ters, and Half- Quarters, into Decimal Parts of a Foot; Or, reducing Inch -Meafure into Foot Meafure. 3 n XT n O 1 n 3 Deci- mal • OCO Parts of an Inch. 1 .010 1 .02 1 . 104 1 8 03I .1 ,5 .190 .28l ■3 6 5 .448 ■53i .615 .698 .781 .865 1 <2 .042 .125 .208 .292 .275 .'4£8 542 .62^ .708 .792 •?75 .052 i ,o6i •073 083 .167 .094 •135 .21 9 302 . 146 ,i 5 6 .177 .26 .271 •35 .43^ .521 .-04 .688 •77 1 ■854 •937 .A 29.24 4 •3^3 •3+4 .385i.396.406 •469J.479 -49 5 .417 .427 •5< 6 7 8 _9 re •5 5521.569 •573 •5 8 3l-59* 6.351-646 .656 .667.677 75 i-/6 •7i9 .802 .885 .969 •729 12 •74 .823 8 3 3«.8^4 .89 1.906 1 1 L917J.927 .94S •95* 979 .99 By this Table } or Half Quarter of an Inch in Foot Meafure in Decimals, is 01, or ,4^. parts of a Foot; j of an Inch is .021, or fhorter .02, that is ,^ parts of a Foot* and fo on in the upper Line of the Table. And in the next Line under it, 1 Inch is A 4 ,083 ( 8 ) .083 of ,4o- of a Foot; Inch 1 i is .094 or ,|-o » Inch i| is, .104 or / ; Inch 1 f is . 115, or .VV of a Foot and fo on; And in the Firft Column under Decimals 1 Inch is .083 - r 2 Inches .167; 3 Inches is .25; 4 Inches .333 ; and fo on to 11 Inches is .917 or ij^- All thefe may be found on the Line of Foot- Meafure on the Back-fide of the Rule. CHAP. II. The Ufe of the Double Scale, /hewing how to find the Area of any Plain Superficies -, and alfo Arithmetically. P R O B. I. To me of ure a Geometrical Square* BY the Sliding - Rule. Let there be a Square whofe Sides are each three Feet and a half. Set 1 on the Line B, to 3 -§■ on the L'ne A, then againft 3 \ on the Line B, is 12 £ Feet on the Line A, which is the Content of fuch a Square. 1 1 * 1 3 1 " 4 1 5 1 6 1 b 7 1 8 1 9 1 / c \ d \ e 1 g A rich- ( 9 ) Arithmetically. F.P. The Side of the Square, Feet 3-t or — 3.5 Multiplied by itfelf . — . 3.5 *75 105 The Product is ■• Feet 12.25 This Arithmetical Operation is thus per- formed, multiply the Sides cf the Square, namely 3^, or 3 ^ into itfelf, the Product is 1225, from which cut off 2 Places, becaufe there are 2 Decimals in both Fractions, and it will (land thus 12 i4v > which is 12 Feet and a Quarter. Note^ The Figure itfelf mews the Content in Feet by Infpcdtion, for if you tell the great- er Squares, you may fee they are 9 in Number; a and h makes 10; c and d makes 11 ; e and/ makes 12 ; and the little Square g in the Corner, makes one Quarter of a Foot, which is in all 12 Feet, More "Examples by the Rule. Length of the Side 5 J Feet Length of the Side 7^ Feet Length of the Side iol Feet Length of the Side 13 Fcetj 27! Feet 56 1 Feet H5I Feec 1 69 Feet PR OB. ( '° ) PR OB. II. To meafure a Long-Square by the Sliding-Rule \ and alfo by Arithme- tic k. LE T there be a Long Square, whofe longed Side is 27 Feet and a Half, the flaorteft 16 Feet and a Quarter : fet 1 on the Line B, to i6£ on the Line A, then againft Q.j\, on the Line B, is 446J Feet the ' tent of the Long Square on the Line A. on- By Decimal Arithmetick. Multiply the Length by the Breadth, and from the Product cut off as many Places to the Right-hand as there are Decimals in the Length and Breadth ; the Integers remaining to the Left-hand are the Square Feet. Example. F.Par. Length ^^\ Feet, or — — 27.50 Breadth 16 1 Feet, or ■ 16.25 13750 5500 16500 2750 446.8750 But ( >I ) But Contracted thus. F. Par. Breadth Feet 16 £ or in, Decimals — 16.25 Length Feet 2-j\ or 27.25, inverted is 5.72 3250 "ST 81 The Product is the Content req. Feet — 446.8 In this Contracted Way, Multiply each Fi- gure of the inverted Number (beginning at the Right-hand) by the Figure over it, and putting the feveral Multiplications even at the Right-hand, as above : Thus multiply- ing by 2, the Product is 3250: Then fay 7 times 5 is 35, leave out 5 and carry 3 in your Mind, then fay 7 times 2 is 14, and 3 I car- ried makes 17, fet 7 juft under the o,. and car- ry the one in your mind ; fay 7 times 6 is 42, and 1 I carried makes 43, fet 3 under 5 and carry the 4 in ycr.r mind •, and (o on to the End of that lline ; Then lay 5 times 2 is IPj reject the o, and carry one in your mind ; then fay 5 times 6 is 30, and 1 carried makes 31, fet 1 juft under the 7, and carry the 3 in your mind j laying 5 times 1 is 5, and 3 T carried makes 8, fettir.g it under the 3, an<£: the multiplying is done : Then add them to- gether and it's 446,8 j that is Feet 446 T -*- or 446 |^as before; Ncte. ( " ) Note* Where the Units place of the invert- ed Number ftands, fo many places are to be cutoff in the Product; As in this Example, 7 the Units place, when inverted ftands under 2, the firft place in Decimals, and therefore one place muJft be cut off in the Product of the Right-hand; fo that rhe Product 4468 and be 446.8 or 446 to ; obferve the like in all that follows. More Examples by the Sliding- Rule. Contents Breadth 11 \ Feet] Length 15 |f 174 \ Feet Breadth 17 \ Feet > Length 21 \\ 371 % Feet Breadth 22 \ Feet J Length 19 J1.676 I Feet P R O B. III. Hoi- to meafure a Rhombus by the Sliding Rule. • Uppofe the Side of a Rhombus be 8 Feet 6 Inches and a quarter, and the Breadth, or Line A B, 8 Feet 4 Inches and a halt : That is, the Length is Feet, 8 iw, and Breadth is Feet rll. Set 1 on the Line B, to 8 Feet iU on the Line A •, then againft 8 Feet 10-Son the Line B, is 71 Feet i-Sff parts of a Foot on the Line A; Now if you would find the Value of the Decimal, or part of the Foot look for via on your Rule, and you will find againft it 4J Inches, fo that the Content of this Rhombus is 71 Feet 4 ^ Inches, By ( >3 ) By Decimal Aritbmetick. Multiply the Breadth AB, by the Length of any of the Sides, and from the Product cut off as many Places to the Right-hand as are Decimals in the Length and Breadth, the In- tegers remaining to the Left-hand are the fquare Feet required. Example. F. P. The Length Feet 8.o6| Inches, or 8.52 Breadth AB, Feet 8.04-!- Inches, or 8.38 a 08 16 3W 6816 The Content is Feet 71.3976 or 7 1 Feet 4 J Inches. And contracted is thus : F.P. Length Feet 8.06 Inches 8£ is in Decim. 8.52 Breadth AB, Feet 8.04 Inches and \ is 7 g g 8.38, inverted is -. \ '^ 6816 *55 68 Then Content required as before is Feet — 71-39 PR OB« f 7 A N 'p ROB. IV. How to meafare a Rhomboides by the Sliding Ruk. A Dalit the Length of a Rbomboides be 17 Feet. 3 Inches, or 17 ,44, and the true Breadth 8 Feet 7 Inches, or 8 ,44, what is the Content ? Set 1 on the Line B, to 17.25 on the Line A-, then, againft 8.58 on the Line B, is 148 Feet on the Line A. By Decimal Arithmetick. Multiply the Length by the Breadth, and from the Product cut off the Decimals, and the Remainder is the Square Feet. Example. F.Par. The Length 17 Feet 3 Inches, or 17.25 The Breadth 8 Feet 7- Inches, or-— 8.58 /]& 17- *5 / I38CO 8625 13800 The Content is Ft. 148.0050 or 1.48 Feet. By Contraction tbus-> F.Par. -Length 1 7 Feet 3 Inches in Decimals — 17.25 .--Breadth 8 Ft. 7 Inch, or 8.58, inverted is 85.8 The Content as before is PR OB. ( *S ) P R O B. V, How to meafure a Triangle hy the Sliding-Rule. Theorem. Fj^Very Triangle is half of that \jj Long-Square whofe Length and Breadth is equal to the Perpendicular and Bafe. Therefore from the greateft Angle let fall a Line Perpendicular, to the Side (op- pofite to the greateft Angle) called the Bale; and then to find the Content of any Triangle, rake half the Length of the Bafe and the whole Perpendicular; or half the Length of the Per- pendicular, and the whole Bafe. Example* Let the Bafe of a Triangle be 4 Feet, t4£ or 4. Feet 1 Inch f, and the Perpendicular 2 Feet-ril, or 2 Feet 1 Inch f, the half of the one is 2 Feet 7 parts, and the half of the o- ther is 1 Foot 7 Parts ; fet one on the Line B, to 4 5 on the Line A, then againft 1.07, half the Perpendicular on the Line B, is 4 Feet and aim oft half for the Content. Or if you fet 1 on the Line B, to 1.07 on the Line A ; againft 4,1 , on the Line B, is 4 and alnoft a half on the Line A. Again, another way, if you kt I on fne. Line B, to 4.1 on the Line A •, then Aguinft 2.15 on the Line B, is 8 Peet -5-! ( which is about ij Inches' on the Line A, 'the half thereol- is 4 Feet ^ j Inches, which is the Con- tent of the Triangle. By i 16 ) By Decimal Arithmetic!: all the three JV-ays, F. P. (The whole Bafe 4. r 5 1.07 i/Way j Thc h aif p cr p CJ ,di C ular is 2905 415 The Content 4 Ft. 5 i Inches or Ft. 4.4405 , ur C The whole Perpendicular — 2.1c 2 meafure any Regular Figure. OF thefe Regular Figures there are feveral Sorts, as the Pentagon, contained under five Sides •, the Hexagon, contained under fix Sides ; the Heptagon, contained under fe\cn Sides •, and the Octagon, contained under eight Sides, C5V. Now to meafure any of thefe Sorts of Fu jgures practically, it is by dividing thern into lij.n^lcs, which is done by drawing Lines from the Center of the Figure to every Angte, then from the Center to the middle of any one of the Triangle Sides, draw a Line, which Line is the Perpendicular. Having the rer- pendicular and Bale of any or thefe Tri- angles (by Prgb. 5.) find the Content of one Triangle, and that multiplied by the Number of Triangles, finds the Content of the Figure. Note, To find the Center of any Regular Figure, of even Number of Sides, draw a Line from one Angle to it's oppofite, the middle of which is the Center •, but if your Figure have any odd Number of Sides, as 5, or 7, &c. draw Lines from any two Angles to the middle of their oppofite Sides, their In- terfecuon is the Center, B 2 CHAP, ( 20 ) CHAP. III. The Ufe of the G\n-\\ne, and Line of Numbers, called the Double-fcale, in meajuring of Circles, and their Parts. Prob. i. ET the Diameter of a Circle be JL/ i Feet T |4, to find the Content, fet ii on the Girt-line D, to 95 on the Double- line C ; then againft 2 Feet ,-Ji on the Girt- line D, is 3 Feet T |4- on the Double-fcale of Numbers C, which is the Content. Arithmetically, 'The Rule. Multiply the Diameter in Inches by itlelf, and then by this fixt Number .7854, and from the Product cut off from the Right- hand 4 Figures, the Remainder to the Left- hand is the Content of the Circle in Inches, which divided by 144, gives the Feet, and the Remainder, if any, divide by 12, gives the Inches. Example. The fixt Number ■ — .7854 The Diameter multiplied ■■ 729 Inches 70686 The Diameter 27 Inches 15708 The Diameter 27 Inches 54978 F.In. 189 144)572.5566(3. 11 54 432 Piam.multip.729 In. 12)140 12 021 OI2 09 The Content is 3 Feet 1 1 J Inches, the ( 21 ) The Rule may be thus, Multiply the Diame- ter in Foot-meafure by itfelf, the Product is called the Square of the Diameter,, then multiply by the fixt Number 0.7854 in the Contracted way. and this hft Product is the Content required, as follows. F.Par. The Diameter 2 Feet T -J4- or 2.25 The Diameter 2 Feet —-£- inverted is 52.2 45° 45 11 The Square of the Diameter is Feet 5.06 The fixt Number 0.7854 inverted is — — 4587 354- 405 2 5 2 Content 3 Feet 11J Inch, almofl, or Feet 3.974 P R O B. II. To find the Content of a Semicircle, BY having the Diameter by the laft Problem, find the Content of the whole Circle, the half thereof is the Content of the Semicircle. I think this needs no Example. PROB. III. To find the Content of a Quarter of a Circle, ammonly called a Quadrant. £w/^-npHERE is a Quadrant, whofe J Semidiameter is 7 Feet, and B 3 the ( 22 ) the Circuit oi the Arrh is 1 1 Feet, what is the Content? Set i on the Double-line B, to 7 the Semii-iameter on the Double-line A, then againft 5 Feet y %~ which is is-.lf the Circuit of the Arch) on B, is 3 hi on A, the Content 38 Feet 6 Inches, or 38 T |. PROB. IV. To find the Content cf a Sector of a Cirele. fls Rule. QET 1 on the Double-line, to ^3> the Semidiameter on the -Double- line, then a^aLnft half" the Circuit of the Arch on the Doubje-iih'e, is the Content on the other Double-line. This needs no Example. PROB. V. How to me af ure the Segment oj a Circi'e. Segment of a Circle, is a ftrait Line drawn ccro' the Circle, but not th-o' the Cen- ter, divides a Circle inio two Parts or Seg- rrer.es \ an J the lelTer is thus meal'ured Let the Seller be meafyred, whereof the Segment is pare, then fub {tract the Triangular Part; the Remainder is the Content of the Segment : But to the greater Segment, the Content of the Tri- angle included is to be added. CHAP. ( n ) CHAP. IV. The Ufe of the Double-Scale •, /hewing how to medjure all manner of Stipe fi .es ; as Board t Glafsy Painting, Plaifierir.g, Wainfcotting t Tyling, Paving, &c. by the Sliding-Rule, and Arithmetically. Firft. In meafuring Boards*. P R O B. I. Let there be a Board whofe Breadth is if\ Inches, or 27 ,|4j ar >d the Length 15 Feet J, or 15 <-§-§-, what is the Con- tent by the Sliding Rule. Example. Set 12 on the Double Scale B, to 27^ on the Double Scale A, then againft 15I Feet on the Double Scale B, is 35 Feet, the Content on the Double Scale A. ■Decimally. The Rule. Multiply the Length in Feet by the Breadth in Inches, and cut off fo man/ Figures from the Right-hand, as are Decimals in your Length and Breadth, the Remainder divided by 1 2, the Quotient is the Feet, and the Remainder (if any) the odd Inches. B 4 Example^ ( 2 4 ) Example, Length is Feet 15.25 Breadth is ■ Inches 27.50 76250 10675 3050 rr *2)4i9-375°(34-i* 36 o59 The Content is 34 Ft. 1 1 Inch. 48 11 Inches The Length 15 Feet J, or Feet 15.25 Breadth 27 Inch. -|, or 27.5 inverted is — 5.72. 3050 1068 7 6 Product is 419 Inch. ,4 as before ; Inches 419.4 But if the Breadth be taken in Foot-meafure, it's done without Divifion as follows, F.Par. The Length as before, 15 Feet £, — — 15.25 Jkeadth 27 Inch. £, or 2 Ft. ,-£§ inverted 92.2 Content is 34 Ft. 1 1 Inch, as before, or Ft. 34 92 And by the Sliding Rule it's thus : Set 1 on the Doyble Scale B, to 22.9 on the Double Scale ( *s ) Scale A; then againft 15.25 on the Double Scale B, is almoft 25 Feet on the Double Scale A, the Content as before. Or, by the Rule of PraRice, when the Length is in Feet and Inches, and the Breadth alio in the fame, counting for 1 Foot in Breadth, the whole Length is the Content, and for the Inches, according to its Aliquot Parts of a Foot, or 12 Inches; as it follows, in the lad Example. A Board whole Length is 1 5 Feec 3 Inches, and Breadth 27-^ Inches, or 2 Feet 3! Inches. F.Inch. The Length ■ ■ — i5-°3 Set it down again — - — - — 15-03 Divide by 4, becaufe 3 Inch, is £ of a Foot 3. 09 J Div. thelaft by 6, becaufe -|In. is 4- of 3 In. o.oy\ And altogether is the Content required — 34.11!- In like manner the Examples following may be done, both by the Foot-meafure, and by this Rule of Praffice, which is worth the Learner's minding. More Exapiples. The Breadth 9 Inch } Length 1 3 Feet C Cont. 9 Feet £ The Breadth 21 Inch £ Length 19 Feet< Cont. 3 3 Feet J The Breadth 25 Inch 3 Length 29^ Feet £ Cont. 61 Feetf P R O B. II. Another Way when the Dimen- fions are Feet and Parts, and the Content is required in Feet and Parts, by the Sliding Rule. Let C 26 ) Let there be a Board whofe Length is 24 Feet f, and the Breadth 1 Foot £, what is the Content ? Example. Set 1 on the Double-Scale B, to i-i on the Double-fcale A; then againlt 24I on the Double-fcale B, is 37 Feet r « on the Double fcale A, which is the Content. More Examples. Breadch 3^ Feet *J Length 20 FcQtC 65; Feet Breadth 7| Feet (Length 25 Feet J 187^ Feet Breadth 12 Feet (Length 30 Feet] 360 Feet Breadth 15 Feet J Length 35 Feet (.525 Feet How to work this fecond Problem- Decimally* as is taught Chap. 1. Prob. 2. Example. F.Par, Length . 24.75 Breadth * ■ ' ■ 1.50 122750 2475 Content is 37 Ft. or Si- Inch or Feet, 37. 1 250 And by Contraction thus, the faid Example. F.Par, Length 24 Feet \ x or ■ 24-75 Breadth 1 Foot ^, or 1.5, inverted is — - 5.1 III! I ■ 2475 I2 37 The Content 27 Feet 1 Inth J-, or Feci — 37. 1 2 Example* ( 27 ) Examples Decimally contrafted. Ereadth 4.27 -^ Len. 06.21 {* 26-^ or 26 feet 6 Inc. 95 feet 9 Inc. 192 feet 5 Inc. ■§ [58 feet 6luc. £ Ereadth 4.27 -^ Len. 06.21 <~ 26-^ or , Breadth 8.46 (Len. 11 32 1 9^ T H or < Breadth 1 1 54 f Len. 1 5 . 35 ) 92 T + ^ or i< Breadth 16.74 * ^ en - 2 i-4 2 ^35^ t'I or 3; PROB. II 1. Directions for the meafuring of mofl Sorts of Artificers Works, and firfi of Glazier's Work, with the Manner of taking Dimenfions. THE beft way to meafure Glazier's Work is by the Sliding Rule, and the Dirr.en- fions are taken very exact even to 4 °f an Inch ; they commonly agree for their Work by the Foot, whether the Glafs be Old or New, Squares or Quarries. Note, Sam-Win- dows are glazed by the Square ; that is, they tell how many Squares there be in all the Lights, and then reckon what they come to a* fo much the Score or Dozen. PROB. IV. Let there be a Pane of Glafs ig\ Inches long, and 7 Inches broad , what is the Content ? Fxample. By the Sliding Rule. Set 144 (reprefented b> 1.44.) or. i!e Line B, to 7 Inches on the Line a, rher. jg§ i^it 20 - on the Line B, is 1 Foot and almoft a Half on the Line A, or Deci- mally i toot A-t-i Mon ( 28 ) More Examples, Breadth 3* Inch -j Length zo\ Inch ("49 parts, or 5 In. -5 Breadth 5I Inch / Length 2jj| Inch \ 92 parts, or 1 1 In.i Breadth 9^ Inch C Length 30I Inch] z feet 1 Inch, £ Breadth 13 £ Inch J Length 35 Inch t 3 feet 2 Inches,^. The foregoing Problem Arithmetically. Multiply the Length in Inches and Parts, by the Breadth in Inches and Parts, and from the Right-hand of the Pro duel - cut off lb many Figures as the Decimals in the Length and Breadth (if there be any) and the Remainder divide by 144, the Quotient is the Feet, and if any thing remain, divide it by 12, the Quo- tient is Inches. Ex empty. Length in Inches and Parts — — 20.5 Breadth in Inches and Parts — -• 7.0 144)206.50(1.5 144 12)062(5 Inches 60 The Content is 1 Foot 5 Inches 02 and Remainder 2. To perform this Decimally, without Divi/ion, Take the Length and Brtadth in Foot mea- fure, and then it's thus j Example. ( *9 ) Example. F. Par. The Length 29^ Inches, or. 2.46 The Breadth 7 Indies or 0.58, inverted is — 85 1230 l 97 The Content Is 1 Foot 5^ Inch, or Feet 1*427 Arter the lame manner may all the foregoing Examples be wrought. P R O B. V. "Dire 3 ions for measuring Joyners and Painters Work; with the Manner of taking their Dimenfions . Oyners and Painters Work, are generally agreed for by the Yard ; and therefore ha- ving caff up their Dimenfions, thty bring the whole Sum into Yards, by dividing the Feet Now in taking the Dimenfions of Joyners or Painters Work, fuch as Polleftion, and Bead- work, you mull by a Line girt Bends and Hol- lows, and fo bring it down to the Bottom, alter which meafure the Length of the Line by your Rule, fctting it down for one of the Di- menfions, then meafure the Length of the for- mer Height, and fet it down for a lecond Dimenfion. Note j That in Joyners Meafure, Window-fJjut- ters and Doors, are Work and haif } becaufe they are worked on both fides. The { 30 ) The Paintings ;of Windows are generally agreed for by fb much the Light , and Cafe- men ts> at fe much a Cafement. In Joyners and Painters IVork, Deductions are made for all Windows and Chimneys. Example. A Joyner hath wainfcotted a Room 44 Feet and a Half in Compafs, 9 Feet 3 quarters in Height j what is the Content by the Sliding- Rule. Set 1 on the Double Scale B, to 44 Feet \ -on the Double Scale A, then againft 9 Feet three quarters on the Double Scale B, is 433 Feet -f.-g- on the Double Scaie A, the Content : Or thus, kt 9 on the Line B, to 44^ on the Line A, then againft 9J on the Line B, is 48 Yards on the Line A, which is the Content. Decimally. F. Par The Compais of the Room 44.50 The Height ■ ^ 9 75 22250 40050 _ _ yds. feet in. 9)433-875°U8 01 io£ 3* The Content is 48 7*Wj 073 1 Foot io Inches -| 72 01 And ( I* ) And by Contra ft ion, the faid Example is thus, F.Par. The Compaft of the Room 44! Feet is 44.5 The Height 9 Feet £ or 9.75 inverted is 57.9 40.05 3" 11 The Product is ■ Feet 433-8 The Feet multiplied by o. 1 1 1 inverted is 1 1 1 .0 4338 434 45 The Content as before in Yards is ■ 48.15 Hozv to meajure Painters JVork by the Sliding- Rule. Example. There is a piece of Painting 13 Feet ■§■ broad, and 23 A Feet long. Set 9 on the Double Scale B, to 13 \ on the Double Scale A, then againft 2*3 -| on the Double Scale B, is 35 Yards £ on the Double Scale A, which is the Content of fuch a Piece of Painting, Decimally ( 3* ) Decimally. F.Par Length 23-I Feet or — — 23.50 Breadth l$l Feet or « }3 m $° 1 17500 7050 2 35° yds.Ft. In. 9)317.2500(35: 2 : 3 27 047 Content is 3 5 Yards 45 2 Feet 3 Inches 02 feet More Examples. Breadth 4 | feet") Length 22 Breadth 15 ^ feet I Length 19 •§• Breadth 10 feet f Length 30 Breadth 19 feet J Length 23 P R O B VI. *the meafuring of Plaijlerers and Painters Work by the Sliding Rule. Example 1 . Of Plaifiering. LE T there be a Ceiling plaiftered, whofe Length is 25 Feet 3 Inches, and the Breadth 15 Feet 6 Inches : Set 9 on the Double Scale B, to 25 \ on the Double Scale A ; then againft \§\ on the Double Scale B, is 43 Yards and almoit a half, on the Double Scale A, which is the Content. Example ( 33 ) Example i. Of Paving, Let there be a Piece of Paving 34.1 Feet long, and i2-£ Feet broad ; fet 9 on B, to 34-*- on A, then againft n\ on B, is almoft 43 Yards the Content. P R O B. VII. TH E meafuring fuch Superficies as are' meafured by the Square of jo Feet (viz. 100 Feet in a Square ; and if you multiply 10 by 10 it makes 100) fuch as Flooring, Roofing, Partitioning, Tyling, &c. By the Sliding- Rule. The Rule. Set 100, or 1 in the middle, to the Length ; againft the Breadth is the Content. Example. There is a Floor whofe Length is 61 7 Feet and a half, the Breadth 14 Feet and a quarter, fet 100 on B, to 6 if on A •, then againft 14^ on the Double-Scale B, is 8 Square J on the Double -Scale A, for the Content being 8 Square 75 Feet - Decimally. Multiply the Length by the Breadth, and from the Product cut off two more Decimals than is in the Length and Breadth, the Figures remaining arc the Squares. C Ex. ( 34 ) Example. F.Par. Length 6\ \ Feet, or in Foot-meafure — 61.50 Breadth 1 *£ Feet, or in Foot-meafure — i4- 2 5 30750 12300 24600 6150 Content 8 J Squares, or y6 Feet 8-763750 By Contraction, the fame Example is thus, Breadth 14^ Feet is in Foot-meafure Feet 14.25 Length 6i| jr eet) or 6j^ inverted is — 5.16 855° 142 7± Product (cutting off 3 places is) Square 8.763 More Examples. Contents. Length 42 feet r Breadth 24I feetjio fquares 29 feet Lengtn 64 feet -j Breadth 29 feetiiS fquares 56 feet Length S6 feet ( Breadth 35! feet } 30 fquares 53 feet Deductions in this Problem. In Carpenters Work, you are to deduct for all Well holes in your Flooring for all Chimney - hearths, and the like. Alfo in Partitioning, Deductions mnft be made for all Doors and Windows that are meafured in. In Rocfiyig make no Deductions for Window- Shafts, or Sky- Lights, becaufe they are more Trouble to the Workman than the Stuff is worth that would cover them. CHAP, ( 35 ) CHAP. V. Having in the former Part of this Treatije fhezved^ the meafuring ef all Manner of Superficies, like- wife all Sorts of regular Figures ; I fhali n;zv give you fome Inflruclions for the taking the Dimenfions cf Trees, or round Timber, and jo proceed to the meafuring of fuch, and all Mi.nuer of Solids. 1"^ H E Tree being cut down, the Cuftom is to Girt, it, as is fhewed in the begining of the next Chapter, and for the Length they account from the But-end, up fo far as the Tree will hold half a Foot Girt, when twice folded. The Dimenfions being taken, the Tree is to be meafured by Chapter the fixth, as fquare Timber. If the Tree have great Boughs that will hold half a Foot Girt, fuch Boughs are called Timber, and they are meafured and are added to the Whole. 7. Note, If round, rough Timber be mea- fured for Sale, the common Way among Artificers in allowing for Rind or Barls. is thus : If the fourrh part of the Girt or Cir- cumference of the Tree be T -§4 or half a Foot, they allow ,*° or half Inch half quar- ter. If the fourth Part of the Circumference C 2 ( 36 ) be a Foot, they allow ,|£ or one Inch and almoft a quarter; if one Foot \ Girt ,,J or an Inch and above three-quarters for the Bark, &c. But for Beach, Elm, Ajh, and fuch that is thin bark'd, then the Allowance muft be a fmall matter lefs. 2. I have feen great Difference in the Girt of a Tree in the Space of two Feet, or lefs, and that hath been generally where one or two Arms have been cut off: In fuch a Cafe it is neceffary to Girt the Tree twice or thrice, if there be any great Difference, or otherwife there will be lofs to the Buyer or Seller. Again, they fay, the Buyer hath privilege to Girt any where between the Middle and the Ground End, if it be for his Advantage. Lafily, The Consent of any Piece of Timber being found in Feet, if divided by 50, you have the Content in Loads : But fome will have a Load to be 40 Solid Feet, therefore you may take which of the two is moft cuftomary with you. The Reafon why the Difference is, they fay, becaufe it is fuppofed that 40 Feet of rou;d Timber, or 50 Feet of hewn Timber, weigh about a Tun, or twenty Hundred Weight, which is commonly accounted a Cart-Load. CHAP. ( 37 ) CHAP. VI. The meafuring of Round Timber the common Way. TAKE the Length in Feet, Half-Feet, and if defired in Quarters, then meafure half way back again, where Girt the Tree with a fmall Chord or Chalk-line : Double this Line twice very even. This fourth-part of the Girt or Circumference meafure in Inches, Halves, and Quarters of Inches •, but be fure the Length be given in Feet, &c, and the Sides of the Square, or one-fourth of the Girt in Inches. So you have always three Numbers given, to find the fourth, viz. 12 on the Girt-line for the firft, the Length in Feet always for the fecond, and the Side of the Square for the third, in Inches, Halves and Quarters. Now we come to the Rule. Set 12 on the Girt-line D, to the Length (in Feet, &c.) on the Double Scale C, then againft the Side of the Square on the Girt line D, in Inches, Halves and Quarters you will find on the Double Scale C, the Content. Note, This Rule is general. Example I. Suppofe the Girt of a Tree in the Middle be 64 Inches, and the Length 31 Feet, what is the Content ? Set 12c the Girt- line D, to 31 Feet on the Double Scale C ; C 3 then ( 3* ) then again ft 16, the one-Fourth of 64, on th Girt-line D, is 55 Feet, the Content on th e Double Scale C. 2. A Piece of Timber is 15 Feet long, and one-fourth of the Girt 42 Inches : Set 1 2 on the Girt-line D, to 15 on the fecond Length of the Double Scale C •, then againft 42 at the Beginning of the Girt-line D, is on the Double Scale C, 184 Feet the Content, reckoning the flrft 1 at the Beginning of the Line of Numbers (or Double-fcale) to be ioo, the 8 Grand Divifions 8o> the two fmall ones 4 ; fo ac- counting all together makes (as I faid before) 184 Feet. 3. The Length is 9 Feet three quarters, and one-fourth of the Girt 39 Inches, fet 12 on the Girt-line D to 9J on the Line of Numbers C, and againft 39 at the Beginning of the Girt- line D, is 103 Feet, the Content of the Double- fcale of Numbers C. 4. The Length 9 Inches j the Quarter of the Girt is 35 Inches. Now becaule the Length being not one Foot, meafure it by your Line of Foot Meafure ; and fee what Part of a Foot it makes, a Foot fuppofed to be divided into a hundred Parts: or do thus, fet 12 on the Double-fcale B, to 100 on the Double- fcale A ; then againft the Length in Inches, namely, 9 on the Double-line B, is 75 on the Double-fcale A, for the Decimal or Part of ( 39 J of a Foot : Then fet 12 on the Girt-line D, to 75 in the fir ft Length of the Double line C, then againft 35 on the Girt-line D, is 6 Feet ,*, or almoft halfaFooton the Double-fcale C, for the Content. 5. A Rail is 16 Feet long, the Quarter of the Girt 3 Inches; fet 12 on the Girt-line D, to 16 on the firft Length of the Doable-Scale C, then againft 30 (now called 3) on the Girt-line D, is juft 1 Foot the Content, on the Double- line C. More Examples. One-fourth "} 1 1 Inches £ 4of _.£ H 34 Feet of the Girt 1 16 Inches a 39^ I g'2 70 Feet T -| or Side of [14 Inches | J3 50 I « ^ 68 Feet the Square f2i Inches w 48 J g =-. 147 Feet T | inthefeEx- I 8 Inches £ ju 2 6ij CJ ^ 19 Feet -j£ amples is J 31 Inches , or Foot 1.33 Parts; what's the true Content ? Se ( 46 ) Set 89 on the Girt line D, to 31 Feet on the Dcub'e line C, then 1.33 on the Girt-line D, points to 70 Feet on the Double line C, the true Content required ; whereas by the firft Example of Chapter the 6th, the Content of this Tree is but 5.5 Feet, by the common Way of meaiuring, which is no lefs than 16 Feet fhort of the truth. I (hall next (hew how to find the true Content of "Timber , Arithmetically, as followeth. By a fixed Number, or Decimal, .2821, which multiply by the Girt of the Tree taken in Inches, and from the Product cut off four Figures to the Right-hand, the remaining to the Left-hand, are the Inches for the Side of the Square equal to the Girt or Circumference ex the Tree. And for understanding it, take thefe two or three Examples, how to find the Side of the Square that (hall be equal to the Girt be Circumference of the Tree, by tWisJixt Num- ber .2821, always to be ufed -, when the Girt is Inches. This Decimal, or fxt Number is .2821 The Girt in the id Example is 168 22568 16926 2821 The Pro.fa<9: is — — — - Inches 47.3928 Here C 47 ) Here the Side of a Square equal to that Gir is 47 Inches and a quarter ; for thefe four De cimals cut off. are fo many parts of ioooq and .3928 is almoft ,f of a Foot, which is more than a quarter of an inch, Another Example. The fixt Number is .2S21 The Girt is 48 Inches 22568 The Side of a Square equal 1 1 2 84 To this Girtis 13I Inc. or Inc. 13.5408 Another Example. The fixt Number is • .2821 The Girtis .^I nches 1 - 284 The Side of a Square equal — 14105 To this Girtis 15^ Inc. or Inc. 15.2334 I now proceed to work the foregoing piece of Timber or Tree. Arithmetically. Multiply the Inches of; the Square (with the firft Left-hand Figure of the four which is cut off) into its felf, and that Product by the Length in Feet, and divide the laft Product by 144, the Quotient is Feet, and the Remain- der, if any, divide by 12, the Quotient is Inches. Exam- ( 4* ) Example. F. Par. The fide of the Square in Inches — 47.3 Multiply by itfelf • ' ■- 47-3 1419 189a Tn the Product cut off 1 Decimals 2237.29 Pt. The Length in Feet 15 11185 2237 F . I. i44)335d5v233-3 288 • • 047 5 43 2 0435 432 The Content of this Timber! is 233 Feet 3 Inches — 3 — 003 Inch Another Example for Praclice, There is a round Tree, whofe Girt is 48 In- ches, and the Length 8 Feet 9 Inches, what is the Content ? Multiply the Inches of the Side of the Square equal to the Girt with the firft Figure on the Left-hand, of the four cut. off) into it iHf, and from the Product cut off two Figures to ihe Right-hand, the Remainder multiply by the Inches of the Length of the Tree (and not by the Length in Feet) becauie the Length is Feet and ( 49 ) and Inches, therefore you muft bring your Length into Inches by multiplying them by 12, this laft Product divide by the Inches that are in a fquare Foot of Timber, namely 1728, the Quotient is the Feet, the Remainder (if any) divide by 144, the Quotient is the odd Inches. Example. The Side of the Square being Inches — 13.5 Multiply it by itfelf 13.5 F.In. 675 The Length 8 9 405 Multiplied by — 12 135 Makes ■ 105 Inches — — 182.25 The Length in Inches is — 105 In. 910 1820 The Content of this") 1728)19110(11 F't. round Tree is 11 Feet, I 1728 and 102 remains, which J "0T830" is not one Inch. J 1728 OIG2 Note, The fix'd NumbeF .282 1 is the Side of a Square, that's equal to a Circle whofe Girt is 1 Inch, 1 Foot, or one any Thing. Alfo the Number 10.635, is the Side of a Square that's equal to a Circle whofe Diameter is 12 Inches. D Alfo ( 50 ) Alio .8862, (or (hotter 89) is the Side of a Square that's equal to a circle whofe Diameter is 1 Foot, or one any Thing. Now to perform this Arithmetically, when all the Dimenfions are taken in Foot-meafure. The Rule. Multiply the whole Girt by the fix'd Number .2821, and the Product is the Side of a Square equal to that Girt •, then multiply that Product by itfelf, and this Iaft Product multiply by the Length of the Tree, and it produceth the Content required. Example. Let the Length be 15 Feet, and the Girt 14 Feet. I demand the true Content ? The fix'd Number is — , .2821 Multiply by the Girt 14 Feet, inverted is 41 2821 1128 The Product is the Side of the Square Ft. 3.949 Multiply by itfelf, inverted is 040.3 '11.847 35* 6 156 27 The Product is the Square ■ Feet 15.576 Multiply by the Length 15 Feet, inverted 51 77*5 The Product is the Content Feet 23 <.6i That is 233 Feet T 44-> or 233 Feet 7 Inches. Another ( 5i ) Another Example for Practice. A round Tree, whofe Girt is 48 Inches, or 4 Feet, and Length 8 Feet 9 Inc. or 8 Ft. T |4. The fix'd Number is — — .282 Multiply by the Girt 4 Feet ■ , 4 Product is the Side of the Square — Foot 1.128 Multiply by itfelf, inverted is 8.21 1 1 1 28 1 12 22 The Product is the Square Foot 1.270 Multiply by the Length 8.75, inverted is 5.78 10160 889 60 The Product is the Content — Feet 11.109 CHAP. VIII. ' To meafure the CUB E. ET there be a Cube whofe Sides are 3 Feet 3 Inches, what is the Content ? By the Sliding-Rule. Set 12 on the Girt-line D, to 3 £ on the Double-fcaie C •, then againft 39 Inches the Side of the Cube on the Girt-line D, is 34 Feet and ^ on the Double-fcale C, the Content required. D 2 De~ ( 5* ) Decimally, F.Par. The Side 3 ft. 3 in. or 3.25 Multipl. by the fame 3.25 1625 650 975 The \ft Prod, is iu.5625 Multipl. by the fa'me 3.25 528125 21 1250 316875 ■ 1 ■■ ■■ ■ ■ *m The Content is 34 ~ Feet or — Feet 34.328 1 25 But by Contraftion thus> F.Par. The Side of the Cube 3 Feet 3 Inches or 3.25 Multiply by 3 Feet 25 Parts, inverted is 52.3 975 16 The firft Product called the Square, is Ft. 1056 Multiply by 3 Feet, 25 Parts inverted is — 52.3 3168 211 S3 The Content 34 J Feet as before < -Ft. 34.32 CHAP. ( 53 ) CHAP. IX. To meafure Timber that is neither Rounds nor Square •, but firjt to find a Mean Proportional between any two Numbers. TO find this Mean Proportional, fct the greater of the two Numbers on the Girt-iine, to the fame on the Double-line of Numbers ; then againft the kffer en the Double- line of Numbers, is the Mean Proportional on the Girtline; or fet the lelfer on the Girt- line, to the fame on the Double-line of Num- bers ; then againft the greater on the Double- line of Numbers, is the Mean Proportional on the Girt-line-, any one of thefe will do. Note x Examples of this are in the next Prob- lem. P R O B. I. To meafure unequal fqttared Timber, that is, when ' the Breadth and Depth are not equal. Meafure the Length of the Piece in Feet, and the Breadth and Depth, (at the End) in Inches. Then find the Mean Proportional between the Breadth and Depth of the Piece as is taught above, and in the Example following. The Mean Proportional is the Side of a Square equal to the End of the Piece j which having found, the Piece may be mea- fured as Square Timber. D 3 Ex* '( 54 ) Example I. In a Piece of Timber whofe Length is 13 Feet, the Breadth 23 Inches, and the Depth 13 Inches; fet 23 on the Girt-line D, to 23 on the Double-icale C; then againfi 13 on the Line C is 17.35, or 17 3 on the Girt-line D, then fetting 12 on the Girt-line D, to 13 Feet the Length on the Line of Numbers C, then againft 17 \ (the Mean Proportion) on the Girt- line D, is 27 Feet the Content required. Example II. In Stone, which let it be 7 Feet 40 Parts, or 7 Feet and about 5 Inches in Length, and 30 Inches in Breadth, and 23-I- deep ; fet 30 Inches on the Girt-line D, to 30 on the Double- fcale C ; then againft 23-t on the Line of Num- bers C, is 2.6j, on 26.50 on the Girt-line D : Then fet 12 on the Girt-line D, 107.40, on the Line C, then againft 26-jr on the Girt-line D, is 36 feet the Content on the Donble-fcale C. The Content of thefe two Examples by Arith- metic, is thus. The firjl Example. Let there be a Piece of Timber whofe Length is 13 Feet, the Breadth 23 Inches, and the Depth 1 3 Inches. ( 55 ) The Rule. Multiply the Breadth in Inches, by the Depth in Inches; and that Product multiply by the Length in Feet, and divide the lafl: Product' by 144, the Quotient is Feet, and the Remainder (if any) divide by 12, the Quotient is Inches. Example. The Breadth 23 Inches. The Depth — 13 Inches. 69 23 Note, This Example by the The Product 299 Sliding Rule was 27 Feet, Multiply by 1 3 Ft. but by Arithmetick it 897 wants fomething-, for it 299 is 26 Feet 11 Inches, and 144)3887, 26Ft. 44 of an Inch. 288 1007 864 12)143(1 1 Inches 12 023 12 11 Remainder is f* of an Inch. D 4 By ( 56 ) By Fcot-meafure it's done thus : F.Par. Breadth 23 Inches in Foot-meafure is 1.92 Depth 13 Inches or Feet 1.08, inverted is 80.1 1.92 l 5 The Produft is Feet — 2.07 Multiply by the Length 1 3 Feet, inverted is 3 1 207 62 Content is 26 Feet 1 1 Inches, or Feet — — 26.9 The Second Example, Decimally. Let there be a Piece of Stone 7 Feet 40 Parts Jong, 1 Feet 50 Parts in Breadth, and 1 Foot 96 Parts in Depth, what is the Content ? The Rule. Multiply the Length by the Breadth, and the Prod uft by the Depth or Thicknefs ; and cut off from the laft Product, as many Places to the Right hand as there are Decimals in the three Dimenfions, the Integers remaining are Feet. Example- ( 57 ) Example. F.Par. The Length — — 7.40 The Breadth — 2.50 37000 1480 ^ju^j^tjk^^ ^he Prod. Feet 18.5000 A ^^feg^ Wg| The Depth Ft. 1.96 I 1 ICOOO 1665 '85 7-90 The Content is 36 Ft. 3 Inc. or Ft. 36.260000 By Foot-meafure it's thus contracted. F.Par. The Stones Length 7 Feet 5 Inches, that is 7.41 Stone Breadth 30 Inches or 2.5 inverted is 5.2 1482 370 The Produft is Feet 18.52 Depth 23^ Inches, or 1.96, inverted is — 69.1 1852 1667 1 1 1 The Content is ^6 Feet 4 Inches, or Feet 36.30 PROB, ( 53 ) P R O B. II. To find the Content of a Piece of 'Timber whofe End is in the Form of a Triangle j and both Ends alike and eqiial. TO find the Content of fuch Timber, firft find a mean Proportional between the Bale and half the Perpendicular of the Tri- angular Fnd, or between the Perpendicular and half the Bale, both rr.eafured in Inches. Then is the mean Proportional the Side of a Square equal to the Triangle. Then to find the Content, fet 1 2 on the Girt- line D, to the Length in Feet on the Line of Numbers C, then agamit the mean Proportional on the Girt line D, is the Content on the Line cf Numbers C. But the Dimenficns being all taken in Foot' tneafure, and the mean Proportional found in the fame -, then fet i on the Girt-line to D, the Length in the Double lint C ; then againft the mean Proportional on the Girt line D,is the Content in the Double-line C. Note, If the two Sides of a Triangle be equal, the other is Side is called the Bafe, but if the three Sides be unequal, the longetl Side is the Bafe : From whence the neareft Diftance, to the oppofkc Angle, is the Perpendicular. Example. ( 59 ) Example. A Piece of Timber 1 9 Feet 6 Inches in Length, the Bale of the Triangle at each End being ar Inches, and the Perpendicular to each Bafe beins; 16 Inches, what's the Content? Set 21 Inches on the Girt-lhe D, to i.\ on the Dnuble-line C ; then againtt 8 on the Line C, is 12 I, or 12.95 on the Girt-line D, the Mean Proportional. Then let 12 on the Girt-line D, to 1 9 §- Feet the Length on the Doable-line C ; and againft 12.95 the Mean Proportional) on the Girt-line D, is 22 ^, or Feet 22 T | (the Content required) on the Double-line C. Or thus, take all the Dimenfions in Foot-meafure, and then the Length 19 Feet 6 Inches is 19.5, the Bafe 21 Inches is 1.75, and the Perpendicular 16 Inches is 1.39; no\v fet 1.75 on the Girt-line D, to 1.75 on the Double-line C, and againft o 6y on the Double- line C, is 1.08 on the Girt-line D, for the Mean Proportional : Then fet 1 on the Girt-line D, to the Length 19.5 on the DouMe line C, and againft 1.08 on the Girt line D, is 22.8 for Feet 22 alinoft n Inches) on the Double-line C, for the Content, very near to that before. Arithmetically. The Rule. Multiply the Bafe by the Per- pendicular (both taken in Foot-meafure) and half that Product multiply'd by the Length, produceth ( <$o ) prockxeth the Content required, as in the forefaid Example. F.Par. TheBafe 21 Inch, or in Foot meafure h — 1,75 The Perpendicular 16 Inch, or 1.33 inver. 33.1 53 5 The Product is—— Feet 2 33 Half the Product is -Feet 1.17 The Length 19 Ft. 6 Inch. 19.5 inver. — 5.91 ) : 7 105 6 The Content is 22 Feet 10 Inches, or Ft. 22.8 P R O B. III. To mecfure Timber that Taper eth Eafure the Length in Feet, and nore, one- third of the Length, which may be found by the Sliding- Rtt'e, thus, Set 3 en the Double-line A, to the Length on the Double line B; then againft 1 on the Double- line A, is the third part on the Double-iine B; then if £he Solid be round, Meafure the Diameter at each End in Inches, fubtraclino- the lefler Diameter from the greater, and add half the Difference to the leffer Diameter, the Sum is the Diameter in the middle of the Piece { 6i ) Piece; then fet 13.540a the Girt line D, to the Length on the Double-line C, then againft; the Diameter in the middle on the Girt-line D, is a fourth Number on the Double-iine C, then fet 13.54, or 13^, of the Girt lineD, to the third part of the Length on the Double-line C-, then againft the half Difference on the Girt-line D, is another fourth Number on the Double-line C, thefe two fourth Numbers added together eive the Content. Example. Let the Length be 27 Feet (the one third by the Rule is 9) the greater Diame- ter 22 Inches, and the leffer 18 Inches. See the Operation. The greater — 22 Inches The teller Diameter 18 Inches The Sum is . < 40 The Difference ■ — 04 The half Difference 02 1 ,, , The lefier Diameter — 18 \ The Diameter in the middle 20 Inches Then fet 13.54 or \$\ on the Girt-line D, to 27 on the Double- line C ; then againft 10 on the Girt-line D, is 58.900, that is 38 Feet 900 P irt-s Again, fet 13.540a the Girt-line D, to 9 on the Double- line C ; then againlt 2 on the Girt-line (reprefented by 20) is 196 Parts of a Foot. The ( 62 ) rThe firft fourth Number- 58.900") ) The fecond fourth Number — 196 I added (.The Sum is the Content 59. 096 J Which is 59 Feet \ Inch. Note, If the Timber be wainy, that is neither Square nor Round, take one Wain in and leave the other out. The fame done by Foot-meajure. F. Par. The greater Diameter 22 Inches, or — — 1.83 The leffer Diameter 18 Inches, or 1.50 The Sum of both Diameters is — Foot 3.33 Half is the middle Diameter ■ ■■ Foot 1.67 Difference of the Diameter is ■■■ • Foot 0.33 Half Difference of the Diameter is Feet— o. 1 7 Then fet 1.13 on the Girt-line D, to the Length 27 Feet on the Double-line C •, and then againft 1.67 on the Girt-line D, is 58 Feet , ?. Then, Set 1. 13 on the Girt-line D, to 9 Feet on the Double-line C ; and then againft 0.17 on the Girt-line D, is .196 Parts of a Foot, and both put togethe, is the Content ; that is, 58.9 and .196 added makes 59.096, or 59 Feet | Inch, as before. Suppofc the Solid Square, and having the fame Dimenfions ; that is Length iy Feet, the Greater End 22 Inches Square, and the Lefifer End 18 Inches Square, what's the Content ? The »-5 3-33 1.07 1 °-33 ■ 0.1 6 ( 63 ) F.Par. The greater Square 22 — Inches or — — 1.83 The letter Square 1 S— Inches or- Sum of both Squares 40 — Inches or- •§- is the mid. Square 20 — Inches or Differ, of the Ends 4 — Inches or •| Differ, of the Ends 2 — Inches or Then to find the Content by the Inch- meafure it's thus. Set 12 on the Girt-line D, to 27 Feet (the Length of the Solid) on the Double-line C ; then againft 20 Inches (the middle Square) on the Girt-line C, is 75 Feet T ±. Again, fet 12 on the Girt-line to 9 Feet (one third of the Length) on the Double line ; and againft 2 Inches the half Difference of the Ends) on the Girt-line, is -,44 parts of a foot •, both together is 75 -^4 parts (which is almoft 8 Inches) the Content of the' Solid : Or thus, by Foot-meafure. Set 1 on the Girt line, to the Length 27* Feet on the Double-line, then againft 1 Foot 6y Parts (the Middle Square) on the Girt-line, ftands j$ Feet -£, and fecting 1 on the Girt-line, to 9 Feet (the one Third of t'.ie Length) on the Double-line; then a- gainft 0.167 (the half Difference ©f the Ends) on the Girt-line is (on the Double- line) rH Parts of a- Foot, and both toge- ther ( H ) ther 575 Feet 7 * and r n is ys Feet rli part of a Foot as before, exactly agreeing both by Inch-meafure and Foot-meafure ; and here note, the Numbers that are fix'd in this kind are thus to be understood : 13.540:57; fcff. in fhort 13.54 is the Diameter of a Circle, whole Area or Content is 144 Square Inches: And 1,1283814, &c. in fhort 1.12 is the Diameter of a Circle whofe Area or Content is 1 Foot, or one any Thing. P R O B. IV. To find hew many Inches in Length will make a Foot Solid at any Girt, being the Side of a Square, not exceeding 40 Inches. LET the Girt or Side of the Square upon the Girt-line be fet to 1 on the Numbers, then againft 41.57 of the Girt line is the Num- ber of Inches on the Numbers that will make a Solid Foot. Example. Let the Side of a Square be 7 Inches, fet 7 on the Girt-line D, to 1 on the Double-line C, then againft 41.57 on the Girt-line D, is 35 § Inches on the Double-line C, for the Length of 1 Foot Solid. To do the fame in Foct-meafnre •, the Side of the Square 7 Inches, in Foot-meajure is .58 Parts ; which faid .58 Parts on the Girt-line P, being fet to 1, on the Double-line C -, then (■ 6 5 y then againfc i on the Girt-line D, is 294, that is 2 Feet 94 parts, or 1 Feet i\\ Inches tor the Length, to make one Foot of Tim.er. P R O B. V. Having the Diameter of a Circle, or Round Piece of Timber ; to find the Side of a Square 'within the Circle, or to know how many Inches the Side of the Square will be when the Round Timber isfquared. The Rule. Set always 8.5 on the Double -line A, to 6 on the Double-line B ; then againft the Dia- meter on the Line A, is the Side of the Square on the Line B. Example, Let the Diameter be 18 Inches ; fet 8.5 on A, to 6 on B i then againft 18 on A, is 12$ on the Line B, for the Side of a Square within that Circle. The la.re done in Foot-meafure, the Dia- meter being hS Bncfefes is in Fcot-meafure 1.5; Then fet 1 on the Double-line A, always to .707 or .71 almoft on the Double-line L ; r .1 -agiinft the Diameter 1.5 on the Double -if,; is 1.7 on the Double-line B ; th-.r is Foot 1.7 Tenths the Side of a Square within Circle, wh:fe Diamecej is Foot 1.5 Tenths And here note ci.e given Numbers 8.5, and ( (but more truly 1, and .707) are, on the Diame- E ( * ) ter of a Circle, the other the Side of a Square within that Circle. P R O B. VI. By having the Girt of a 'Tree, or round Piece of limber ; to find the Side of a Square within. The Rule. Set 10 to 9 on the Line A and B, then a- gainft the Girt on the Line ., is the Inches for the Side of the Square on the Line B. Example. Let the Girt be 16 Inches, fet to on the Line A, to 9 on the Line B-, then againft 16 on the Line &, is 14^ on the Line B, for the Side of the Square. By Foot-meajure it's thus: the Girt 16 In- ches., is Foot 1.33 pans; then fet 10 on the Do .^le line A, to 9 on the Double line B, and agai.id: the Girt s.33 on the ~D§uble-line A, is on the Double-line B, 1.19; that is 1 Foot 19 Parts, wjiicb is 1 Foot 1 % Inches, for the Side of the Square within. And Note, the Numbers 10 and 9, or 1 and q, (hews when the Square within the Circle is 1, the fourth part of the Circumfe- rence parts of the fame. Bv ( 6 7 ) By thefe two laft Problems yon mav know (before a Piece of Timber is hewn) hovs many Boards or Planks of any Thicknefs it will make. P R O B. VII. By the fourth Part of the Girt ef Round Timber, to find the Side of a Square equal to it. The Rule. Set i on the Line A, to 1.128 on the Line B ; then againft the (one-fourth of the whole) Girt on the Line A, is on the Line B, the Side of the Square equal to it ? Example Let the Girt (that is the one-fourth, of the whole Girt) be 16 Inches ; how much is the Side of the Square equal to it ? Set 1 to 1. 128, but fhorrer to 1.13, on the Lines A and B ; then againft 16 on the Line A, is 1 3 on the Line B j which fheweth that a Square whole Side is 18 inches, is equal to a Circle whofe Girt is 64 Inches, or one-iourt^h of its Girt is 16 Inches. By Foot mea/ure, the Rule being fet as before, 1 againft 1 13, then againft Foot [.3.3 Pirts (equal to 16 Inches) you will find 1.5, that is 1 Foot t4j equal to 1 Foot 6 inches, the Side of a Square equal to the given Girt, die fame as before. E 2 C H A P. ( 68 ) CHAP. X. To meafure Brick- Work. Rick Work is meafured by the Rod of 1 6 Feet and a half, whofe Square is 272 £ which (heweth that the Point on the Double- tine is 272 £, but 272 being marked on the Rule will ferve (without any confiderable Error) a Brick and a half thick, and is the fixed Number. An Example. There is aBrick Wall whofe Length is 564Feet the Height is 10 Feet and \. Set 272 on the Double-line A, to 564 on the Line B; then againft \o\ on the Line A, is 21 T £ R°d> the Content on the Double-line B. Note 1 . Always the fix'd Number that goeth with the Queftion is called the firft Number, which you may fet to either of the other Numbers. Nile 2. That 272 J Feet makes a Rod of B^ick -work, at a Brick and a half Thicknefs ; if it be thicker fewer Feet go to a Rod ; if thinner the more. If you demand how many Feet makes a Rod at t vvo Bricks Thicknefs : Two being the firft or fixed N umber that goeth with the Queftion muft be fet toi}; then againft 272 \ is about 204, viz 204 Feet , 0°, and fo for any other Thick- nefs, for which Fake ( 69 ) take this General Rule. Set \\ to any Thicknefs ; then againft 272 | on the fecond Length is the Number of Feet that makes a Rod to that Thicknefs, and is called the firft Number for that Thicknefs, as in the table following. By this Table, made by the fame Rule, by the half Brick's Thicknefs of the Wall, you have the firft Number by Inflection ; as thus, againft 1 Bricks i or 3 half Bricks thick, is 163 Feet, for a Square Rod, and fo for any other. Half Square Feet in Half Square Feet in' Bricks z Rod on the Bricks a Rod on the thick. Superficies. thick. Superficies. f. 1 817 Feet 3 b 7 1 1 7 Feet ]. 2 408 Feet 4. 8 102 Feet « h 3 272 Teet 4 1- 9j 91 Feet a. 4 204 Feet 5. 10 82 Feet 2 I- 5 163 Feet 5 i- J1 74 Feet 3. 61 136 Feet 6. 12 68 Feet CHAP. XL Meafuring of LAND. LA N D is ufually meafured by the Pole, Perch or Rod, which is 1 6 -| Feet long j 40 Poles in Length and 4 in Breadth makes an Acre,, fo that an ^*r |°S 16.42 S 7 \ Acres almoft. tt$ L ! o,2tf J k3 C72.5 r J 1 18 Acres. Note, Gunter's Chain containeth four Statute Poles in 'oo Links, fo that any Number of Chains is no more than fo many 100 Links; as 4 Chains is 400 Links, 6 Chains is 600 Links, ( u ) Links, l$c. and 160 Square Statute Poles is an Acre, each Pole being 16 Foot and a Half. Therefore in a fquare Chain is 16 fquare Poles. If vou divide 160 (the Square Poles in an Acre) by 16 i^the Square Poles in a Chain) the Ouotient is 10, the fquare Chains in an Acre. A Square Chain contains ioodo Square Links (for it is 100 multiplied by ioo) and confequently an Acre is iooooo square Links. Therefore the Reafon of the foregoing O peration is evident., for if you multiply 5730 Links by 1625 Links, the Product will be 9311250 Square Links. Divide 9311250 by locooo (the Square Links in an Acre) the Quotient is Acres, and the Remainder is Parts of an Acre. But to divide by icoooo is no more than to cut off five figures to the Right-hand, fo will the Remaining Figures to the Left-hand be Acres, and thofe cut off to the Right hand 11250, are. Parts of an Acre : Again, multiply thofe five Figures cut off from the Right-hand of the Product by four, and from the Product cut off five Figures to the Right-hand, thofe on the Left- hand will be roods, and .if thofe cut off be multiplied by 40, and from the Pioduct five Figures be cut off from the Right-hand, thole en the Left-hand, are Poles or Perches. So C 74 ] So in the preceding Operation there is 93 Acres, no Roods, 1 8 Perches for the Content. P R O B. III. How to reduce Statute Meajurt to Cujiomary, and the contrary. Ccording to a Statute made in the 33d of Edward the Firft, and likewife in the 24th of Elizabeth, a Statute Pole is 16 Feet and a Half long. Now divers Parts of England ufc a Pole of 18 Feet long, and fome a Pole of 21 Feet long, and others a Pole of 24 Feet long. Therefore to turn one Sort of fvTeafure into another-, fuppofe a Statute in- to Cuftomary, do thus j multiply any Number of Acres, Roods and Perches Statute Meajure, by the Square half Yards, or Square half Feet, in a Square Pole of statute Meajure, the Pro- duct divide by the Square half Yards, or Square half Feet in a Pole of the Cuftcmary Meafure, the Quotient gives you the Av res, Roods and Perches of that Cufiomary Meafure -, as for Ex- ample. In 172 \ cres Statute Meafure, how many Acres of 18 Feet to the Pole or Perch? In a Statute Pole or Perch, there is five Yards and a Half, that is 1 1 half Yards ; con- fequently in a fquare Statute Pole there is 121 fquare half Yards, and in a fquare Pole of 18 Feet, there is j'44 fquare half Yards, 172 Yards [ 75 ] 172 Acres Statute Meafure 1 2 1 Square half Yards in a Statute Pole. 172 344 172 144)20812(144 Acres Cuftomary, and ^. • 14 0041 0652 576 076 Overplus Therefore the Cuftomary Acres (at 18 Feet to a Perch) is 144 ^ of an Acre, the Over- plus multiply by four, and the Product divide by 144 gives Roods; again, multiply the Remainder by 40, and the Product divide by 144, the Quotient will be Perches ; as by the following Operation of the preceding Re- mainder. 76 Overplus, or Remainder. 4_ 144,1304(2 Roods 288 016 40 144)640(4 Perches 064 So i 7<5 ] So that 172 Acres Statute Meafure is 144 Acres, 2 Roods and 4 Perches, with a fmail Remainder, not worth Notice, of Cujlomary MeafurCy of 18 Feet to the Pole or Perch. If there had been any given Roods and Perches with the Acres, then you mud turn ail into Perches, and having multiplied and divided as before directed, divide the Quotient hy 1 6o, or 40, and that Quotient again by 4 PROB, [ 77 ] P R O B. IV. In 543 Cujicmary Acres, at iS Feet *e ike Penb, boiv many Acres {/Statute-Meafure ? 543 Cuftomary Acres. 144 fquare \ Yards in a cuftomary Acre, 2172 2172 543 121)78 '92(64.6 Statute Acres* 726 0559 4S4 0752 726 26 Remainder, multiplied by 4 .11 ■ 1 * 104 no Roods 40 121)4160(34 Perches 53° 484 046 Remains Here ( 78 ) Here you fee that in 543 Cuftomary Acres at 18 Feet the Perch, there are 646 Acres, no Roods, 34 Perches Statute Meafurr, and ,*?. Cuftorrary Acres as well as Statute Acres, contain 160 Square Perches, the Difference is only in the bignefs of the Perch. CHAP. XII. The Description and Conflrutlicn of Scamozzi'j Lines upon the Two Foot Rule-, with their Ufe in all the necefjhy Problems preparatory to the Traftice of Building. i.^T^ H E mod ufeful \ ine of all is a Scale of equal Parts, beginning at o at the Center or Joint of the Rule, and proceeding to 30 near the remote End of it oft both Sides -, at which on each Leg of the Rule there is a Brafs Point or Center- Pin fix'd, and as tnefe 30 Divifions are intended in general to reprefent 30 Feet, they are every one iubdivicled into 12 equal Parts, every one of which reprefenfs an Inch ; but they may be applied to any other Meafure. 2. Next to this, and above the 30 Scale is the 40 Scale, or the fame Length divided into 40 equal Parts, and each into 6 Subdlvifions, fo that if the Divisions be Feet, the Subdivisions are each of them 2 Inches. 3. On ( 79 ) 3- On the omer Joint of the Rule, and next without the 30 Scale is a Line of Chords with a Center-Pin, at 60 equal to 15 on the 30 Line, and fo proceeding to 180, where there is ano- ther Center-pin at the end of the Line. 4. Nexr to the Line of Chords is a Line of Tangents, beginning at o, and continuing to the Tangent of 45, which is the Radius of every Citcle. 5. The Vacancy left by the fpreading of the 30 Scale is fuoply'd by the Line of Polygons^ it begins at 2 at the remote End of the Scale, and reckons inwards in fome Scales to 20 ; but it is now thought fufficient to continue it to 12, to prevent the Confufion upon the Scale which would otherwife be occafioned by the Contrac- tion of the Lines, and the decreafing Dirtance of the Numbers towards the Center. Before I give a further Account of the Ufc of Scammozzis Lines, it will be proper to ex- plain the Terms made uieofin Buildings, fo far as it relates to the Ufe of thole Lines ; as 1. A Model or Module is the Diameter of a Pillar or Column, aid 1 Minute is a ooth Part of a Model in any Order; as fu^pofe the Dia- meter of the Column or ?illar be 1 8 Inches, then 18 Inches is a Model, and 3 Tenths is a Minute: and h?nce, if a Column of 18 Inches Diameter be 7 Models, \i Minutes, it's Height is 10 Foot, 9 Inches, 6 Tenths. 2. la ( So ) 2. In the Uie of the Sector, or which is the fame of Scamozzfs Lines, which come to the Center or Joint of the Rule, the Word Lateral fignifies an Extent taken from the Center of the Scale to the Number propos'd, fet off up- on either Limb of the 30 Scale as Lateral 17 , is the Extern from the Joint or Center of the Rule to 17 on the 30 Seek on either Limb; but a parallel Extent is from any Number up- on one Limb to the fame upon the other : Thus, at whatever Opening the Rule is kt^ the Extent from 17 on one Limb to j 7 on the other is called a Parallel Extent of 17, &fc. The Pro- portion between the Enteral and Parallel Ex- tents upon the Sector, and confequently upon Scamozzis Lines are fully demonstrated, Euclid Lib. VI. Prob. )I. and IV. and upon that Foundation we may build the following ge- neral Rule, which is univerfally true in the Uie of all Sectoral Lines, whether equally divided or not, viz. That every Lateral is to its Parallel, as any other Lateral is to its Parellel; The Sect©* 1 or Rule being kept at the fame Exten* or Opening : Therefore in all Proportions in Common Numbers to be performed by Scamozzfs 30 Scale-, the Rule is this; Make the Lateral Second Term a Parallel in the Firjl, then Jhall Parallel Third Term be Lateral Fourth required. Example. ( 8' ) Example. As 18 to 24, fo is 12 to what? Make Lateral 24 a Parallel in 18, then ihall Parallel 12 be Lateral 16; the Anfvver, or to be more plain, extend the Compafles from the Center in the Joint of the Rule to the fecond Term 24, and keeping that Extent in the Cornpafles, open the Seftsr fo as that Ex- tent may reach from 18 on the 30 Scale on one Leg, to 18 on the 30 Scale on the other Leg of the Rule •, then keeping the Rule at the fame Opening, extend the Compafles from the Third Term 12, on one Limb of the Rule, to 12 on the other Limb, that Extent apply'd from the Center or Joint of the Rule along the 30 Scale, will reach to 16 the Anfvver required; and this being the Method of working all di- rect Proportions either in Lines or Numbers, I need not repeat Examples. The Ufe ofScamozzi's Lines in feveral necef- fary Problems. F R O B. I. T0 divide a Given Line lefs than two Foot into any Number of equal Parts, asjuppofe a Line of 7 Inches is to be ■ divided into 17 equal Parts. TAke 7 Inches in your Compafles, and make it a Parallel in 17 upon the 30 Scale, then fhall the Parallel of 10, 12, 15, &c. reprefent their refpe<5Uve Numbers of Parts upon the Line of 7 Inches propofed to F be ( »2 ) be divided. The Ufe of this is, if I would re- prefent a Garden of 28 Yards fquare upon a piece of Paper of 3 Inches fquare, take 3 In- ches off the Scale and make it a Parallel in 28, and keeping the Scale at the Opening the Parallel of 5, 6, to, 4sfV. fhall reprefent the fame Number of Yards upon the Paper, or to any Number under 30 ; but if a Number a- bove 30 be required to be laid down upon a Line of 7 Inches : As fuppofe 48, take the half of it, viz. 24 ; makes the given Line 7 Inches a Parallel in 24 ; then fhall every Pa- rallel reprefent its double, as the Parallel of 9 reprefents 18 •, and if the Line was to be continued beyond 48, the Method is the fame, for the Parallel of 30 would be the Meafure of 60, &c. and by this Means without a Multi- plicity of Problems any Scale to any Draught may be made either to meafure, or lay it down by. P R O B. IT. Tofetthe%o Scale at any An- gle propojed, or being fet in any Pofition to know what Angle one 30 Line makes with the other,; and firft to Jet it at Right Angles, or a per feci Square. Make Lateral 90 of the Chords, a Parallel in 15 on the 30 Scale ; then does the two 30 Scales make a Right Angle. Or take 10 of the 30 Scale in your Compaf- fes, and letting one Foot in 8, open the Ri^e till ( 8 3 ) till the other Foot will fall in 6 on the fame Scale, and then the two 30 Lines ftand at Right Angles. Euclid. Lib. 1. Prob. 46. Or make Lateral 21 Foot 2 Inches a Parallel in 15, it produces the fame. To fet the Rule to any Angle propofed ; as fuppofeof50 Degrees, make Lateral 50 of the Chords a Parallel in 15 on the 30 Scale, and it is done ; and to know what Angle the Rule lies open at in any Pofition, make Pa- rallel 15 on the 30 Line a Lateral upon the Chords, and that gives the Angle requir'd. PROB. III. To find a mean Proportio- nal between any two Numbers given, or in ■plainer Terms, to find an Intermediate Num- ber, whoje Square is equal to the Producl of the two given Extreams. Set the 30 Scale at Right Angles by Prob. 2 and take off the Lateral half Sum of the two given Numbers, and with this Extent, and one Foot in the Lateral half Difference of the two given Numbers, obferve where the other Foot falls upon the 30 Scale on the other Le°- of the Rule, for that is the mean Proportio° nal required. Thus the mean Proportional be- tween 4 and 16 is 8, becaufe 4 times 16 is equal to the Square of 8, or as 4 to 8, fo is 8 to 16. F 2 Jsote, ( 84 ) Note, One Ufe of the mean Proportional, among many others is, in meafuring Solids, vvhofe Breadth and Depth are not equal, for in that Cafe the mean Proportional between the Breadth and the Depth is equal, to the Side of the Square, and to be ufed as fuch in all Cafes where the Side of the Square is a Term ufed to find the Content. P R O B. IV. Of Superficial Meafure, and fir ft the Breadth of a Plank given, to find bow much in Length makes a Foot. Make Lateral 12 a Parallel in the Breadth, and letting the Rule continue fo, take off Pa- rallel 12, and that apply 'd laterally gives the Length of a Foot. Example. At 9 Inches broad make Lateral 12 a Parallel in 9, then (hall Parallel 12 be La- teral i6 } the Length of a Foot required. 2. The Breadth of a Board given in Inches, and the Length in Feet, to find the Content in Feet. Make Lateral Length in Feet a Parallel in 12, then mall Parallel Breadth be Lateral Con- tent in Feet. Example. Bieadth 8 Inches, Length 18 Feet, make Lateralis a Parallel in 12; then will Parallel 8 make Lateral 12, the Content in Feet. P R O B. £ 85 ] P R O B. V. Of folid Meafure fr Timber*, and firfi at any Number of Inches fquare, to know how much in Length makes a Foot. N. B. If a Piece of -Timber be not fquare, make it a fqnare by help of a tman Propor- tional, as in Prob. 3. and ufe that as if it was the fide of the fquare. Make Lateral Side of the Square a Paral- lel in 12 (alwiys) and keeping the Rule at that Openirg makes Parallel Side of the Square a Lateral fourth Number. Again, make La- teral 12 a Parallel in that fourth Number, then will Parallel 12 be laterally the Number of Inches to make a Foot. Example. At 9 Inches fquare, how much in Length makes a Foot. Make Lateral 9 a Pa- rallel in 12, then will Parallel 9 be Lateral 6|. Again, make Lateral 12 a Parallel in 6\, then will Parallel 12 be Lateral 21 J Inches to make a Foot of Timber. 1. The Side of the Square in Inches, and the Length in Feet given, to find the Content in Feet and Inches. Make Lateral Side of a Square a Parallel in 12 (always) then is the Parallel Length a La- teral fourth Number. — Again, make the La- teral fourth Number a Parallel in 12, then is Parallel Side of the Square the Lateral Content in Feet. F 3 Ex- [ 86 ] Ex. Suppofe a Piece of Timber be 20 Feet long, and 9 Inches fqnare, make Lateral 9 a Parallel in 12, then is Parallel 20 (jhe Length) a Lateral fourth Number, viz. 15. Then make Lateral 15 a Parallel in 12, then is Parallel 9 a Lateral 11 J the Content in Feet required. CHAP. XIII. The JJfe of the foregoing Problems in BuildLg, viz. in Jin ding the Lengths and Angles of Rafters, Hips and Collar- Beams in any Square, or Bevelling Roof at any Pitch. N order to the better underftanding the fol- lowing Directions, it is neceflary to ob- ferve that True Pitch is 'when the Rafter is juft three Quarters the Breadth of the Frame or Bou/e 3 and to find it readily feek the Breadth of the Houfe, upon the 40 Scale, and againfl it you will find on the 30 Scale the Length of the Rafter required, three Quarters of which is (nearly) the Height of the Perpendicular from the raifing Piece to the /fop of the Gable, which is found on the Rule, by looking for the Length of the Rafter on the 40 Scale, and a- gainit it on the 30 Scale in the Perpendicular of the Gable required. Example ( 8; ) Example. Let the Houfe be fuppokd \6 Feet broad, the Rafters according to this Proportion are to be 12 Feet long, and the Perpendicular will be 9 Feet. Note, This Proportion is not exactly or ma- thematically true for making half the Breadth of the Houfe which is 8, the Bale of a Right- angled Tnangle ; the Length of the Rafter 12, the Hypotenufe, and the Perpendicular of the Gable 9, the Perpendicular of the Triangle : The Sum of the Squares of the Legs is 145, but. the Square of the Hypotenufe 12, is but 144, whereas they mould be equal, Euclid. Lib. 1. Prob. 47. but the Difference is fo in- ienfible that it is thought fufficiently exact in Practice. To avoid Multiplicity of Schemes and Pro- blems, as to Square, Bevelling or Hipt Roofs, I {hall include as much Variety as can be intel- ligibly done in one i and fir ft, PROB. I. The Conjlrucfion cf a Roof or Frame that has one End Square aud Upright^ and the other Bevel and Hipt. See Fig. 1 . Let PMFG reprefent the Frame of a Houfe whele Gable End PM is fquare, and the other End FG is Bevel and Hipt, and let the Square End PM be 12 Foot the laid MG 23 Feet, the B vel End GF 13 Feet 6 Inches, and the Side F 4 FF ( 88 ) F P 16 Feet 6 Inches ; take P v three-fourths of the Breadth of the Frame PM in your Companies, and one Foot in P draw the Arch by and with the fame Extent and one Foot in M crofs the Arch b in b, and draw M b and P b for the Length of the Rafters 9 Feet, biiTcct B M in a, and F G in n, and draw n a continued to b, then n a reprefents the Ridge of the Houfe, and a b 6 Feet 9 Inches (J of the Rafter P b which is 9 Feer) is the Perpendi- cular of the Gable. Take F » or n G in your CompafTes, and fet it off from n to C, and through C draw KF and G I, which will always crofs each other at Right Angles in C, draw S T perpendicular to a #, and continued both ways to R and V, and draw k I parallel to F G, both through the Center C. To find the Length of the Hips. Take the Perpendicular a b, and fci from C to I and K 6 Feet 9 Inches, and draw K G and F I -, with the Extent F I, and one Foot in F make the Arch Y ; and with the Extent G K and one Foot in G crofs the Arch Y in Y, then draw G Y, it is the longer Hip, and FY is the Ihorter. Continue the Line PM both ways to Q_and Z, and fet the Rafters Length M B or PB from M to Z and from P to Q,.as alfo from S to R, and from T to V, and draw Z V and Q^R, then is P M b the Gable end, PF R Q^ one ( 8 9 ) one Side of the Roof-, M Z V G the other Side of the Roof, and GYF the Hipt Roof of the Bevel-end ; and if the Frame PMGF were fuppofed to lie horizontally, and the Ga- ble P M b fet perpendicular upon the Line P M s the Side of the Roof M Z V G folded in the Line M G, and the Side P QJR. F fold up in the Line P F, the Line Z V and Q^R will co-infide and form the Ridge of the Roof, and the Points R and V will meet perpendicu- lar to the Point C •, and if the Roof of the Hipt Bevel end F G Y be folded up in the Line F G, the Point Y will alio meet with R and V, and compleatly cover the Roof, and the Rafter, Hips, &V. may be cut to their re- fpeclive Lengths by the Scale by which the Frame was made, and the Rafters at 12 or 14 Inches afunder, or more or lefs may be repre- 11- n ted by the Lines m n, m n, &c. parallel to QlP and M Z. To find the Back of the Hips to make them anj'wer both the Sides and End of the Roof \ and Firfi, to find the Back of the lougeft HipG K. Lay a Scale from n to /, it cuts the Diago- nal C G, in x ; fet one Foot of the Corn- pa fifes in x and extend the other to the near- eft Diftance to the Line K G, and fet that Extent from x toy, and draw the Lines I y and ( 9° ) and y n which will form the Angle of the Back of the longer Hip, and in like manner lay a Scale from k to n, it cuts F G in d, fet the Compaffes from d the neareft Diftance to F I and fee that Extent from d to r, and draw k r and r n, they make the Angle of the Back of the leffer Hip. But if for private Reafons the perpendicular Height of the Roof and the Breadth of the Frame be given or determin'd, and by that Means the Roof is above or under true Pitch, increafe or diminifh the Perpendicular a b ac- cording to the given Perpendicular of the Gable, and form the Rafters, Hips, 15 V. ac- cordingly. Example. Let the Breadth of the Frame be P M as before 1 1 Foot, but the Perpendicular is limited to 4 Feet 9 Inches, then is a d 4 Feet 9 Inches the Perpendicular, and M d 7 Feet 7 Inches the Length of the Rafter, equal to which MZ muft be made-, and from this Foundation proceed in all Refpecls as above directed. Frcm 'what has been /aid a General Rule may be formed for Roofing of all Frames, whether above or under true Pitch, the Perpendicular and Breadth being given ; or if true Pitch, the Perpendiculvr being found by the foregoing Directions, which being known t ( 9i ) - known, the Method of performing by Sea- rrozzi's Lines is as follows ; And in order thereto, fet the 30 Scales at Right Angles, by Prob. 2. Page S2. then, 1. To find the Length of the Rafters. Count half the Breadth of the Houfe on one Leg, and the Length of the given Per- pendicular on the other Leg (both on the 30 Scale f the Parallel Diftance between them mea- fured Laterally gives the Length of the Rafter required. 2. To find the Length of the Hip. Reckon the Length of the Rafter now found on one Leg-, and half the Breadth of the Houfe on the other Leg, the parallel Diftance between them meafured laterally gives the true Length of the Hip required. 3. To find the Diagonal Line. Set off Half the Breadth of the Houfe on both Legs, and the parallel Diftance between them meafured laterally gives the Diagonal re- quired. 4. By the Diagonal given to find the Length of the Hips, by a Method different from that in Article 2. Set ( 9* ) Set off the Diagonal on one Leg, and the perpendicular Height on the other, and the parallel Diitance between them meafured late- rally gives the Length of the Hip required. tfhefe four laft Articles are fo plain that I fupprje they need no Example, But if the fquare End be to be hipt, ob- ferve the following Rules, continue the Per- pendicular a b to q, making a q equil to M Z, or Q^P the Length of the Rafter, and draw M q and P q, and fet the half Breadth a M or a P from a to /, then is a q the longed Rafter of the hipt End, M q and P q the Hips, / the Foot of the King-poft, perpendicular to which is /, q y and /, will all meet in a Point when folded as before pro- poled, and then will the Length of the Ridge be only V ;', which will coincide with R /, as will i P vvith P q, and * w with M q, then will m o and n o be fnort Hip Rafters, and ■ P Q^f and M z z, which were of ufe only •when the fquare end was fuppofed Perpendi- cular, are now of no uie at all. By Scamozzi's Lines, to find the Hips, and Height of the King pofr. Set the 30 Lines at Right angles, then fet half the Breadth of the Houfe a M on one Leg, and the Length of the Rafter a q equal to MZ on the other Leg (both laterally from the Center (the parallel Diitance between thefe tv\o Poiiits meafured laterally give tl: e Length of ..- - ■: '■-""------. 'I.'- h ( 93 ) of the Hip requir'd. But to find the Height of the King-poft (having the 30 Scale let at Right Angles (take the Lateral Length of the Rafter a q in your Compafles, and fetting one in the half Breadth of the Houfe reckon'd la- terally from the Center, turn the other Foot, and obferve where it cuts the 30 Scale on the other Leg, for that reckon'd from the Center laterally is the Height of the King-poll re- quir'd. To find the Rafters, Hips and Angles in Frames that Bevel and are Hipt at both Ends, and are broader at one End than the other. See Fig 2. In thofe fort of Frames the middle Breadth is the Guide for the Length of the Rafters, for the Perpendiculars at each End are to be alike and equal to the middle Perpendicular ; but 'in this Cafe the Rafters at the broad End muft be lefs, and thofe at the narrow End more than three fourths of the Breadth of their re- fpective Lnds, if they ftand fquare to keep the Ridge of the Houfe upon a Level or Parallel to the Horizon ; in order to which let A8CD reprefent the Frame of the Houfe or Plan it Hands upon, bevelling at both Ends both at A B and C D ; where note if the two Sides and two Ends be given as A O and B C, as al- fo A B and D C, vet it is neceflary to have one of ( 94 ) of the Angles to regulate the reft. Then bifleCb A B in G, and D C in Z, and draw Z G the Line over which the Redge of the Houfe is to be, and upon that Line fet A G from G to T, and CZ from Z to F; through F draw r q parallel to D C, and through I draw / v patallel to A B, then through I draw the Di- agonals A m and B k } which will always cut each other at Right Angles •, through F alfo draw the Diagonals D n and C K, which will alfo cut each other at Right Angles. Find the middle of the Line Z G as at g y in which crois ZG at Right Angles with the Line x b, three fourths of which, x s is the Length of the middle Rafter /; C or C d. Now to find the Length of the Hips, and firlT. for the Hips Handing upon the Angles A and B, A I and H I reprefents the Diftance of the Foot of the King-pod from the relpeftive Angles A and B, alfo ( H is, by Conftructions equal to the Perpendicular of the Roof or Height of the Kingpofti therefore in the Right Angled Triangle A i H, in which A I is the Bale I H is the Perpendicular, A H is the Hypotenufe, from thence called the Hip of the Houfe, there- fore A H and B H are the Length of the two Hips for the Angles- A and B, and if AV be made equal to A II, and B V equal to B H ; thefe two Lines join'd in V the Tri- angle, A B V exactly reprefents the Roof of the Bevel End A B from Hip to Hip, and by the ( 95 ) the fame Rule the Hips of the other End D C are found to be D E and C E, and the End of the Roof to be reprefented by the Triangle CED; and if the Hips be meafured in the Scheme by the fame Scale by which they were protracted, their Length may be exactly deter- mined from thence. For the Angles for the Back of the Hips, I refer the Reader to the Directions for the Bevel End of a Houfe in the Explanation of Fig. I. The following Proportions perform'd by Scamozzi's Lines. To find the Length and Angles of every prin- cipal particular Rafter in Frames broader at one End than the other. The Perpendicular being the fame in all parts of the Houfes as hinted before, open the 30 Scale fqnare, and fetting one Foet of the CompafTes in the Height of the Perpendi- cular in one Leg of the 3c Scale, and fet the other Point in the half Breadth of the Frame on the other Leg, that Extent of the Compares meafured laterally from the Center, gives the Length of the Rafter required, and to find the Angles, lay a Rule to the Compafs Points, whether Scamozzi's y ' or any other joint Rule, and fet a Bevel, and it fhews the Angle at the Raifing piece, or at the Kidge of the Houfe, to cut the Ends of the Rafters by. Ts ( 9« ) ¥o find the Length and Angles of Collar Beams in any Roof. Take the whole Breadth of the Frame in your Compafles meafured laterally on the 30 Scale, and keeping the CompaiTes at that Ex- tent, let one Foot in the Length of the Rafter on one Leg, and the other in the Length of the Rafter on the other Lepr, that Extent mea- fured laterally on the 30 Scale gives the Length of the Raifing- piece within the Rafters ; then at what Height above the Raifing-piece you intend the Collar-Beam mall be, if you lay a Ruler parallel to the Line between the two Points of the Compalles, that Rule [hall repre- fent the Collar- Beam (allowing fpare Wood for the Tenons) or if it be an Extent of another pair of CompafTes it is the lame, meafured late- rally on the 30 Scale gives the Length of" the Collar-Beam requir'd. For the fides CBTSandDQP A with the length of their Rafters, &V. See Prob. 1. Page 87, the Manner of performing being much the fame as in zhc bevelling Hipt Roof. Fig. I. FINIS. OC SOUTHERN REGIONA A 000 018 674 •