The description and use of the carpenters-rule together with The use of the line of numbers commonly called Gunters-line : applyed to the measuring of all superficies and solids, as board, glass, plaistering, wainscoat, tyling, paving, flooring, &c., timber, stone, square on round, gauging of vessels, &c. : also military orders, simple and compound interest, and tables of reduction, with the way of working by arithmatick in most of them : together with the use of the glasiers and Mr. White's sliding-rules, rendred plain and easie for ordinary capacities / by John Brown. Brown, John, philomath. 1688 Approx. 207 KB of XML-encoded text transcribed from 107 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2004-08 (EEBO-TCP Phase 1). 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A29760) Transcribed from: (Early English Books Online ; image set 105159) Images scanned from microfilm: (Early English books, 1641-1700 ; 1089:4) The description and use of the carpenters-rule together with The use of the line of numbers commonly called Gunters-line : applyed to the measuring of all superficies and solids, as board, glass, plaistering, wainscoat, tyling, paving, flooring, &c., timber, stone, square on round, gauging of vessels, &c. : also military orders, simple and compound interest, and tables of reduction, with the way of working by arithmatick in most of them : together with the use of the glasiers and Mr. White's sliding-rules, rendred plain and easie for ordinary capacities / by John Brown. Brown, John, philomath. [205] p., [1] leaf of plates : ill. Printed for W. Fisher and R. Mount ..., London : 1688. Special t.p. (p. [169]): The use of the line of numbers on a sliding (or glasiers) rule in arithmatick & geometry ... / first drawn by Mr. White ; ... made easie and useful by John Brown. London printed : [s.n.], 1688. Imperfect: some pages tightly bound, with slight loss of print. Reproduction of original in the Huntington Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). 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Keying and markup guidelines are available at the Text Creation Partnership web site . eng Mensuration -- Early works to 1800. Mathematical instruments. 2004-01 TCP Assigned for keying and markup 2004-03 Apex CoVantage Keyed and coded from ProQuest page images 2004-04 Emma (Leeson) Huber Sampled and proofread 2004-04 Emma (Leeson) Huber Text and markup reviewed and edited 2004-07 pfs Batch review (QC) and XML conversion The Description and Use OF THE CARPENTERS-RULE : Together with the vse of the LINE of NUMBERS Commonly called GUNTERS-LINE . Applyed to the Measuring of all Superficies and Solids , as Board , Glass , Plaistering , Wainscoat , Tyling , Paving , Flooring , &c. Timber , Stone , Square or Round , Gauging of Vessels , &c. ALSO Military Orders , Simple and Compound Interest , and Tables of Reduction , with the way of working by Arithmatick , in most of them . Together with the Use Of the Glasiers and Mr. White 's Sliding-Rules . Rendred plain and easie for ordinary Capacities . By Iohn Brown. London , Printed for W. Fisher , and R. Mount , at the Postern on Tower-hill , 1688. Courteous Reader , THis Little Book was first written by me several years since , and hath been accepted of among many , that ●ave had the Perusal-thereof . And several Impressions , in that time , be●●g sold off ; and it being now out ●f Print , and none to be had , I have ●●vised it , and left out what might ●ell be spared ; and added that which ●ight make it more plain , and easier 〈◊〉 be remembred . As for Instance , in the using of ●●e Line of Numbers ( commonly ●●lled Gunters Line ) for the mea●●ring of Board , or Timber , or Stone , ●●e fixed Points or Centers is only 10 ●●d 12 , for square Timber or Stone . And in measuring of round Timber or Stone ( as round Timber ) ●●ere is used only 13. 54 for Inches , ●●d 1. 128 for Foot-measure , being ●e Diameter in Inches , and Foot●easure of any round solid , when one ●oot in length makes one solid Foot of 12 Inches every way , or 1728 so ▪ lid Cube Inches , which is a foot 〈◊〉 Timber , or Stone . And if the Circumference , or Gi●● of the Piece about is given , then t●● fixed Point or Center used , is at 4●● 54 the Inches , and 100 part of ●● Inch about , when one Foot , or 1●● Inches long makes one Foot solid . Or else at 3.545 the Feet , and 100● parts about , when one Foot long mak● one solid Foot equal to 1728 Cube I● Also , after every Problem , is t● brief way of working it by the Pen , 〈◊〉 a proof of the truth of every Oper● ▪ tion by the Rule ; being more th●● was before in the former Impression● Also , the Line of Pence is a● ded to the Line of Numbers , and plain way set forth of the use there by the Line on the Rule . Or the m● plain description thereof in a Pr● of Gunters Line 11 times repeate● which may be had with the Book , Tower-Hill , or the Minories . The Use of the Gunters Line ●he Art of Gauging , is here but brief●y hinted , because there are several Books of Gauging , purposely made for that Imployment , more compleat ●han can be expected in this short Discourse . And Lines of Area's of Circles , in Ale-gallons , at any Diameter given , from 1 Inoh to 200 In●hes ; which may be used for round ●r square Vessels , to give the content ●f every Inch deep in any taper Vessel , as fast as any one can write it down . And Directions for the ready measuring the Drip , or stooping Bottoms of round or square Tuns , and the Liquor about the Crowns of Coppers . Which Books are to be had at the Postern at Tower-Hill , or at the Authors , in the Minories . To this Impression is added the Use of the Lines called Diameter , Circumference , Square-Equal , and Square within a Circle ; and to find the Circles , Area , or Content by them ; or , having the Area , to find the Diameter , or the Circumference or the Square-equal , or Square-within . Also , to this is added the way b● the Pen , to multiply Feet , Inche● and 12 parts of an Inch together whereby any Superficies , as Boar● Floor , Wall , Yard , or Field , way ●● exactly measured by the Pen. Als● by a second Operation , or Multipl● cation , may any Solid , as Timber , 〈◊〉 Stone , or round Vessel , be measured the Arithmetical way whereof , 〈◊〉 worded as plain as in any Bo●● whatsoever . Also the Use of the Glasier's ▪ Sl● ▪ ding-Rule , to measure Glass , or a●● Superficies . And Mr. White 's Sliding-Ru●● to measure Timber ; being as ne●● and ready a way as ever was used . Thus you have a brief Account 〈◊〉 what is contained in this little Boo● and I wish it may be helpful to ma● a Learner , for whom it is prepare● So I remain ready to serve you in th● or the like . John Brown. From my House at the Sphere an● Sun-Dial in the Minories , Londo● ●he Description and Use of the Carpenters Rule . CHAP. I. IT is call'd a Carpenters Rule , ( rather then a Ioyners , Bricklayers , Masons , Glasiers , or the like ) I suppose , because ●●ey find the most absolute necessi●● of it in their way , for they have 〈◊〉 much or more occasion to use 〈◊〉 than most other Trades , though ●●e same Rule may measure all kind ●f Superficies and Solids , which ●wo Measures measure every visi●●e substance which is to be mea●●red . And it is usually made of ●●ox or Holly , 24 Inches in length , ●nd commonly an Inch and half , or 〈◊〉 Inch and quarter in breadth ; ●nd of thickness at pleasure ; and ●n the one side it is divided into ●4 equal Inches , according to the ●tandard at London●nd ●nd every one of those 24 Inches is divided into eight parts , that i● Halfs , Quarters , and Half-quarter● or ten parts , as you please and th● Half-inches are known from th● Quarters , and Quarters from th● Half-quarters , by short , longer and longest strokes , and at ever● whole inch is set figures , proceedin● from 1 to 24 , from the right hand toward the left , and these part● and figures are on both edges 〈◊〉 one side of the Rule both way● numbred , to the intent that howsoever you hold the rule you have th● right end to measure from , provided you have the right side . On the other side you have th● Lines of Timber and Board measure , the Timber-measure is tha● which begins at 8 and a half , tha● is , when the figures of the Timber line stand upright to you , then I sa● it begins at the left end at 8 and 〈◊〉 and proceeds to 36 within an Inc● and ⅜ of an Inch of the end . Also a● the beginning end of the Line o● Timber measure is a Table of figures , which contains the quantity of the under-measure from one Inch square to 8 Inches square , for the figure 9 comes upon the Rule , as you may see near to 8 in the Table . On , or next the other edge , and same side you have the Line of Board-measure , and when those figures stand upright , you have 6 at the left , or beginning end , and 36 at the other ( or right end ) just 4 Inches off the end unless it be divided up to a 100 , then it is nigh an Inch and half off the end . This Line hath also his Table of Under-measure at the beginning end , and begins at 1 and goes to 6 , and then the divisions on the Rule do supply all the rest to 100. Thus much for Description : Now for Use. The Inches are to measure the length or breadth of any Superficies or Solid given , and the manner of doing it were superfluous to speak of , or once to mention , being not only easie , but even natural to every man , for holding the Rule in the left hand , and applying it to the board , or any thing to be measured ▪ you have your desire : But now for the use of the other side , ● shall shew it in two or three examples in each measure , that is , Superficial or Solid . And first in Superficial , or Board-measure . Ex. 1. The breadth of any Superficies ( as Board , or Glass , or the like ) being given , to find how much i● length makes a Square Foot , ( or i● equal to 12 inches broad , and 12 Inches long ; for so much is a ●rne Foo● Superficial . ) To do this , look for the number of Inches your Superficies is broad in the Line of Board-measure , an● keep your finger there , and right against it , on the Inches side , you have the number of Inches that goe● to make up a Foot of Board o● Glass , or any Superficies , Suppose ● have a peice 8 Inches broad , How many Inches make a Foot ? I look for 8 on the Board-measure , and just against my finger ( being set to 8 ) on the Inch side , I find 18 , and so many Inches long , at that breadth , goes to make a Foot Superficial . Again , suppose it had been 18 Inches broad , then I find 8 Inches in length , to make a Foot superficial ; but if 36 Inches broad , then 4 Inches in length makes a Foot. Or you may do it more easie thus : Take your Rule and hold it in your left hand , and apply it to the breadth of your Board or Glass , making the end that is next 36 even with one edge of the board or glass , & the other edge of the board shews how many Inches or Quarters of an Inch go to make a foot of the board or Glass . This is but the converse of the former , and needs no example , for laying the Rule ●o it , and looking on the Board-measure , you have your desire . Or else , you may do thus in all narrow peices under 6 inches broad . As suppose 3¼ , double 3¼ it makes 6½ , then I say , that twice the length from 6½ to the end of the Rule , shall make a Foot Superficial , or so much in length makes a foot . Ex. 2. Having A Superficies of any length and breadth given , to find the Content , that is , how many Foot there is in it . Having found the breadth , and how much makes one Foot , turn that over as many times as you can , for so many Foot is there in that Superficies : But if it be a great breadth , then you may turn it over two or three times , and then take that together , and so say 2 , 4 , 6 , 8 , 10 , &c. or 3 , 6 , 9 , 12 , 15 , 18 , 21 , and till you come to the end of the Superficies . Note that the three short strokes between figure and figure , are the Quarters ; as thus , 8 and a quarter , 8 and a half , 8 and three quarters , then 9 , &c. till you come to 30 , and then 30 and a half , 31 , &c. to 36. And if it be divided any further , it is to whole Inches only to 100. The use of the Table at the beginning end of the Board-measure , First , you have five ranks of figures ; the first , or uppermost is the number of Inches that any Superficies is broad , and the other 4 are Feet , and Inches , and parts of an Inch that goes to make up a Foot of Superficial measure : As for example , at 5 Inches broad you must have 2 Foot , 4 Inches , and 4 Fifths of an Inch more , that is , 4 parts of 5 , the Inch being divided into 5 parts ; but where you have but two figures beside the uppermost , and Ciphers in the rest , you must read it thus , At two Inches broad you must have six Foot in length , no Inches , no parts . Thus much for the use of the line of Superficial or Board measure . The Use of the Line of ( Solid ) or Timber-measure . The use of this Line is much like the former : For first , you must learn how much your piece i● square , and then look for the same number on the Line of Timber-measure , and the space from thence to the end of the Rule , is the tru● length , at that squareness , to make a Foot of Timber . Ex. 1. I have a piece that is 9 Inches square , I look for 9 on the Line of Timber-measure , and ther● I say , the space from 9 to the end of the Rule , is the true length to make a Foot of Timber , and it i● near 21 Inches , 3 eights of an Inch. Again , suppose it were 24 Inches square , then I find 3 Inches i● length makes a Foot , for so I find 3 Inches on the other side , just against 24 : But if it were small Timber ▪ as under 9 Inches square , then you must seek the square in the upper rank in the Table , and right under you have the Feet Inches , and parts that go to make a Foot square , as was in the Table of Board-measure . As suppose 7 Inches square , ●hen you must seek the square Inches in the upper rank in the Table , and right under you have the Feet Inches , and parts that go to make a Foot square , as was in the Table of Board-measure . As suppose 7 Inches square , right under 7 , I find in the Table 2 Foot 11 Inches , and 2 sevenths of an Inch , divided into 7 parts , and at 8 Inches square you find only 2 Foot , 3 Inches , 0 parts , and so for the rest . But if a piece be not just square , but broader at one side than the other , then the usual way is to add them both together , and to take half for the square , but if they differ much , then this way will be very erroneous , and therefore I refer you to the following Rules : But if it be round Timber , then take a string and girt it about , and the fourth part of this is usually allow'd for the side of the square , and then you deal with it as if it were just square . Thus much for the Use of th● Carpenters Plain-rule . I have also added a Table fo● the Under-measure for Timber & Board , to Inches and Quarters ; an● the use is thus : Look on the left side for the number of Inches an● Quarters , your Timber is square● or your Board is broad , and right against it you have the Feet ; Inches tenth part of an Inch , and tenth of ▪ tenth ( or hundredth part of an Inch ) that goeth to make a Foot o● Timber or Board . Ex. 3. A piece of Timber 3 Inches 1 quarter . F. Inch. 10. 100 ▪ square will have 13 7 5 9 parts to make a Foot. And a Board of 3 Inch ▪ and a quarter broad must ▪ F. Inc. 10. 100. have 3 8 3 0 in length to make a Foot ; and so of the rest , as is plain by the Table , and needs no further explication , being common to most Artificers . A Table for the under Timber-measure , to Inches and quarters . in qr . feet . in . 10p . 100   1 2304 0 0 0   2 576 0 0 0   3 256 0 0 0 1 1 144 0 0 0   1 92 1 9 7   2 64 0 0 0   3 47 0 2 4 2 2 36 0 0 0   1 28 4 3 3   2 23 0 4 1   3 19 0 3 1 3 3 16 0 0 0   1 13 7 5 9   2 11 9 0 6   3 10 1 8 8 4 4 9 0 0 0   1 7 11 6 6   2 7 1 3 3   3 6 4 5 9 5 5 5 9 1 2   1 5 2 6 9   2 4 9 1 2   3 4 4 2 6 6 6 4 0 0 0   1 3 8 2 3   2 3 4 9 0   3 3 1 9 3 7 7 2 11 2 8   1 2 8 8 6   2 2 6 7 2   3 2 4 7 7 8 8 2 3 0 0 8¼   2 1 3 9 A Table for the under Board-measure , to Inches and quarters .   fe . in . 10. 100   48 0 0 0   24 0 0 0   16 0 0 0 1 12 0 0 0   9 7 2 0   8 0 0 0   6 10 2 9 2 6 0 0 0   5 4 0 0   4 9 6 0   4 4 3 6 3 4 0 0 0   3 8 3 0   3 5 1 4   3 2 4 0 4 3 0 0 0   2 9 8 8   2 8 0 0   2 6 3 1 5 2 4 8 0   2 3 4 2   2 2 1 8   2 1 0 4 6 2 0 0 0   1 11 0 5   1 10 1 5   1 9 3 3 7 1 8 5 8   1 7 8 6   1 7 2 0   1 6 5 8 8 1 6 0 0 8¼ 1 5 4 5 Note also , that this Table , or any smaller part of under-measure , may be supplyed by the divisions of the board and timber-measure only as thus ; Double the inches and parts of breadth for board-measure , or of squares for timber-measure , and seek it in the Lines of board or timber-measure , and count twice from thence to the rules end , for board , or 4 times for timber , and that shall be the true length that makes a foot of board or timber . Ex. 4. At 4 inches and ½ square or broad 4½ doubled , is 9. then look for 9 on the board-measure , and two times from thence to the end , shall make a foot of board . Or look for 9 on the Line of timber-measure , and 4 times from thence to the end of the Rule , shall be the true length to make a foot of timber , at 4 inches ½ square . But if it be so small a piece , that when it is doubled , the number is not on the divided part of the rule , then double it again , and count 4 times for board-measure , and 16 times for timber . Ex. 5. At 2 inches and a half , a quarter broad , or square , that doubled , is 4¼ , which is not on the rule , therefore I double it again , saying , 4 ¼ and 4¼ is 8½ which , is on the rule ; then for board , count 4 times from 8½ on the board-measure , to the upper end by 36 , to make a foot of board at 2⅛ broad : And for timber , count 16 times from 8½ near the beginning of timber-measure , which will be near 32 foot , to make a foot of timber at 2● square : But if twice doubling will not do , then double again , and count 8 times for board , and 64 times for timber , as in the Table you may see , which will be very slender timber . Also between the two lines of Inches , is set four scales of equal parts , called Circumference Diameter , Square-equal , and Square-within . Whose Use may be thus . The Diameter of any circle being given , to find the circumference or the side of a Square-equal , o● the side of the square within . Example , suppose the Diameter of a circle be 15 inches . Take 15 from the scale called Diameter , and measure it in the scale called , circumference and i● gives 47. 10. Also the same extent measured in the line called square-within and it gives 10. 55. For the side of a square-within in that circle of 51 inches Diameter . Again , the same extent being measur'd in the scale call'd square-equal and it gives 13. 27 for the side of ● square equal to a circle of 15 inches Diameter . Lastly , this 13. 27 the side of the square-equall multiplyed by it sel● gives 176 , the Area of that Circle in Inches , whose Diameter is 15 Inches . The same may be done , if the Circumference be first given , then that taken first from that Line , and measured in the other Lines , you shall have the respective Answers , as before . But if the Area be first given , then to find the Square-equal , find the Square-root of the Area , and that root shall be the side of the Square-equal . Example . Suppose the Area of a Circle be 288 , what is the side of the Square-equal . The middle between 1 and 288 , is at near 17 the side of the Square-equal , for 17 squared is 289. Then 17 taken from the Scale call'd square equal , gives you any of the rest . These Scales serve to draw any Platform of a House or Field very convenlently , being of several bignesses . The Description and Use of the Line of Numbers , ( commonly called Gunter's Line . ) CHAP. II. The definition and description of the Line of Numbers , and Numeration thereon . THE Line of Numbers is only the Logarithmes contrived on a Ruler , and the several ranks of figures in the Logarithms are here express'd by short , and longer , and longest division ; and they are so contrived in proportion one to another , that as the Logarithmes by adding together , and substracting one from another produce the quesita , so here by turning a pair of Compasses forward or backward , according to due order , from one point to another , doth also bring out the quesita in like manner . For the length of this Line of Numbers , know , that the longer it is the better it is , and for that purpose it hath been contrived several ways , as first into a Rule of two Foot long , and three Foot long by Mr. Gunter , and I suppose it was therefore called Gunter's Line . Then that Line doubled or laid so together , that you might work either right on , or cross from one to another , by Mr. Wingate afterwards projected in a Circle , by Mr. Oughtred , and also to slide one by another by the same Author ; and ●ast of all projected ( and that best of all hitherto , for largeness , and conseqenly for exactness ) into a Serpentine , or winding circular Line , of 5 , or 10 , or 20 turns , or more or less , by Mr. Brown , the uses being in all of them in a manner the same , only some with Compasses , as Mr. Gunter's and Mr. Wingat's ; and some with flat Compasses , or an opening Index , as Mr. Oughtred's and Mr. Browne's , and one without either as the sliding Rules ; but the Rules or Precepts that serve for the use of one , will indifferenly serv● for any : But the projection that ● shall chiefly confine my self to , i● that of Mr. Gunter's ; being the most proper for to be inscribed o● a Carpenters Rule , for whose sake● I undertake this collection of the most useful , convenient , and proper applications to the uses in Arithmatick and Geometry . Thus much for definition of what manner of Lines of Numbers there be and of what I intend chiefly to handle in this place . The order of the divisions o● this Line of Numbers , and com● monly on most other , is thus , i● begins with 1 , and proceeds wit● 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 ; and then ● 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ; whos● proper power or order of numeration is thus : The first 1 dot● signifie one tenth of any whole number or integer ; as one tenth o● a Foot , Yard , Ell , Perch , or the like ; or the tenth of a penny , shil●●g , pound , or the like , either in ●●eight , or number , or measure ; ●●d so consequently , 2 is 2 tenths three tenths ; and all the small ●●termediate divisions , are 100 ●●rts of an integer , or a tenth , of ●●e of the former tenths ; so that 1 the middle , is one whole integer , ●●d 2 onwards two integers , 10 at ●●e end is 10 integers : Thus the ●●e is in its most proper acception natural division . But if you are to deal with a ●●ater number then 10 , then 1 at beginning must signifie● inte●● , and in the middle 10 integers , 10 at the end 100 integers . But if you would have it to a fi●●e more , then the first 1 is ten , the ●●nd a hundred , the last 10 a ●●usand . If you proceed further , ●n the first 1 is a 100. the middle 1000 , and the 10. at the end is ●●00 , which is as great a number you can well discover , on this or most ordinary lines of numbers and so far with convenient car● you may resolve a question very exactly . Now any number bein● given under 100●0 , to find the point representing it on the rul● do thus . Numeration on the Line of number PROB. I. Any whole number being given und●● four figures , to find the point the Line of numbers that doth r●present the same . First , Look for the first figure your number , among the long ●● visions , with figures at them , a●● that leads you to the first figure your number : then for the seco●● figure count so many tenths fro● that long divisions onwards , as th● second figure amounteth to ; th●● for the third figure , count from 〈◊〉 last tenth , so many centesmes the third figure contains ; and for the fourth figure , count fr●● the last centesme , so many millions , as that fourth figure hath unites , or is in value ; and that shall be the point where the number propounded is on the line of numbers : Take two or three Examples . First , I would find the point upon the line of numbers representing 12 , now the first figure of this number is one , therefore I take the middle one for the first figure ; then the next figure being 2 , I count two tenths from that 1 , and that shall be the point representing 12 , where usually there is a brass pin with a point in i● . Secondly , To find the point representing 144. First , as before I take for 1 , the first figure of the number 144 , the middle Figure 1 then for the second Figure ( viz. 4. ) I count 4 , tenths onwards for that : Lastly , for the other 4 , I count 4 , centesmes further , and that is the point for 144. Thirdyl , To find the point representing 1728. First , As before , for 1000. take the middle 1 , on the line Secondly , For 7 , I reckon seventenths onward , and that is 700. Thirdly , For 2 , reckon two centesmes from that 7th . tenth for 20 ▪ And lastly , For 8 , you must reasonably estimate that following centesme , to be divided into 10 parts ( if it be not express'd , which in lines of ordinary length cannot be done ) and 8 , of that supposed 10. is the precise point for 1728 the number propounded to be found , and the like of any number whatsoever . But if you were to find a fraction , or broken number , then you must consider , that properly , or absolutely , the line doth express none but decimal fractions : thus , 1 / 10 or 1 / 100 or 1 / 1000 and more nearer the rule in common acception cannot express ; as one inch , and one tenth or one hundredth or one thousandth part of an inch , foot , yard , perch , or the like , in weight , number , or time , it being capable to be applyed to any thing in a decimal way : ( but if you would use other fractions , as quarters , half quarters , sixteens , twelves , or the like , you may reasonably read them , or else reduce them into decimals , from those fractions , of which more in the following Chapters ) for more plainness sake , take two or three observations : 1. That you may call the 1. at the beginning , either one thousand , one hundred , or one tenth , or one absolutely , that is , one Integer , or whole number , or tenintegers , or a hundred , or a thousand integers , and the like may you call 1 , in the middle , or 10 at the end . 2. That whatsoever value or denominations you put on 1 the same value or denomination all the other figures must have successively , ei●●er increasing forward , or decreasing backwards , and their intermediate divisions accordingly , as for example ; If I call 1 at the beginning of the line , one tenth of any integer , then 2 following must be two tenths , 3 three tenths , &c. and 1 in the middle 1 integer , 2 two integers , & 10 at the end must be ●en integers . But if one at the beginning be one integer , then 1 in the middle must be 10 integers , and 10 at the end 100 integers , and all the intermediate figures 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 integers , and every longest divsion between the figures 21 , 22 , 23 , 24 , 25 , 26 ; &c. integers , and the shortest divisions tenths of those integers , and so in proportion infinitely : as [ 1 10 1. 10 ] [ 1. 10. 100. ] [ 10. 100. 1000 ] [ 100. 1000. 10000. ] in all which 4 examples , the first order of Figutes , viz. 1 / 10 1. 10. 100. is represented by the first 1. on the line of of numbers , the second order of figures , viz. 1. 10. 100. 1000. is represented by the middle 1. on the ●ine of numbers : the last order or Place of figures , viz. 10. 100. 1000. 10000. is represented by the 10 , at the end of the line of numbers . 3. That I may be plain ( yet further ) if a number be propounded of 4 figures , having two cyph●rs in the middle , as 1005 , it is expressed on the line between that prime to which it doth belong , and the next centesm or small division next to it ; but if you were to take 5005 , where there are not so many divisions , you must imagin them so to be , and reasonably estimate them accordingly . Thus much for numeration on the line , or naming any point found on the Rule , in its proper value and signification . Note , that since this Book was first written , I have provided a Print of the Line eleven times repeated , where the Numerator is certainly expressed from 0001 to 10000000 ; and may be had here , or in the Minories . CHAP. II. PROB. I. Two Numbers being given , to find third Geometrically proportiona● unto them ; and to three , a fourth ▪ and to four , a fifth , &c. GEometrical proportion is when divers numbers being compar'd together , diffe● among themselves , increasing o● decreasing , after the rate or reaso● of these numbers , 2. 4. 8. 16. 32 ▪ for as 2 is half 4 , so is 8 half 16 and as this is continued , so it may be also discontinued , as 3. 6. 14 ▪ 28 ; for though 3 is half 6 , and 14 half 28 , yet 6 is not half 14 ▪ nor in proportion to it , as 3 is to 6. There is also Arithmetical and Musical proportion ; but of that in other more large discourses , being not material to our present purpose ( though I may hint it afterward . ) To find this by the numbers , extend the Compasses upon the ●●ine of numbers , from one number ●o another ; this done , if you ap●ly that extent ( upwards or downwards , as you would either increase ●r diminish ) from either of the numbers propounded , the moveable point will stay on the third proportional number required . Also , the same extent applyed the ●ame way from the third , will give ●ou a fourth , and from the fourth 〈◊〉 fifth . Example . Let these two numbers 2 and 4 be propounded to find a third proportional to them ( that is , to find a number that shall bear the same proportion that 2 doth to 4 ) and then to that third , a fourth , fifth , and sixth , &c. ) Extend the Compasses upon ●he first part of the line of num●ers , from 2 to 4 ; this done , if ●ou apply the same extent upwards ●om 4 , the moveable point will fall upon 8 , the third proportional required ; and then from 8 , it will reach to 16 , the fourth proportional ; and from 16 to 32 , the fifth ; and from 32 to 64 , the sixth proportional . But if you will continue the progression further , then remove the Compasses to 64 in the former part of the line , and the moveable point will stay upon 128 , the seventh proportional ; and from 128 to 256 , the eighth ; and from 256 to 512 , the ninth , &c. Contrarily to this , if you would diminish , as from 4 to 2 , extend the Compasses from 4 to 2 , and the moveable point will fall on 1● ; and from 1 to 1 / 10 , or 5 of ten , which is one half ( by the second Problem of the first Chapter ) and from 5 to 25 , or ¼ , and so forward . But generally in this , and most other work , make use of the small divisions in the middle of the line , that you may the better estimate the fractions of the numbers you make use on ; for observe , look how much you miss in setting of ●he Compasses to the first and se●ond terms , so much , or more will ●ou err in the fourth : therefore ●he middle part will be most use●ul : as for example , as 8 to 11 , ●o is 12 to 16 , 50 , or 5 , if you do ●magine one integer to be divided ●ut into 10 parts , as they are on ●he line on a two foot Rule ; but on the other end you cannot so well express a small fraction , as there you may . PROB. II. One number being given , to be multipl●ed by another number given , to find the Product . Extend the Compasses from 1 to ●he Multiplicator , and the same ex●ent applyed the same way ▪ from ●he Multiplicand , will cause the moveable point to fall on the product . Example ▪ Let 6 be given to be multiplyed by 5 , here if you extend the Compasses from 1 to 5● the same extent will reach from ● to 30 ; which number 30 , though it be numbred but with 3 , yet you● reason may regulate you , to call i● 30 , and not 3 ; for look what proportion the first number bears to 1 , the same must the other number ( or Multiplicand ) bear to the Product ; which in this place , cannot be 3 , but 30. Another Example , for more plainness ▪ Let 125 be given to be multiplied by 144 ; extend the Compasses from 1 to 125 , and the same extent laid the same way from 144 , the moveable point will fall on 18000 : now to read this number 18000 ( so much , and no more ) you must consider , that as in 125 there is two figures more than in 1 , so there must be two figures more in the Product 18000 , than in the Multiplicand 144 : and as for the order of reading the numbers , you may consider well the first Problem of the first Chapter . Some other Examples for more light . 3. As 1 to 25 , so 30 to 750 : as 1 to 8 , so 16 to 48 : as 1 to 9 , so 9 to 81 : as to 12 , so 20 to 240. One help more I shall add , as to the ●ight computation of the last figure in 4 figures ( for more cannot ●e well exprest , on ordinary lines , as that on a two foot Rule is ) but for the true number of figures in the Product , note , that for the most part , there is as many as there ●s in the Multiplicator and Multiplicand put together , when the ●wo first figures of the lesser of ●hem , do exceed so many of the ●●rst figures of the Product ; but ●f the two first figures of the least of them , do not exceed so many of ●he first figures of the Product , then ●t shall have one less than the Multiplicator and Multiplicand put together ; as , 92 and 68 multiplied , ●akes 6256 , four figures , because 68 is more than 62 ; and 12 multiplied by 16 , makes but 192 , three figures , for the reason abovesaid , because 19 is more than 16. Now for right naming the last figure , write them down ; as thus , 75 by 63 ; now you multiply 5 by 3 , that is 15 , for which you , by vulgar Arithme●ick , set down 5 , and carry 1 ; therefore 5 is the last figure in the Product , and it is 4725. PROB. III. Of Division . One number being giuen to be divided by another , to find the Quotient . Extend the Compasses from the Divisor to 1 , and the same extent will reach , the same way , from the Dividend to the Quotient ; or , extend the Compasses from the Divisor to the Dividend , the same extent shall reach , the same way , from 1 to the Quotient ; as for Example . Let 750 be a number given to be divided by 25 ( the Divisor ) I extend the Compasses from 25 to 〈◊〉 , then applying of that extent the same way from 750 , the other ●oint of the Compasses will fall up●n 30 , the Quotient sought : or ●ou may say , as 25 is to 750 , so is to 30. 2. Let 1728 be given , to be di●ided by 12 , say , as 12 is to 1 , so is ●728 to 144. Extend the Compas●es from 12 to 1 , and the same ex●ent shall reach the same way from ●728 to 144 : or as before , as 12 to 1728 , so is 1 to 144. 3. If the number be a decimal fraction , then you work as if it were an absolute whole number ; ●ut if it be a whole number joyned to a decimal fraction , it is wrought here as properly as a whole number : Example , I would divide 111.4 by 1.728 , extend the Compasses from 1.728 to 1 , the same extent applyed from 111.4 , shall reach to 64.5 : so again , 56.4 ▪ being to be divided by 8.75 , the Quotient will be found to be 6.45 . for the extent from 8.75 to 1 being laid the same way from 56.4 , will reach to 6.45 , the Quotient , viz. 6 , and a decimal fraction 45. To reduce this decimal fraction to the vulgar fraction found by the Pen ; the same extent of the Compasses laid the contrary way from 45 the decimal fraction , gives 39 / 75 ▪ the vulgar fraction found by the Pen , as is more apparent in the next Example . In Division by the Pen , the fraction remaining is the Numerator , and the Divisor is the Denominator ; to find which by the line , do thus , with the same extent of the Compasses that wrought the Division , laid the contrary way from the decimal fraction , gives the usual fraction when you work by the Pen. Example . To divide 522 by 34. The extent from 34 to 1 reaches from 522 to 15.35 ; then the same extent laid the contrary way from 35 , gives 12.15 , as by the Pen. Now to know , of how many figures any Quotient ought to consist , it is necessary to write the Dividend down , and the Divisor under it , and see how often it may be written under it , for so many figures must there be in the Quotient ; as in dividing this number ●2231 by 27 , according to the Rules of Division , 27 may be written three times under the Dividend , therefore there must be 3 figures in the Quotient ; for if you extend the Compasses from 27 to 1 ●t will reach from 12231 to 453 , the Quotient sought for . Note , that in this , and also in ●ll other Questions , it is best to ●rder it so , as that the Compasses ●ay be at the closest extent ; for ●ou may take a close extent more easily and exactly than you can a ●rge extent , as by experience you will find . PROB. IV. To three numbers given to find a fourth in a direct proportion ( of the Rule of Three Direct . Extend the Compasses from the first number to the second , that done , the same extent applied the same way from the third , will reach to the fourth proportional number required . Example . If the Circumference of a Circle , whose Diameter is 7 , ●● 22 , what Circumference shall a Circle have , whose Diameter is 14 ? Extend the Compasses from ●● in the first part , to 14 in the second , and that extent applyed the same way from 22 , shall reach t● 44 , the fourth proportional required ; for so much shall the Circumference of a Circle be , whose Diameter is 14. — And the contrary , if the Circumference were given again . Ex. 2. If 8 foot of Timber b● worth 10 shilling , how much is 1● foot worth ? Extend the Compasses from 8 to ●0 ( either in the first part or second ) the same extent applyed ●e same way from 12 , shall reach ●o 15 ; which is the answer to the ●uestion ; for so many shillings is ●2 foot worth at that rate . PROB. V. Three numbers being given , to find a fourth in an inversed proportion , ( or the Back Rule of Three . ) Extend the Compasses from the first of the numbers given , to the second of the same denomination , ●● that distance be applyed from the third number backwards , it shall reach to the fourth number ●ought . If 60 pence be 5 shillings , how much is 30 pence ? facit 2.5 . two ●illings , 5 tenths of a shilling , that ●● , being reduced , 2 s. 6 d. Again , If 60 Men can raise a ●rest-work of a certain length and ●readth , in 48 hours , how long will it ●e ere 40 Men can raise such another ? Extend the Compasses from 60 ▪ to 40 , numbers of like denomination , viz. of Men ; this done , that extent applyed the contrary wa● from 48 , will reach to 72 , the fourth number you look for ▪ Therefore I conclude , that 40 ▪ Men will perform as much in 72 hours , as 60 Men will do in 48 hours . Note , that this Back Rule o● Three , may for the most part , b● wrought by the Direct Rule of Three . If you do but duely consider the order of the Question , for you must needs grant , that fewer Men must have longer time , and the contrary therefore the Answer must be , in proportion to the Question , which might have been wrought thus as well : The extent from 40 to 60 will reach the same way , from 48 to 72 in direct proportion ; or contrarily , as 60 to 40 , so is 72 to 48 : which you see , is but turning the question to ●ts direct operation , according to the true reason of the question . Thus you have the way for the Direct and Reverse Rule of Three : for the Double Rule of Three , and Compound Rule of Three , this is the Rule for it . Always in the Double Rule of Three 5 terms are propounded , and a sixth is required , three of which terms are of supposition , and two of demand ; now the difficulty is in pla●●ng them , which is ●●est done thus , as in this Example . Example . If 5 spend●● ●● l. in 3 months , How many pounds will serve 9 Schollars for 6 months ? Note , here the terms of supposition are the first three , viz. 5 , 20 , and 3 , and the terms of demand ●●e 9 and 6. Then next , for the right placing them , observe which of the terms of supposition is of the same denomination with the term required , as here the 20 l. is of the same denomination with the how many pounds required : set th● always in the second place , and the two terms of supposition o● above another in the first place and the terms of demand one above another in the last place , th●● 5 — 20 — 9 3 — pounds — 6 Then the work is performed ●● two single Rules of Three Dir●● thus : Extend the Compasses from 5●● ▪ 20 , the same extent applyed t●● same way from 9 , shall reach 36 , a fourth ; this is the first o●●ration : Then as 3 to 36 , the 4● so is 6 to 72 , the number of pou●● required . By the Line of Nu●bers , the Double Rule is wroug●● as soon as the Compound : the●●fore I shall wave it now . Four Questions and their Answe●● to shew the various forms of work●● on the Line of Numbers . Quest. 1. If 12 Men raise a Frame in 10 days , in how many days might 8 Men raise the like Frame ? Reason tells me , that fewer Men must have longer time ; therefore the work is thus , as 12 is to 8 , so is 10 to 15 , by the last Rule ; or , as 8 to 10 , so is 12 to 15. Quest. 2. If 60 yards of Stuff , at 3 quarters of a yard broad , would hang a Room about ; how many yards of Stuff , of half a yard broad , will serve to hang about the same Room ? The work is thus ; as 5 10th . to 7 10th . ½ , so is 60 to 90 ; or , as 75 to 5 , so is 60 to 90 , wrought backwards . Quest. 3. If to make a Foot Superficial 12 Inches in breadth , do require 12 Inches in length , the breadth being 16 Inches , how many Inches in length must I have to make a Foot Superficial ? The work is thus ; as 16 is to 12 , so is 12 to 9 , the number of Inches to make a Foot. Quest. 4. If to make a Foot Solid , a Base of 144 Inches require i● Inches in height , a Base being given of 216 Inches , how much in heighth makes a Foot Solid ? The work then is , as 216 is to 144 , so is 12 to 8 : or otherwise thus , as 12 is to 216 , so is 144 to 8 , the heighth sought . PROB. VI. To three numbers given , to find a fourth in a doubled Proportion . This Problem concerns Questions of proportions between Lines and Superficies : now if the denominations of the first and second terms be of Lines , then extend the Compasses from the first term to the second ( of the same kind o● denomination ) this done , that extent applyed twice , the same way from the third term , the moveable point will stay upon the fourth term required . Example . If the Content of a Circle whose Diameter is 14 , be 154 , what will the Content of a Circle be , whose Diameter is 28 ? Here 14 and 28 having the same denomination , viz. of lines , I extend the Compasses from 14 to 28 , then applying that extent the same way , from 154 twice , the moveable point will fall on 616 , the fourth proportional sought , that ●s , first from 154 to 308 , and from ●08 to 616. But if the first denomination be ●f superficial content , then extend the Compasses unto the half of the ●istance between the first and second of the same denomination ; of the same extent will reach from the third to the fourth . Example . being●54 ●54 . have a Diameter that is 14 , ●hat shall the Diameter of a Circle ●e , whose content is 616 ? Divide the distance betwixt 154 and 616 into two equal parts , then ●et one foot in 14 , the other shall each to 28 , the Diameter required . The like is for Squares : For if a Square , whose side is 40 foot , contain 1600 foot ; how much shall ● Square contain , whose side is 60 foot . Take the distance from 40 to 60 , and apply it twice from 160● and the moveable point will sta● on 3600 , the content sought for . PROB. VII . To three numbers given , to find fourth in a triplicated proportion . The use of this Problem consisteth in Questions of proportion between Lines and Solids , wherei● if the first and second terms ha● denomination of Lines , then extend the Compasses from the fir●● to the second , that extent applye● thtee times from the third , w● cause the moveable point to st●● on the fourth proportional required . Example . If an Iron Bullet , whose Diameter is 4 Inches , shall weigh 9 pound what shall another Iron Bullet weigh whose Diameter is 8 Inches ? Extend the Compasses from 4 to 8 , that extent applyed the same way three times from 9 , the moveable point will fall at last on 72 , the fourth proportional & weight required , that is in short , thus , as 4 to 8 , so 9 to 18 , so 18 to 36 , so 36 to 72. But if the two given terms be weight or contents , of solids , and the Diameter or side of a Square or ) a Line is sought for , then divide the space between the two given terms of the same Denomination into three parts , and that ●●sistance shall reach from the third ●o the fourth proportional . Example . Divide the space between 9 and 72 in three parts , that third part shall reach from 8 to 4 , ●or from 4 to 8 , as the Question ●as propounded , either augmenting or diminishing . ) Example . If a Cube whose side is Inches , contains 216 Inches , how many Inches shall a Cube contain whose side is 12 Inches ? Extend the Compasses from 6 to 12 , that extent measured from 2● in the first part of the Line of Numbers three times , shall a last fall upon 1728 , in the second part of the Line of Numbers ; for note , if y●● had begun on the second part , y●● would at three times turning , hat● fallen beyond the end of the line and the contrary , as above , hol● here in Squares also . PROB. VIII . Betwixt two numbers given , to 〈◊〉 a mean Arithmetically Proportional . This may be done without ●● help of the Line of Numbers ; 〈◊〉 vertheless , because i● serves to 〈◊〉 the next following , I sh●ll here 〈◊〉 sert it , though I thought to 〈◊〉 both this and the next over i●●● lence ; yet to set forth the exc●● lency of Number , I have set th● down . and the Rule is this : Add half the difference of the ●iven terms to the lesser of them , and ●hat aggregate ( or summe ) is the Arithmetical mean required : or add ●hem together , and the half summe is the same . Example . Let 20 and 80 be the terms given , now if you substract one out of the other , their difference is 60 , whose half difference 30 , added to ●● the lesser term , makes 50 ; and that is the Arithmetical mean ●ought : also 20 and 80 is 100 , the ●alf is 50 ▪ as before . PROB. IX . ●etween two numbers given , to find a mean Musically proportional . Multiply the difference of the ●erms by the lesser term , and add likewise the same terms together ; ●is done , if you divide that product by the summ of the terms , ●●d to the Quotient add the lesser ●●rm ; that last summ is the Musi●●l mean required . Or shorter , ●●us ; Multiply the terms one by another , and divide the product by their summ , and the Quotient doubled is the Musical mean required ▪ Example . The numbers given being 8 and 12 , multiplied together , make 96 , that divided by 20 ▪ the summ of 8 and 12 , the Quotient is 4 80 , which doubled , i● 9-6 10 s , the Musical mean required . This may be done by the Line of Numbers ; otherwise thus find the Arithmetical mean between 8 and 12 , and then the Analogy or agreement is thus ; As the Arithmetical mean found viz. 10 , is to the greater term ●● so is the lesser term 8 to the Musical mean required , 9 6 / 10. PROB. X. Betwixt two numbers given , to find mean Geometrically proportional ▪ Divide the space on the Line ●● Numbers , between the two e● tream Numbers , into two eq●● parts , and the point will stay at t●● mean proportional required . So the extream numbers being 8 and 32 , the middle point between them will be found to be 16. PROB. XI . Betwixt two numbers given , to find two means Geometrically proportional . Divide the space between the two extream numbers , into three equal parts , and the two middle points dividing the space , shall shew the two mean proportionals . As for Example ; let 8 and 27 be two extreams , the two means will be found to be 12 and 18 ; which are the two means sought for . PROB. XII . To find the Square-Root of any number under 1000000. The Square-Root of every number is always the mean proportional between 1 , and that number for which you would find a Square-root ; but yet , with this general caution , if the figures of the number be even , that is , 2. 4. 6. 8 or 10. &c. then you must look for the unit ( or one ) at the beginning of the Line , and the number in the second part , and the Root in the first part ; or rather reckon 10 at the end to be the unit , and then both Root and Square will fall backwards toward the middle 1 in the second length or part of the line : but if they be odd , then the middle 1 , will be most convenient to be counted th● unity , and both Root and Square will be found from thence forward towards 10. So that according to this Rule , the square of 9 will be found to be 3 , the square of 64 wil● be found to be 8 , the square of 14● to be 12 , the square of 1444 to b● 38 , the square of 57600 to be 240● the square of 972196 will be found to be 986 : and so for any other number . Now to know of how many figures any Root ought to consist , put a prick under the first figure , the third , the fifth , and the seventh , if there be so many ; and look how many pricks , so many figures there must be in the Root . PROB. XIII . To find the Cube-Root of a Number under 1000000000. 1111 The Cube-root is always by the first of two mean proportionals between 1 and the number given , and therefore to be f●und by dividing the space between ●●em into three equal parts : so by this means , the Root of 17●8 will be found to be 12 , the Root of 17280 is near 26 , the Root of 17●800 is almost 56 , although the p●i●● on the Rule repres●ming all t●e square numbers , is in one place , y●● by altering the unit , it p●oduc●th v●rious points and numbers for t●eir respective prop●● Roots . The Rule to find which unit , is i● this manner : You must set ( or suppose to be set ) pricks under the first figure to the left hand , the fourth figure , the seventh , and the tenth ; now , if by this means , the last prick to the left hand shall fall on the last figure , as it doth in 1728 , then the unit will be best placed at 1 in the middle of the line , and the Root , the Square , and the Cube , will all fall forwards toward the end of the line . But if it fall on the last but one , as it doth in 17280 , then the unit may be placed at 10 in the beginning of the line , and the Cube i● the second length . But if the las● prick fall under the last but two● as in 172800 , it doth then pla● the unit always at 10 in the end o● the line ; then the Root , the Square and Cube , will all fall backward and be found in the second par● between the middle 1 and the end of the line . By these Rules it doth appear , that the Cube-root of ●● is 2 , of 27 is 3 , of 64 is 4 , of 125 is 5 , of 216 is 6 , of 345 is 7 , of 512 is 8 , of 729 is 9 , of 1000 is 10. As you may see by this Table of Square and Cube roots . Thus you have the chief Use of the Line of Numbers in general , and they that have skill in the Rule of Three , and a lirtle knowledge in Plain Triangles , may very aptly apply it to their particular purposes ; yet for their sakes for whom it is intended , I shall inlarge , to some more particular applica●ions , in measuring all sorts of Superficies and Solids ; wherein I do judge it will be most serviceable to them that be unskilful in Arithmetick , as before said . A Table of the Square and Cubique Roots Root . Square . Cube . 1 1 1 2 4 8 3 9 27 4 16 64 5 25 125 6 36 216 7 49 343 8 64 512 9 81 729 10 100 1000 12 144 1728 26 676 17576 56 3136 175616 204 41616 8489664 439 192721 84604519 947 896809 849278123 1000 1000000 1000000000 CHAP. III. The Use of the Line of Numbers , in measuring any Superficial measure , as Board , Glass , Plaistering , Paving , Painting , Flooring , &c. THe ordinary measure , and most in use , is a Two-foot Rule divided into 24 Inches , and every Inch in●o 8 parts , that is , Halfs , Quarters , and Half-quarters ; but these parts not agreeing with the parts on the Line of Numbers , which are Decimals or tenth parts , is bred very much trouble ; and there cannot be exactness without taking of small parts , as Ha●f quarters of Inches , or else using o● Reduction ; and it is also as troub●esome by Arithmetick as by the Line of Numbers . To avoid which , I would advise either to measure altogether by Foot-measure ( that is , a Foot divided into 1000 parts ( or rather , as i● sufficient for ordinary use , 100 ) and then the divisions on the numbers will agree fitly to the parts on your Rule , without any trouble for Fractions ; for so doing , Fractions do become whole Numbers as it were , and are wrought accordingly : But if you use i● not in measuring , yet you may have it set for to help you for the ready reducing of such Numbers as shall require it , thoug● I shall apply it to Inches also as it is commonly used , that i● may appear useful both ways ▪ accordingly as any man shall b● affected . The like reason holdeth for inches , Yards , Ells and Perches , o● any other measure ; for thereb● the work is made more easie , a● shall appear anon . First , by Foot-measure only . PROB. I. The breadth of any Oblong Superficies , as a Table , Gravestone , or the like , given in Foot-measure , to find how much in length makes one Foot. Extend the Compasses from the ●readth in Foot-measure to 1 , the ●ame extent applyed the same way ●rom 1 , shall reach to the length ●equired , to make one Foot Su●erficial in Foot-measure . Exam. At 8 tenths broad . Set one point of the Compasses ●n 8 , and extend the other point ●o 1 ; the same extent being laid ●he same way from 1 , shall reach ●o 1.25 , the length required , be●ng 1 Foot , and 25 parts of a hun●red : For 1.25 multiplied by 08 , ●● 100.0 , or just one Foot in De●●mals . PROB. II. The breadth and length of any Superficies given in Foot-measure , to find the superficial Content in Foot-measure . Extend the Compasses from 1 to the breadth , the same extent applyed the same way from the length in Foot-measure , shall reach to the superficial Content in Foo●● measure . Ex. At 8 tenths broad , & 15 f. long Extend the Compasses from 1 t● 8 , the same extent laid the sam● way from 15 foot the length , shal● reach to 12 foot , the superficia● content required . Again , at 1. 75 broad , and 25. 30 lon●● Set one point of the Compass●● in 1 , and open the other to 1. 7 then the same extent laid the sam● way from 25. 30 , will reach to 4● feet 275 parts , the superficial co●tent required . Note , the 275 parts coun●ed ●● foot measure , right against it inches is 3 inches 3 tenths . PROB. III. The breadth of any Oblong Superficies being given in inches , to find how many inches in length makes one foot . The extent from the inches ●road to 12 being laid the same way from 12 , shall reach to the inches long to make 1 foot . Examp. At 9 inches broad . Set one point in 9 , open the other to 12 , then the same extent said the same way from 12 , reaches to 16 inches , the length required to make 1 foot superficial : for 9 multiplied by 12 is 144 , the number of superficial inches in one foot . PROB. IV. The breadth and length of an Oblong Superficies given in inches , to find the superficial content in like inches . Examp. At 30 inches broad , and 83 inches long . Set one point of the Compasses in 1 , and open the other to 183 then the same extent laid the same way from 30 , shall reach to 5490 the true content in inches . Now to call this 5490 , and no more or less , observe , as 183 is two figures more than 1 , so must 549● be two figures more than 30 , as i● observed in the Rule of Multiplication . PROB. V. The length and breadth given i● inches , to find the content in Superficial Feet . The extent from 144 ( th● inches in one foot ) to the breadt● in inches , being laid the same wa● from the length in inches , sha●● reach to the content in feet . Examp. At 30 inches broad , a●● 183 long . Set one point in 144 , and op●● the other to 183 , the same exten● laid the same way from 30 , reache● to 38 foot 12 / 000 , the near conte●● in feet required . BROB. VI. The breadth given in inches , and the length in feet , to find the content in Superficial Feet . The extent from 12 to the breadth in inches , shall reach the same way from the length in feet to the superficial content in feet required . Examp. At 30 inches broad , and 15 foot 3 inches long . Set one point in 12 , and open the other to 30 , the breadth in inches ; then the same extent laid the same way from 15 foot 3 inches the length , reaches to 38 foot 1 inch and ½ , the true content . PROB. VII . The length and breadth being given in feet and inches , to find the superficial content in feet & inches . As 1 to the breadth or length in feet and inches , so is the length or breadth in feet and inches to the superficial content in like feet and inches . Examp. At 18 foot 9 inches broad , and 30 foot 7 inches long . Set one point always in 1 , open the other to the ( length or ) breadth 18 foot 9 inches , then the same extent laid the same way from 30 foot 7 inches the length reaches to 573 foot 5 inches , the superficial content . Note , if you have the line of Numbers divided into twelves for inches as before said , you may work this question more readily , and truely as I have often times made them for my own use and others also . Note , that how broad soever any superficies is , so much is there in a foot long of that Superficies . Examp. If a board be 3 feet 9 inches broad , there is 3 feet 9 inches in every foot long of that board , therefore ( 3 feet 9 inches or ) the breadth in feet and parts , multiplied by the length in feet and parts , give the superficial content . PROB. VIII . Of a Circle and his parts , as Diameter , Circumference , Square-equal and Square within , and Area of a Circle . For the ready finding of any of these , any one being given , there is found out five numbers , in whose places on the li●e may be set five Center-pins for the more ready finding of them and readiness to use ; which are as followeth , Viz. Diameter — 10. 000 Circumference — — 31. 416 Square equal — — 8. 862 Square within — — 7. 071 Area or Content — 78. 538 Thu if the Diameter of a Circle be 10 inches , then the Circumference is near 31 inches and 416 parts of 1000. The side of the Square-equal is 8 inches 862. The side of the Square within is 7. 071. The Area or Content is 78 inches 538 parts . And any one of these being given , all the other four may be readily found by the Line of Numbers , as followeth . Any Diameter of a Circle given to find the Circumference , &c. The extent from 10 a fixed Diameter , to any other Diameter given , shall reach the same way from 31. 416 the fixed Circumference , to the Circumference required for the given Diameter And from the fixed side of th● Square-equal to the required sid● of the Square-equal . And from the fixed side of the Square-wit● in to the required side of th● Square within , &c. 1. Examp. Let the given Diameter of a Circle be 15 , what is t●● Circumference answerable to i● &c. Set one point of the Compass● in 10 the fixed Diameter , and t●● other in 15 the given Diameter ; then the same extent applyed the same way from 31. 416 the fixed Circumference , shall reach to 47. 14 the Circumference required . 2. To find the side of a Square-equal to a Circle of 15 inches Diameter . Also the same extent applyed the same way from 8. 862 , the fixed number for the side of the Square-equal , shall reach to 13. 30 , the ●ide of the Square-equal to the Circle required . To find the side of the Square within . The same extent from 10 to 15 being laid the same way from 7. 071 the fixed side of the Square within , shall reach to 10. 61 , the ●●de of the Square within in a Circle of 15 inches Diameter required . To find the Area or Content of the Circle whose Diameter is 15 inches . The same extent from 10 to 15 being twice repeated , the same way from 78.538 will reach to 176 inches and 7 tenths , the Area of a Circle of 15 inches Diameter ▪ PROB. IX . The Circumference of any Circle being given , to find the Diameter ▪ the Square-equal or Squar● within and the Area . Examp. If the Circumference give● be 47.14 , what is the Diameter the side of the Square-equal , ●● Square within , and Area . 1. Set one point in 31.416 the fixed Circumference , and the other in 47.14 the given Circumference ; then the same extent lai● from 10 the fixed Diameter , sha●● reach to 15 the Diameter required for a Circle of 47 ▪ 14 Circumference . 2. Also the same extent la●● from 8.862 the fixed side of t●● Square-equal , shall reach the sa●● way to 13.30 , the side of t●● Square - equal required . 3. Also the same extent laid the same way from 7.071 , shall reach to 10.61 , the side of the Square-within in a Circle of 15 inches Diameter required . 4. The same extent from 31. 416 the fixed Circumference , to 47.14 the given Circumference , being twice repeated the same way from 78.538 , the fixed Area for 10 inches Diameter , shall reach ●o 176 inches and 7 tenths , the Area of a Circle of 47.14 inches about . PROB. X. The Area of any Circle being given , to find the Diameter , Circumference , Square-equal , or the side of the Square within required . The exact middle , or half di●tance measured on the Line of Numbers , between the fixed Area ●8 . 538 and the given Area , shall reach from the fixed Diameter to the inquired Diameter ; and from the fixed Circumference to the inquired Circumference , and from the sides of the fixed Square-equal or within to the inquired side● of the Squares-equal or within . Examp. Let the given Area ●● 176 inches ●● . The exact middle between 78 ▪ 538 the fixed Area , for a Circle o● 10 inches Diameter , and 176 inches and 7 tenths the given Area measured on a Line of Numbers . 1. Then that extenr laid th● same way from 10 the fixed Diameter , reaches to 15 the inquire● Diameter . 2. Also the same extent lai● the same way from 31.416 the fi●● ed Circumference , gives 47. 1●● the Circumference required . 3. Also the same extent lai● the same way from 8.862 the fix●● side of the Square-equal , reach● to 13.30 , the side of the Squar●● equal required . 4. Lastly , the same extent l●●● the same way from 7.071 the fixed Number for the side of the Square within , shall reach to 10.61 the side of the Square within required for a Circle whose Area is 176 inches and 7 tenths . PROB. XI . Having the Area of any Circle , to find the side of a Square equal so it . The Square root of the given Area ( found by Problem 12 of the 11th . Chapter ) is always the side of the Square-equal to the Circles Area given ; thus the Square-root of a Circle whose Area is 144 is 12 , the Square-root of 144. PROB. XII . To find the Content of a Circle two ways . Multiply the Diameter by it self , and multiply that product by 11 , and divide this last product by 14 , and the Quotient shall be the content required . Or else multiply half the Diameter and half the Circumference together , and the product is the content required . PROB. XIII . How to measure a Circle , a Semicircle , or a quarter of a Circle , or any part that goeth to the Centre of the supposed Circle . First for a Circle . Take half the Diameter and half the Circumference , and measure it then as an Oblong Square ; for the half Circle take half the Diameter and half the Semicircumference and do likewise . Thirdly , for the quarter of a Circle , take half the Arch of that quarter , and the Radius or Semidiameter of the whole Circle , and work as you would d● with an oblong square piece , and you shall have your desire . PROB. XIV . How to measure a Triangle . Take half the Base and the whol● Perpendicular , and work with them two as if it were an oblong square figure , or you may take the whole Base and half the Perpendicular . Or , by the Line of Numbers , the extent from 2 to the Base shall reach from the Perpendicular to the Content . PROB. XV. How to measure a Rhombus , or a Rhomboides . A Rhombus is a Diamond-like figure , as a quarry of Glass is , containing 4 equal sides , and two equal opposite angles : but a Rhomboides is a figure made of two equal opposite sides , and two equal opposite angles : and to measure them you must take any one side , and the nearest distance from that side to its opposite side for the other side , and then reckon it as an oblong Square . PROB. XVI . How to measure a Trapezium . A Trapezium is a figure comprehended of 4 unequal sides , and of 4 unequal angles ; and before you can measure it , you must reduce it into two Triangles , by drawing a line from any two opposite corners , then deal with it as two Triangles ; or you may save some work thus , the line you draw from corner to corner will be the common base to both Triangles : then say , as 1 is to half the Perpendiculars of both the Triangles put together , so the whole Base to the Content . PROB. XVII . How to measure a many sided irregular figure or Polygon . You must reduce it into Triangles or to Trapeziums , by drawing of lines from convenient opposite corners ; and then the work is all one with that of the last Problem . PROB. XVIII . How to measure a many sided regular figure , commonly called a regular Polygon . Measure all the sides , and take the half of the summ of them for one side of a square , and the nearest distance from the center of the Polygon to the middle of one of the sides for the other side of a square ; and with them two numbers work as if it were a square oblong figure , and it will give the content of the Polygon desired . PROB. XIX . How to reduce Feet into Yards , Ell● , or other parts . First for yards , if 9 foot make one yard , how many shall 36 foot make . The extent from 9 to 2 will reach from 36 to 4 , for so many yards is 36 foot . But if you were to measure any quantity by the yard , as the plaistering or painting of a House , then I would advise you to have a yard to be divided into a hundred parts ( which is as near as commonly Workmen go to , or else into 1000 , if you do require more exactness ) and measure all your lengths and breadths with that , and set them down thus , 2. 25 ( or by 1000 , thus 2. 250 ) and the length thus 10. 60 , and multiply them together , and the product is the true content of that long square : the like holds fo● Ells , or Poles , Furlongs , or any other kind of measure . Again , fo● a yard in length , if 3 foot make one yard , then what shall 30 make 〈◊〉 it maketh 10 : or the contrary , i● 10 yards make 30 foot , what shall 12 make ? The extent from 10 to 12 will reach from 30 to 36 foot 〈◊〉 but if it be given in feet or inches ▪ then say , as 9 to the breadth , so i● the length in feet and inches ( o● decimal parts ) to the content i● yards required . CHAP. IV. The Use of the Line of Numbers i● measuring of Land by P●rc●●● and Acres . PROB. I. Having the breadth and length of an Oblong Superficies given in Perches , to find the content in Perches . As 1 perch to the breadth in perches , so the length in perches to the content in perches . Examp. As 1 is to 30 , so is 183 to 5490 perches . PROB. II. Having the length and breadth in perches , to find the content in square acres . As 160 to the breadth in perches so is the length in perches to the content in acres . As 160 unto 30 , so is 183 to 34. ●1 ( in acres and 100 parts . PROB. III. Having the length and breadth of an oblong Superficies given in Chains , to find the content in Acres . It being troublesome to divide ●he content in perches by 160 , we may measure the length and breadth by Chains , each Chain being 4 perches in length , and divided into a hundred links , then the work will be more easie in Arithmatick or by the Rule ; for as 10 to the breadth in Chains , so the length in Chains to the content in Acres . Examp. As 10 to 7. 50 , so is 45. 75 to 34. 81 ( 100 parts of an Acre . ) PROB. IV. Having the Base and Perpendicular of a Triangle given in Perches , to find the content in Acres . If the Perpendicular go for the length , and the whole Base for the breadth , then you must take half of the oblong for the content of the Triangle , by the second Problem , as 160 to 30 , so 〈◊〉 183 to 34. 31 ; or else , without halsing , say , as 320 to the Perpendicular , so is the Base to the content in Acres ; as 320 unto 30 , so is 183 to 17. 15. PROB. V. Having the Perpendicular and Base given in Cains , to find the content in Acres . As 20 to the perpendicular , so is the Base to the content in Acres . As 20 to 7. 50 , so is 45. 75 to 17. 15 parts . PROB. VI. Having the content of a Superficies after one kind of perch , to find the content of the same Superficies according to another kind of perch . As the length of the second perch , is to the first , so is the content in Acres to a fourth number , and that 4th . number to the content in acres required . Suppose a superficies be measured with a chain 66 feet , or with a perch of 16½ , and it contains 34. 31 , and it be demanded how many acres it would contain if it were measured with a perch of 18 foot ? These kind of proportions are to be wrought by the Backward Rule of Three , after a duplicated proportion : wherefore I extend the Compasses from 16.5 unto 18.0 , and the same extent doth reach backward , first from 34.31 to 31.45 , and then from 31.45 to 28.84 , the content in those larger acres of 18 foot to a perch . PROB. VII . Having the plot of a Field with the content in Acres , to find the Scale by which it was plotted . Suppose a plain contained 34 acres 31 centesmes , if I should measure it with a Scale of 10 in an inch , the length should be 38 chains and 12 centesmes , and the breadth 6 chains and 25 centesms , and the content according to that dimension , would by the 3d. Problem of this Chapter be found to be 23 , 82. whereas it should be 34 ▪ 31 ; therefore to gain the truth ▪ I divide the distance between 23 ▪ 82 and 34.31 into two equal parts , then setting one foot of the Compasses upon 10 , the supposed true Scale , I find the other to extend to 12 , which is the length of the Scale required . PROB. VIII . Having the length of the Oblong , to find the breadth of the Acre . As the length in perches to 160 so is one acre to the breadth in perches . As 40 to 160 , so is 1 to 4. Again , as 50 to 160 , so is 1 to 3.20 , so is 2 to 6.40 ; or again , if you measure by chains , as the length in chains to 10 , so is 1 acre to his breadth in chains ; as 12.50 unto 10 , so 1 to 0.80 ; or if the length be measured by foot measure , then as the length in feet unto 43560 , so is 1 acre to his breadth in foot measure . So the length of the oblong being 792 feet , the breadth of one acre will be found to be 55 foot , the breadth of two acres 110 feet . The use of this Table is to shew you how many inches , centesmes of a chain , feet , yards , paces , perches , chains , acres , there is in a mile , either long or square , or consequently any of them all , in any of the other that is less ; as for example , I would know how many inches there is in a long perch , I look on the uppermos● row for perches , and in the nex● row under I find 198 for the quantity of inches in a long perch . B●● if I would know how many inches there is in a square perch , the● look for perch on the left hand ▪ and in the inch column you ha●● 39204 , for if you multiply 196 b● 196 , it will produce 39204. A necessary Table for Mensuration of Superficial Measure .   Inch. Centesme . Feet . Yard . Pace . Perch . Chain Acre . Mile . Inch. ( 1 ) 7 92 12 36 60 198 792 792● 6●360 Centes . 62 7264 ( 1 ) 1●515 4 545 7 575 25 〈◊〉 〈◊〉 8000 Feit . 144 2 295 ( 1 ) 3 5 16 5 66 〈◊〉 5280 Yards . 1296 20 655 9 ( 1 ) 1 66 5 50 22 220 1760 Pace . 3600 57 381 25 2 778 ( 1 ) 3 30 13● 1●2 1056 Perch . 39204 625 272.25 30 25 1● . 89 ( 1 ) 4 40 ●●20 Chain . 627264 10000 4356 484 174. 24 16 ( 1 ) 10 ●0 Acre . 6272640 100000 43560 4840 1742. 4 160 10 ( 1 ) 8 Mile . 101448960 64000000 27878400 3097●00 1115136 1●2400 6●00 640 ( 1 ) Square . Inches . Centesmes . Feet . Yards . Paces . Perch . Chain Acre . Mil● . The like is for any other number in the whole Table , and is of very good use to reduce one number into another , or one sort of measure into another ; as inches into feet , and feet into yards , and yards into perches , and perches to chains , and chains into acres , and acres into miles , or the contrary , either long-wise or square-wise ; as is well known to them that have occasion for these measures . Thus much shall suffice for superficial measure , the practice of which will make it plain to any ordinary capacity . CHAP. V. The Use of the Line of Numbers in measuring of solid Measure , as Timber , or Stone , or such like solids . PROB. I. The breadth and thickness of any solid given , to find the side of a square that shall be equal in area to it . Divide the space on the Line of Numbers , between the breadth and thickness , into two equal parts and the Compass point shall stay at the side of the Square-equal required . Examp. At 6 inches thick , and 18 inches broad . The exact middle between 6 and 18 on a true Line of Numbers , will be at 10 . 39● , which is a Geometrical mean proportion between 6 and 18 , and the side of a Square-equal , to a piece 6 inches thick , and 18 inches broad ; for 10.393 squared , viz. multiplied by 10.393 , produceth 108 very near , being the product of 6 times 18. PROB. II. The side of the square of any solid given , to find how much in length makes a foot of solid or Timber-measure , by foot measure or inches . The extent from the side of the square given in foot-measure to 1 , being twice repeated or laid the same way from 1 , shall reach to the length required to make a foot solid . Examp. At 2 foot and 20 parts square , how much in length makes on● foot solid . The extent from 2.20 to 1 being laid or extended twice the same way from 1 , will stay at o● 2065 , the true length to make ● foot solid . For 2.20 the side of the square , multiplyed by it self is 4.84 , the● 4.84 multiplied by 2065 , the product is 999.46 , which is near one foot solid . 2. If the side of the square b● given in inches , then the exten● from the inches square to 12 , being twice repeated the same wa● from 12 , shall reach to the inche● long to make one ▪ foot . Examp. At 15 inches square . Set one point in 15 and the o●●er in 12 , the same extent laid ●●ice the same way from 12 , shall ●●ach to 7 inches 68 parts , the true ●●●ngth to make one foot solid . For ●●5 inches squared is 225. Then 225 multiplied by 7.68 , ●●oduceth 1728 , the cube-inches 〈◊〉 a foot-solid . 3. For small Timber work thus . The extent from the inches ●●uare to 12 being laid two times ●●e same way from 1 , shall reach ●● the feet and parts in length to ●●ake one foot solid . Examp. At 3 inches and a half ●●uare . The extent from 3 inches and a ●alf to 12 being laid twice the ●●me wa● from 1 shall stay at ●1 foot 76 par●s , the length in ●eet to make a foot solid . For 3½ squared is 12.25 , which ●ultiplied by 11 foot 76 parts , ●roduceth 144 , the number of long inches in a foot solid , fo● 144 pieces a foot long , and on● inch square is a true solid foot o● 1728 cube inches . PROB. III. The breadth and thickness of any solid given in inches or foot measure , to find the length of one fo● solid . 1. The extent from 12 to th● breadth in inches shall reach fro● the depth in inches to a fourth . 2. The extent from that fourt● number to 12 laid the same wa● from 12 , shall reach to the inch●● long to make one foot solid . Examp. At 20 inches broad a●● 9 inches thick , how many inches lo● makes one foot . 1. The extent from 12 to 9 t●● inches thick , being laid the sa● w●y from 20 , the inches broa● gives 14.98 a fourth number . 2. The extent from 1498 t● 12 being laid the same way fro● 12 , gives 9 inches and 60 p●r●● the length to make one foot solid . But for small Timber work ●hus . 1. The extent from 1 to the breadth , shall reach from the depth to a fourth . 2. The extent from that fourth ●umber to 12 , laid the same way from 12 , gives the length in feet ●nd parts to make one foot solid . Examp. At 3 inches thick and 5 inches broad . 1. The extent from 1 to 3 the ●●ickness , laid the same way from 〈◊〉 inches the breadth , reaches to ●4 . 95 a fourth . 2. The extent from 14.95 the ●ourth to 12 , being laid the same ●ay from 12 , gives 9 foot 60 parts ●●e length in feet to make one foot ●●lid . For the product of 20 multipli●d by 9 is 180 , and 180 multiplied ●y 9 inches 6 tenths will produce ●728 the cube-inches in one foot . So also for the small piece . The 3 inches multiplied by is 15 , then 15 multiplied by 9 fo● 60 parts , will produce 144 , as b● fore , the long inches in a foot 〈◊〉 lid . PROB. IV. 1. At any squareness , or brea● and thickness given , to find 〈◊〉 much is in a foot long . The extent from 12 to the 〈◊〉 ches square , laid the same 〈◊〉 from the inches square , gives 〈◊〉 quantity in one foot long . Examp. At 4 inches squan●● Set one point of the Compa● in 12 , and the other point i● the same extent laid the same 〈◊〉 from 4 the inches square , giv● inch and 33 parts , the quantity one foot long . Which being multiplied by 〈◊〉 length of the Tree , gives the 〈◊〉 true content . 2. If the Timber is not sq●● then thus , The extent from 12 to the breadth , shall reach the same way from the depth or thickness to the quantity in one foot long . Examp. At 14 and●8 ●8 inches thick . The extent from 12 to 14 inches the breadth , shall reach the same way from 8 the inches thick , to 9 inches and 33 parts , the quantity or content in 1 foot long . Then if the Tree be 20 foot long , 20 times so much is 15 foot ●7 inches ⅔ , the true content . PROB. V. being most used . The side of the Square given in inches , and the length in feet , to find the content in feet and parts required . The extent of the Compasses from 12 to the inches square , be●ng twice repeated the same way from the length , gives the true content of the Tree required . Examp. At 15 inches square , ●nd 2C foot long , how much Timber 〈◊〉 there . Set one point in 12 , and extend the other to 15 , the inches square . Then the same extent laid twice the same way from 20 the feet long , shall reach to 31 foot 3 inches , the true content required . For 15 inches , or 1 foot 3 inches squared ( or multiplied by it self● is 1. 6. 9 or 1 foot 6 inches ¾ , an● 20 times so much is 31 foot 3 inches . PROB. VI. The breadth and thickness of any piece given in inches , and the length in feet , to find the soll●● content in feet . 1. The extent from 12 to the breadth in inches , laid the same way from the depth in inche● gives a fourth number . 2. The extent from that fourth number to 12 shall reach the same way from the length to the content required . Examp. At 8 inches thick an● 13 inches broad , and 19 foot long what is the solid content in feet . 1. The extent of the Compasses from 12 to 8 inches the thickness , being laid the same way from 13 inches the breadth , sh●ll reach to 8.66 a fourth . 2. The extent from 12 to that fourth , being laid the same way from 19 foot the length , gives 13 foot 73. ( or near 9 inches ) the true content in feet and parts . For 8 inches the thickness , multiplied by 1 foot 1 inch the breadth is of . 8 inc . 8.12 pts . then 19 foot the length , multiplied by of . 8. in . 8.12 makes 13 foot 8 inches and 8.12 , the exact content of a peece 8 inches thick , 13 inches broad , and 19 foot long . PROB. VII . The length , breadth , and depth of any solid given in inches to find the solid content in cube-inches . 1. The extent from 1 to the breadth in inches , being laid the same way from the depth in inches , shall reach to a fourth . 2. The extent from 1 to that fourth , shall reach the same way from the length in inches to the content in cube-inches required . Examp. At 9 inches thick , 15 inches broad , and 40 inches long . 1. Set one point of the Compasses in 1 , and the other in 9 the inches thick , then the same extent laid the same way from 15 the breadth in inches , gives 13.50 a fourth . 2. Then the extent from 1 to 13.50 the fourth , laid the same way from 40 the length in inches , gives 5400 the content in inches . For 9 multiplied by 15 is 135 , and 135 multiplied by 40 , is 5400. PROB. VIII . The length , breadth , and depth of a great solid given in feet and decimals , or in feet and inches , to find the content in feet and parts . 1. the extent from 1 to the breadth in feet and parts , shall reach the same way from the depth in feet and parts to a fourth . 2. The extent from 1 to that fourth being laid the same way from the length in feet and parts , shall reach to the content in feet and decimal parts required . Examp. Suppose a Cistern be 3 foot 75 parts ( or 3 foot 9 inches ) broad , and 4 foot 50 parts ( or 4 foot 6 inches ) deep , and 7 foot 66 parts , or 8 inches long , how many solid feet will it hold . 1. The extent from 1 to 3.75 being laid the same way from 4.50 gives 16.875 for a fourth . 2. The extent from 1 to 16.875 the fourth , being laid the same way from 7.666 the length , gives 129 foot 3637 parts , or 4 inches and ⅓ . For 3.75 multiplied by 4.50 , gives 16.875 the fourth , then 16.875 multiplied by 7.66 the length , gives 129 foot 364 parts , or 4 inches and ⅓ , which you may reduce by a look of your eye on inches and foot measure , laid one by the other on your Rule . Note , every 6 foot is a full Beer Barrel , therefore 129 foot is 11 Barrels and ½ . Note also , if the space on the Line of Numbers between every figure , is divided into 12 proportional parts by small pricks , they will represent inches , and be very ready in use , for any Lea●ner especially . CHAP. VI. To measure round Timber , by having the Diameter given in foot-measure , or inches , and the length in feet . PROB. I. At any Diameter given in inches , to find how many inches in length will make one foot solid . The extent from the Diameter in inches to 13.54 ( the Diameter when one foot long makes 1 foot of Timber ) being laid twice the same way from 12 , shall reach to the inches in length ●o make 1 foot . Examp. At 10 inches Diameter . The extent from 10 to 13.54 , being laid twice the same way from 12 , gives 22 inches , the length to make one foot solid . For 78.54 the area of a Circle of 10 inches diameter , multiplied by 22 produceth 1728 , the cube-inches in 1 foot solid . PROB. II. The Diameter given in foot measure , to find how much in length make one foot solid . The extent from the Diameter in foot-measure to 1.128 ( the Diameter in foot-measure , when one foot long makes one foot ) being laid two times the same way from 1 , gives the length in foot measure to make 1 foot solid , at that given Diameter . Examp. At 0.833 Diameter , how much in length makes 1 foot solid . The extent from 0.833 the given Diameter to 1.128 , laid twice the same way from 1 , gives 1.833 the near length in foot measure to make 1 foot solid . For 54.56 the area of a circle whose diameter is 0.833 , being multiplied by 1.833 , the product is 1000.045 , the decimals in 1 foot solid . PROB. III. The Diameter given in inches , to find how much is in one foot long . The extent from 13.54 , to the Diameter in inches given , being twice repeated from 12 , shall reach to the inches of Timber contained in one foot long . Or the same extent laid twice the same way from 1 , will reach to the feet and inches contained in one foot long . Ex. At 8 inches diameter , how much is in one foot long . Set one point in 13.54 , and open the other to 8 , the given Diameter in inches . Then the same extent laid twice the same way from 12 , gives 41 inches ¾ the long inches of Timber in one foot long , 144 be-being one foot . Or the same extent laid twice the same way from 1 , reaches to 0.348 , 1000 being 1 foot . Then 0.348 multiplied by the length of the Tree in feet gives the true content . Examp. At 15 inches diameter , and 20 foot long . The extent from 13.54 to 15 being laid twice the same way from 12 , gives 14 inches 66 parts , or laid twice from 1 , gives 1 foot 223. Then 20 times so much is 24 foot 56 parts , or 6 inches ¾ . Also 20 times 14.66 inches is 293.20 inches , 12 being one foot , which divided by 12 makes 24 foot near 7 inches . PROB. IV. The Diameter in inches being given of any tree or cillander , & the length in feet , to find the content in feet . The extent from 13.54 to the Diameter in inches , being laid twice the same way from the length , gives the content in feet . Examp. At 15 inches Diameter , and 20 foot long . The extent from 13.54 to 15 , being laid twice the same way from 20 , gives 24 foot and 42 pts . the content in feet . Note , that right against 42 in feet measure is 5 inches in the inch measure . For the area of a Circle 15 inches diameter is 177 , and 20 times this is 3540 , which divided by 144 , hath 24 foot 84 cube inches a Quotient ( for 7 inches the near content . ) PROB. V. The Diameter given in inches , and the length in inches , to find the content in inches . The extent from 1.128 to the Diameter in inches , being laid twice the same way from the length in inches , gives the true content in cube-inches . Examp. A Well , or a Tubb of 40 inches Diameter , and 50 inches deep . The extent from 1.128 to 40 the inches Diameter , being laid twice the same way from 50 the length in inches , gives 62900 inches , the near content in cube-inches . Or the extent from 1.128 to 3 foot 33 parts ( or 4 inches ) the given Diameter laid twice from 4 foot 166 ( or 4 foot 2 inches ) gives 36 foot 40 parts , the content in feet and parts . For 36 times 1728 inches is 62208 inches , then 692 inches in the 40 part over being added makes 62900 the cube inches in that Cilander . PROB. VI. Having the Circumference given in inches , to find how many inchee long makes one foot , or how many feet and parts long . The extent from the Circumference ( or girt ) about any Cillander given in inches , to 42.54 ( the inches about , when one foot long makes one foot solid ) the same extent laid twice the same way from 12 , reaches to the inches long , to make one foot solid measure . And the same extent laid twice the same way from 1 , gives the feet long to make one foot solid . Examp. Suppose a Tree be 30 inches about , how many inches , or feet and inches long , makes one foot solid measure . The extent from 30 inches the girt , to 42.54 , laid twice the same way from 12 , gives 24 inches 1 / 10 , the inches long to make one foot . Also the same extent laid twice the same way from 1 , gives 2 foot 01 or 1 / 100 part , the length in feet and parts to make 1 foot . For 71.7 the area of a Circle of 30 inches about , being multiplied by 24. ● , produceth 1728 , the cube inches in a foot solid . PROB. VII . The Girt of any Cillander given in inches , to find how many inches , or feet and inches is in one foot long . The extent from 42.54 to the girt in inches , laid twice the same way from 12 , gives the number of inches in one foot long . Or the same extent laid twice from 1 the same way , gives the feet and parts in a foot long . Examp. At 40 inches about . The extent from 40 the inches about to 42.54 , being laid twice the same way from 12 , gives 13 inches and 63 parts , the content of one foot long . Or the same extent laid twice the same way from 1 , gives one foot , and 136 parts of 1000 , the quantity in feet and parts in one foot long ; which 136 parts is 1 inch , 6 tenths and a half , as is seen presently by the inches and foot measure laid together on your two foot Rule . Then this 13 inches 63 parts , multiplied by the length of the Tree , gives the true content . PROB. VIII . The Circumference or Girt given in inches , and the length in feet , to find the content of any Cillender in feet and parts . The extent from 42.54 to the inches about , being laid twice the same way from the length in feet , gives the content in solid feet required . Examp. At 36 inches about , and 30 foot long . The extent from 42.54 to 36 the inches girt , being laid twice the same way from 30 the feet long , gives 21 foot and a half , the near content in solid feet . For the Area of a Circle 36 inches about , is 103.1 inches , which multiplied by 30 the length of the Cillender in feet , gives 3093 which summ divided by 144 , produceth 21 foot 69 cube inches , the near content . PROB. IX , Having the Girt and length given in foot measure , to find the content in feet . The extent from 3.545 ( the feet and 100 parts about , when one foot long makes one foot of Timber ) to the Girt in feet and parts . The same extent laid twice from the length in feet , gives the content in feet . Examp. A Brewers Tun of 20 foot about , and 4 foot and ½ deep , how many solid feet is it . The extent from 3.545 to 20 , shall reach the same way at two turnings from 4 foot and ½ to 143 foot and 10 parts , the solid content in feet : and 6 foot being a full Beer-barrel , it contains 24 barrels , Beer-measure . For 31 foot 8 parts , the area of a Circle of 20 foot about , being multiplied by 4 foot 5 parts the depth , gives 143 foot and 10 pts . as before . PROB. X. Having the Girt in inches , and the length of a Cillander given in inches , to find the solid content in cube inches . The extent from 3.545 to the Girt in inches , being twice repeated the same way , from the length in inches , gives the content in inches . Examp. At 48 inches about , and 24 inches in length , how many cube inches is it . The extent from 3.545 to 48 the inches girt , being twice repeated from 24 the length in inches , gives 4398 the content in solid inches . For 183.2 the area of a Circle of 48 inches about , multiplied by 24 the inches deep , produceth 4397 the near content as before . To insure you the number of places , the Print 11 times repeated , doth certainly direct you . PROB. XI . It being an ordinary way in measuring of round Timber , such as Oak , Elme , Beech , Pear-Tree , and the like , ( which is sometimes very rugged , uneven , and knotty ) to take a line and girt about the middle of it , and then to take the fourth part of that for the side of a Square-equal to that Circumference : but this measure is not exact , but more than it should be . But either because of allowance for the faults abovesaid , or for Ignorance , the custome is still used , and Men commonly think themselves wrong'd , if they have not such measure . Therefore I have fitted you with a Proportion for it , both for Diameter and Circumference . And first for Diameter . The Diameter given in inches , and the length in feet , to find the content . As 1.526 to the Diameter , so is the length to a fourth , and that fourth to the content in feet , according to the rate abovesaid . The extent from 1.526 to 9.53 being twice repeated from 8 , shall reach to 3.12 , the content . PROB. XII . Having the Circumference in inches to find the content in the abovesaid measure . As 48 to the inches about , so is the length to a fourth number , and that fourth to the content . The extent from 48 to 30 being twice repeated from 8 , shall fall upon 3. 12 , the content required . PROB. XIII . How to measure Taper Timber , that is , bigger at one end than at the other . The usual way for doing of this is to take the Circumference of the middle or mean bigness ; but a more exact way , is to find the content of the base of both ends , and add them together ; and then to take the half for the mean , which multiplied by the length , shall give you the true content . Examp. A round Pillar is to be measured , whose Diameter at one end is 20 inches , at the other end it is 32 inches Diameter , and in length 16 foot ( or 192 inches ) the content of the little end is 314. 286 , the area or content of the greater end is 773.142 , which put together , make 1087.428 , whose half 543.714 multiplied by 192 the length , gives 104393.143 Cubical inches , which reduced into feet , is 60 feet and 713 cubical inches for the solid content of the Pillar . PROB. XIV . To measure a Cone , such as is a Spire of a Steeple , or the like , by having the height and Diameter of Base . Examp. Let a Cone be to be measured , whose Base is 10 foot , and the height thereof 12 foot , the content of the Base will be found by the 14th . Problem of superficial measure , to be 78. 54 ; then this 78. 54 multiplied by 4 a third part of 12 , the perpendicular or height of the Cone will give 314. 4 , for the content of the Cone required . By the numbers work thus ; the extent from 1 to 4 will reach from 78. 54 to 314. 4. But because there may be some trouble in getting the true perpendicular of a Cone , which is its height , take this Rule : First , take half the Diameter , and multiply it in its self , which here is 25 , then measure the side of the Cone 13 , and multiply that by it self which here is 169 , from which take the Square of half the Base , which is 25 , your first number found , and the remain is 144 , the Square-root of which is the height of the Cone , or length of the perpendicular . PROB. XV. To measure a Globe or Sphere Arithmetically . Cube the Diameter , then multiply that by 11 , and divide by 21 , gives you the true solid content ; let a Sphere be to be measured whose Axis or Diameter is 14 , that multiplied by it self , gives 196 , and 196 again by 14 gives 2744 , this multiplied by 11 gives 30184 , and this last divided by 21 gives 1437. 67 , for the content of the Sphere whose Diameter is 14. But more briefly , by the Numbers thus , the extent from 1 to the Axis , being twice repeated from 3. 142 , will reach to the superficial content , that is , the superficies round about . But if the same extent from 1 to the Axis be thrice repeated from 5238 , it will reach to the solid content ; as 1 to 14 , so 3. 142 to 617 being twice repeated , as 1 to 14 , so 5278 to 1437 being thrice repeated . As for many sided figures , if they have length , you have sufficient for them in the Chapter of superficial measure , to find the base , and then the base multiplied by the length , giveth the content . But as for figures of roundish form , they coming very seldom in use , I shall not in this place trouble you with them , for they may be reduced to Spheres or Cones , or Trirngles , or Cubes , and then measured by those Problems accordingly . And so much for the mensuration of Solids . CHAP. VII . The Use of the Line of Numbers in Questions that eoncern Military Orders . PROB. I. Any number of Souldiers being propounded , to order them into a square Battle of Men. Find by the 12th . Problem of the second Chapter , the square-root of the number given ; for so much as that root shall be , so many Souldiers ought you to place in Rank , and so many likewise in File , to make a square Battle of Men. Examp. Let it be required to order 625 Souldiers into a square Battle of men ; the square-root of that number is 25 ; wherefore you are to place 25 in rank , and as many in file ; for Fractions in this practice are not considerable . For had there been but 3 less , there would have been but 24 in rank and file . PROB. II. Any number of Souldiers being propounded , to order them into a double battle of Men : that is , which may have twice as many in rank as file . Find out the square-root of half the number given , for that root is the number of men to be placed in file , and twice as many to be placed in rank , to make up a double Battle of Men. Examp. Let 1368 Souldiers be propounded to be put in that order : I find by the 12 aforesaid , that 26 , &c. is the square-root of 684 ( half the number propounded ) and therefore conclude , that 52 ought to be in rank , and 26 in file , to order so many Souldiers into a double Battle of men . PROB. III. Any number of Souldiers being propounded , to order them into a quadruple battle of men ; that is , four times so many in Rank as File . Here the Square-root of the fourth part of the number propounded will shew the number to be placed in File , and four times so many are to be placed in Rank . So 2048 being divided by 4 , the quotient is 512 , whose root is 22 ( 6 ) and so many are to be placed in file and 88 in Rank , being four times 22 , &c. PROB. IV. Any number of Souldiers being given , together with their distances in Rank and File , to order them into a Square battle of ground . Extend the Compasses from the distance in File to the distance in Rank ; this done , that extent applyed the same way from the number of Souldiers propounded , will cause the moveable point to fall upon a fourth number , whose square-root is the number of men to be placed in file ; by which , if you divide the whole number of Souldiers , the quotient will shew the number of men to be placed in Rank . Examp. 2500 men are propounded to be ordered in a Square-battle of ground , in such sort that their distance in File being seven foot , and their distance in Rank 3 foot , the ground whereupon they stand may be a just square : To resolve this question , extend the Compasses upon the Line of Numbers downwards from 7 to 3 ( then because the fourth number to be found , will in all likelihood consist of 4 figures ) if you apply that extent the same way from 2500 , in the second part among the smallest divisions , the moveable point will fall upon the fourth number you look for , whose square-root is the number of men to be placed in file . By which square-root , if you divide the whole number of Souldiers , you have the number of men to be placed in Rank . As 7 to 3 , so 2500 to 1072 , whose biggest square-root is 32 , then as 32 is to 1 , so is 2500 to 78. PROB. V. Any number of Souldiers being propounded , to order them in Rank and File , according to the reason of any two numbers given . This Problem is like the former ▪ for as the proportional number given for the file , is to that given for the Rank , so is the number of Souldiers to a fourth number ▪ whose square-root is the number of men to be placed in Rank , by which , if you divide the whole , you may have the number to be placed in File . Examp. So if 2500 Souldiers were to be martialled in sueh order , that the number of men to be placed in File , might bear such proportion to the number of men to be placed in Rank , as 5 bears to 12 , I say then , as 5 is to 12 , so is 2500 to 6000 , whose square-root is 77 the number in Rank ; then as 77 is to ● , so is 2500 to 32 ▪ &c. The number of men to be placed in File . CHAP. IX . The use of the Line in questions of Interest and Annuities . PROB. I. A summ of Money put out to Use , and the Interest forborn for a certain time , to know what it comes to at the end of that time , counting Interest upon Interest at any rate propounded . Take the distance with your Compasses between 100 , and the increase of 100 l. for one year , ( which you must do very exactly ) and repeat it so many times from the principal as it is forborn years and the point of the Compasses will stay on the Principal with the Interest , and increase according to the rate propounded . Examp. I desire to know how much 125 l. being forborn 6 years , will be increased according to the rate of 6 l. per Cent. reckoning Interest upon Interest , or Compound Interest . Extend the Compasses from 100 to 106 , that extent being 6 times repeated from 125 , shall reach to 177 l. the Principal increased with the Interest at the term of 6 years at the rate propounded . But if it were required for any number of months , then first find what 100 is at one month , then say thus , If 100 gives 10 s. at one month , what shall 125 be at 6 months end ? facit 75 s. And the work is thus : First say , If 100 gives 10 s. at one months end , what shall 125 ? and it makes 12 s. 6 d. then say , If one month require 12 s. 6 d. what shall 6 months require ? facit 75 s. that is three pound fifteen shillings , the thing required to be fonnd . PROB. II. A summ of money being due at any time to come , to know what it is worth in ready money . This question is only the inverse of the other ; for if you take the space between 106 and 100 , and turn it back from the summ proposed as many times as there are years in the question , it shall fall on the summ required . Examp. Take the distance between 106 and 100 , and repeat it 6 times from 177 , and it will at last fall on 125 , the summ sought . PROB. III. A yearly Rent , Pension , or Annuity being forborn for a certain term of years , to find what the Arrears come to at any rate propounded . First , you must find the Principal that shall answer to that Annuity , then find to what summ the Principal would be augmented at the rate and term of years propounded ; then if you substract the Principal out of that summ , the remainder is the Arrears required . Examp. A Rent , or Annuity , or Pension of 10 pound the year , forborn for 15 years , what will the Arrears thereof come to at the rate of 6 per Cent. Compound Interest ? The way to find the Principal that doth answer to 10 l. is thus : If 6 l. hath 100 for his Principal , what shall 10 have ? facit 166 l. 16 s. or 166 l. 8 s. for the extent from 6 to 10 will reach from 100 to 166.8 . which is 166 l. 16 s. Then by the first Problem of this Chapter , 166 l. 16 s. forborn 15 years , will come to 398 l. then substract 166 l. 16 s. out of 398 l. and the remainder , viz. 231 l. 4 s. is the summ of the Arrears required . But note , in working this question , your often turning unless your first extent be most precisely exact ▪ you may commit a gross errour , to avoid which , divide your number of turns into 2 , 3 , or 4 parts , and when you have turned over one part , as here 5 , for 3 times 5 is 15 , open the Compasses from thence to the Principal , and then turn the other two turns , viz. 10 — 15 , and this may avoid much errour , or at the least , much mitigate it ; for in these questions , the larger the Line is , the better . PROB. IV. A yearly Rent or Annuity being propounded , to find the worth in ready Money . First , find by the last what the Arrears come to at the term propounded , and then what those Arrears are worth in ready Money , and that shall be the value of it in ready money . Examp. What may a Lease of 10 l. per annum , having 15 years to come , be worth in ready money ? I find by the last Problem , that the Arrears of 10l . per ann . forborn 15 years , is worth 231 l. 4 s. And likewise I find by the second Problem that 231 l. 4 s. is worth in ready money 96 l. 16 s. and so much may a man give for a Lease of 10 l. per ann . for 15 years to come , at the rate of 6 l. per cent . But if it were not to begin presently , but to stay a certain term longer , then you must add that time to the time of forbearance ; as suppose that after 5 years it were to begin , then you must say , 231 l. 4 s. forborn 20 years is worth in ready money , and it is 72 l. 8 s. and that shall be the value of the Lease required . PROB. V. A sum of money being propounded , to find what Annuity to continue any number of years , at any rate propounded , that summ of Money will purchase . Take any known Annuity , and find the value of it in ready money ; this being done , the proportion will be thus ; as the value found out is to the Annuity taken , so is the summ propounded to the Annuity required . Examp. What Annuity to continue 15 years , will 800 l. purchase after the rate of 6 l. per Cent. Here first I take 10l . per ann . for 15 years and find it to be worth in ready money 96 l. 16 s. by the last Problem ; then I say , as 96 l. 8 s. is to 10 , so is 800 to 82-7 , which is 82l . 14 s. and so much near do I conclude will an Annuity of 82 l. 14 s. per ann . be worth for 15 years , after the rate of 6 l. per Cent. viz. 800 l. Also in this Impression is added the use of the Line of Pence , which is added to the Line of Numbers next to it , when it shall be desired by any one , being very convenient for casting up small summs of Money in any concern whatsoever . The Line of Numbers and Pence together , do give the Decimal Fraction of any summ under 20 s. very near , as by the Print to 11 Radixes is most plainly seen . Where 20 s. or 1 l. in Money , is at 1 in the fourth line , and 2 s. one tenth of a pound , at 1 right over it in the third Line ; 9 farthings and 6 tenths of a farthing , being one tenth of two shillings , at 001 in the second line right over 01. Lastly , 96 / 100 parts of a farthing at 0001 in the first Line of Numbers towards the right hand , or the 96 / 1000 parts , at the 00001 at the left end of the same Line . Therefore note , if 20 s. is 1 , 15 s. is 075 , 12 s. is 06 , 10 s. is 05 , 5 s. is at 025 , and 2 s. or 24 d. at 01 , which on Two-foot Rules is set at ▪ 10 at the end of the Rule , though in this case called but one tenth , when 1 is one pound . Then if 1 or 10 at the end , be 24 pence , 18 pence is at 75 , 12 d. at 5 , 6 d. at 25 , 2 d. at 8.33 in the first part , and 1 d. at 4166 , and 1 farthing at 1042 , a little beyond the first 1 on Two-foot Rules . From hence you may see that the Print of the eleven Lines of Numbers sheweth the right Decimal Fraction of any summ under 1 pound Sterling , & by consequence , for any summ above , and was purposely made to explain this on Two-foot Rules . The Use of the Line of Numbers and Pence laid together on Two-foot Rules , may be in a brief manner , thus ; 1. At any price 100 , or 5 score , what cost 1 , counting 10 at the end of the Line , 10 l. Examp. At 5 l. per 100 , counted at 5 in the second part , because 10 is called 10 l. just over or under it in the line of pence , is 12 d. the exact answer , for 100 s. is 5 l. 2. Examp. At 2 l. 10 s. per 100 , right against 2.5 on Numbers representing 2 l. 10 s. right against it on the Line of Pence is 6 d. the Answer , for 100 6 d. is 2 l. 10 s. Now observe , if 1 in the middle is called 1 l. then in the first part , 9 is 18 s. 9.5 is 19 s. 5 towards the beginning-end is 10 s. and 1 at the very beginning end is 2 s. Then to supply all under 2 s. to 1 farthing , begin again at the upper end at 10 , where is set 24 d. the same with 2 s. and back again , 23. 22. 21. 20. 19. 18 pence at 75 , then 17. 16. 15. 14. 13. 12 d. and 5 , then 11 10. 9. 8. 7. 6 d. at 25 ; then 5. 4. 3. 2 d. at 0.834 , then 1 penny at 0.4166 , then 3 farthings , 2 farthings , 1 farthing at 0.1042 , and every 10th . of a farthing in pricks between the farthings . 3. Examp. At 1 l. per 100 , counted at the middle 1 , is 2 d. 1 far . 6 tenths of a far . for 1. 4. Examp. At 10 s. per 100 , counted at 5 in the first part , just against it in the Line of pence is 1 penny ( or 4 farthings ) & 8 tenths of a farthing , the true answer . 5. Examp. At 2 s. per 100 , counted at 1 at the beginning of the Line of Numbers is no farthings , but 96 parts of a farthing in 100 pts . the answer for the half of 96 is 48 , the farthings in 2 s. 6. Examp. But for any price under 2 s. per 100 , count thus , as 12 d. per 100 , just against 12 d. on the Line of pence is 5 on numbers , for 5 tenths of a farthing the near answer ▪ for 5 tenths of a farthing is half a farthing , and 100 half farthings is 50 farthings , being two farthings above 48 , the farthings in one shilling . 7. Examp. At 6 d. per 100 , right against 6 in the Line of pence is 025 , or one quarter of a tenth of a farthing , on the Line of Numbers , the answer near the truth . For a hundred quarters of a farthing is 50 half farthings , or 25 farthings , one more than 6 d. But if you count 1 farthing less for every 6 d. you shall have it right . Example at 18 d. per 100 , count 17 ▪ d. and 1 far . 3 far . less than 18 d. and just against it on Numbers is 12 / 100 pts . of 1 farthing , the true price of 1. Again , at 12 d. counted at 11 d. 2 far . just against it on numbers is 48 / 100 pts . of a farthing , the price of 1. Again , for 6 d. counted at 5 d. 3 far . just against it is 24 / 100 pts . of a farthing , the true answer or price of 1. I have been large in this , that you might see the reason of it the plainer . Again , on the contrary upwards , at any price for 1 ▪ what cost 100 , or 5 score . At 48 / 100 pts . of a farthing for 1 , the price of 100 , is just 12 d. the contrary to the last example but one . Again , at 72 / 100 pts . of a farthing for one , 100 will cost just 18 d. as in the last but two foregoing . Again , at 69 / 100 pts . of a farthing for 1 , 100 is just 2 s. Note , the 96 is just against 23 d. being 4 far . less than 2 s. that is , 4 farthings abated for the four 6 pence's in 2 s. But for all summs between 2 s. at 1 at the beginning-end , and 10 l. at the end of the Line . This in 3 Examples . 1 Ex. At 4 s. per 100 counted at 2 in the first part of the Line of Numbers , just against it on the Line of Pence , is 1 farthing and 92 / 100 more for 1. 2 Ex. At 13 s. per 100 , counted at 6.5 on the Line of Numbers , just against it on the Line of Pence , is 6 farthings , 2 tenths and ● , the answer . 3 Ex. At 3 l. per 100 , counted at 3 in the second part , is 7 d. o far . ● of a farthing , the answer . But if your 100 be 5 score and 12 , then count thus with a pair of Compasses . The extent from 112 to 3 l. counted as before , laid the same way from the middle 1 , gives 6 d. 1 far . 6 / 10 of a farthing more for 1 l. Again , on the contrary , If 1 cost 7 d. farthing , and half a farthing , what cost 112 ? The extent from the middle 1 to 7 d. 1 far . and half a far . ( half way beyond the prick ) being laid the same way from 112 , will reach to 3.45 , or 3 l. 9 s. the answer . For if every figure in the second part be 1 l. then every 10th . cut between is 2 s. then 4 and ● and 45 is 9 s. Also , if every figure in the first part is 2 s. then 5 cuts or tenths beyond any figure is 12 d. 2½ is 6d . 1 is 2 d. farthing , half farthing . Just as you see by counting 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10. in the second part on the Numbers , just against it on the pence is the exact reduction to pence and farthings required . Ex. Against 3 is 7 d. near a farthing , against 6 is 14 d. near 2 farthings ; against 7.5 is just 18 d. And so for all other . Again , when the price of 100 costs above 10 l. then count thus . Ex. At 65 l. 15 s. and 6 d. per 100. First , 65 doubled is 13 s. then the 15 s. 6 d. counted on the first part of the Line of Numbers at 7. 75 , and just against it on the Line of pence is 7 far . and 44 pts . the true answer is 65 l. 15 s. 1d . 3 far . and 44.100 pts . of 1 farthing . Again , at 18 s. 7 d. 2 far . for 1 , what is the price of 100 ? Note , 18 s. is at 9 on the first part of the Numbers , which here stands for 90 l. then seek the odd 7 d. 2 far . on the Line of pence , and just against it on the Line of Numbers is 3.125 , the 3 is 3 l. the 1 more is 2 s. and the 25 more is 6 d. in all 93 l. 2 s. 6 d. the price of 100. For 1800 s. is 90 l. 100 times 7 d. is 2 l. 18 s. 4 d. 100 half pence is 4 s. 2 d. in all 93 l. 2 s. 6 d. Once more for a greater summ at 8l . 5 s. 6 d. for 1 , what costs 100. First , to 8 l. add 2 Ciphers , and it is 800 l. then add 2 ciphers to the 5 s. and it is 500 s. or 25 l. then just against 6d . in the line of pence in the line of Numbers is 2 l. 10 s. In all , 827 l. 10 s. the true price of 100 at 8 l. 5 s. and 6 d. for 1. A further use of the Lines of Pence and Numbers in several Questions . If 2 pound and ½ of any Commodity cost 7 d. 3 f. and ¼ of a far . what cost 32 pound ¾ ? The extent from 2.5 on Numbers to 7 d. 3 f. and ¾ on the Line of Pence , being laid the same way from 32 75 , shall reach to 4.27 . Note , the 4 is 8 s. and the 27 more counted on the line of Numbers beyond the middle 1 , gives on the line of Pence near 6 d. 2 f. in all 8 s. 6 d. 2 f. the true answer . If 3 yards and ½ of Ribbon cost 3 d. 3 f. and ½ , what cost 17 yards and ● ? The extent from 3.5 on the Line of Numbers to 3 d. 3 f. and half farthing on the Line of pence , being laid the same way from 17.5 on the Line of Numbers reaches to 19 d. 1 f. and half on the Line of pence , the true answer . 3. If 1 yard of plaistering cost 8 d. half penny , what cost 142 yards and a half . The extent from 1 on the Line of Numbers to 8 d. half penny on the Line of pence , laid the same way from 142½ on Numbers , reaches to 5.047 ; the 5 is 5l . and the 047 more , counted on Numbers , on the pence , is 11 d. farthing , the true answer . 4. If 112 l. cost 2 l. 10 s. 6 d. what cost 21 C. 3 q. and 12 lb ? The extent from 112 to 2.525 , for 2 l. 10 s. 6 d. laid the same way from 2448 , the pounds in 21 C. 3 q. and 12 lb. shall reach to 55 lb 4 s. the near true answer . For if 2.525 the decimal for 2 l. 10 s. 6 d. be multiplied by 2448 , the pounds in 21 C. 3 q. 12 lb , the product is 6181200 , which divided by 112 , the quotient is 55 lb. 189 , which reduced , is near 4 s. or 3 s. 38 farthings , or 9 d. half-penny . 5. If 5 ounces and a half of Nntmegs cost 20 d. 3 far . what shall 1 C. 1 q. 15 lb come to ? The extent from 5½ on Numbers to 20 d. 3 q. on the Line of pence , shall reach from 16 the ounces in 1 lb , to 02518 , the 25 is 5 s. and the 18 more is 1 far . 7 tenths , the price of 1 lb. Then the extent from 1 on Numbers to 2518 , laid the same way from 155 , the pounds in 1 C. 1 q. and 15 lb , shall reach to 39 l. 0 s. 10 d. the near true answer to the Question . CHAP. X. The application of the Line of Numbers to use in Domestick Affairs , as in Coals , Cheese , Butter , and the like . I have added this Chapter , not that I think it absolutely necessary but only because I would have the absolute applicableness of the Rule to any thing behinted at ; for it may be the answer of some , Do you come with a Rule to measure my Commodities which are sold by weight ? Yea , so far as there is proportion , it concerns that and any thing else , the application of which I leave to the industrious Practitioner ; only I here give a hin● ; as much as ●o say , here is a Treasure , if you dig , you may find ; for some may be apt to think , it being a Carpenters-Rule , it is fit but for Carpenters use only : but know , that in all measures which are either lengths , Superficials or Solids , or as some call , Longametry , Planametry , and Solidametry : and in all Liquids by weight or measure : and in all time , either by Years , Months , Weeks , Days , Hours , Minutes , and Seconds , and almost , ( I think , I may say ) in all things number is used , and in many things , proportional numbers ; why then may not this Line put in for a share of use , seeing it is wholly composed of , and fitted for proportional Numbers , & of so easie an attainment for any ordinary capacity , and chiefly intended for them that be ignorant of Arithmetick , & have not time to learn that noble Science , as some have . And first , for more conveniency of Reduction , take these Rules of Reduction . Rules for English Money . Note , that 4 farthings make a penny ; 16 farthings , 8 half pence , or 4 pence , make a groat ; 48 farthings , 24 half pence , or 12 pence , make a shilling : 40 pence , or 10 groats , is 3 shillings 4 pence ; 80 pence , 20 groats , or 6 shilling 8 pence , is a Noble ; 160 pence , 40 groats , or 13 shillings 4 pence , is a mark ; 20 shillings , 4 crowns , 3 nobles , or 2 angels , is a pound sterling . Rules for Troy-weight Note , that 24 grains is a penny-weight ; 20 penny-weights , or 24 carrots is an ounce Troy ; 12 ounces is a pound ; 25 lib. a quarter of a hundred , 50 lib. half a hundred , 75 lib. three quarters of a hundred , and 100 lib. is a hundred weight Troy. Rules for Aver-du-poize weight . Note , that 20 grains make a scruple ; 3 scruples is a dram ; 8 drams is an ounce ; 16 ounces is a pound ; 8 pound a stone ; 28 lib. a quarter of a hundred ; 56 lib. half a C. 84 lib. 3 quarters of a C. and 112 lib. or 14 stone , or 4 quarters of a C. is an hundred weight : 5 C. is a hogshead weight : 19½ C. is a Fother of lead : and 20 C. is a Tun weight . And note , that l. signifies a pound in money , and lib. signifies a pound in weight , either Troy or Aver-du-poize . Rules for concave Dry-measure . Note , that 2 pints is a quart , 2 quarts a pottle , or quarter of a peck ; 8 pints 4 quarts ; 2 pottles is one gallon , or half a peck ; 2 gallons is a peck ; 2 pecks make half a bushel ; 4 pecks or 56 lib. make a bushel ; 2 bushels is a strike ; 2 strikes a coomb or half-quarter ; 2 coombs 4 strikes ; or 8 bushels , make a Quarter , or a Seame ; 10 Quarters , or 80 bushels make a. Last . Rules for Concave Wet-measure . Note , that 2 pints is a quart ; 2 quarts a pottle ; 2 pottles 4 quarts or 8 pints make a Gallon ; 9 Gallons make a Firkin , or half a Kilderkin ; 18 Gallons make 2 Firkins , a Kilderkin , or a Rundlet ; 36 Gallons is 2 Kilderkins , or a Barrel ; 42 Gallons make a Terce ; 63 Gollons or 3● Rundlets make a Hogshead ; 84 Gallons , or 2 Terces make a Tercion or Punchion ; 126 Gallons is 3 Terces , 2 Hogsheads , 1 Pipe , or a But. A Tun is 252 Gallons , 14 Rundlets , 7 Barrels , 6 Terces , 4 Hogsheads , 3 Punchions , 2 Pipes , or Buts . Note , that in sweet Oyl , 236 Gallons make a Tun ; but of Whale-Oyl 252 goes to the Tun. Water Measure . Note , that 5 pecks is a Bushel ; 3 Bushels a Sack , 4● Bushels a Flat , 12 Sacks , 4 Flats , or 36 Bushels , make a Chaldron of Coals . Rules for Long-Measure . Note , that 3 barley-corns make an inch ; 2¼ inches make a Nail ; 4 Nails , or 9 inches make a quarter of a yard ; 12 inches make a Foot ; 3 foot , 4 quar . 16 nails , or 36 inches make a yard ; 45 inches , or 5 quarters of a yard , make an Ell ; 5 foot is a pace ; 6 feet , or 2 yards is a fathom ; 5 yards and a half , or 16 feet and a half , is a Pole , Rod , or Perch ; 160 perches in length , and 1 in breadth ; or 80 perches in length and 2 in breadth ; or 4 in breadth and 40 in length , make an Acre . 220 Yards , or 40 poles is a furlong ; 1760 yards , 320 poles , or 8 furlongs is an English mile ; 3 miles is a League ; 20 Leagues , or 60 miles is a degree in ordinary account , and every mile a minute . Rules for Motion and Time in Astronomy and Navigation . Note , that a minute contains 60 seconds , and 60 minutes is 1 degree ; and 30 degrees is 1 sign ; 2 signs , or 60 degrees is a sextile ⚹ , 3 signs , or 90 degrees is a Quadrant , or quartile □ ; 4 signs , or 120 degrees a trine △ ; 6 signs , or 180 degrees is one opposition ☍ ; or semicircle ; 12 signs or 360 deg . is a conjunction ☌ , and the Suns Annual or Moon 's monthly motion . Note also , every hour of time hath in motion 15 degrees : and a minute of time hath 15 minutes of motion , and one degree of motion is 4 minutes of time . Note further , that every hour of time hath 60 minutes , therefore 45 is 3 quarters , 30 is half , 15 is a quarter of an hour ; 24 hours a day natural , 7 days a week , 365 days and about 6 hours is a year . Hence it follows , that ¼ of a degree in the Heavens is 5 Leagues on the Earth , or 15 minutes of motion above , is one minute of time below , therefore a degree or 60 minutes of motion is 4 minutes of time , as before is said . All these Rules I shall express more largely and in shorter terms , by these following Tables . Equation for Motion .   Signs . Deg. minutes . Seconds . Note , that the twelve Signs is 12 360 21600 1296000 One Sign is 1 30 1800 108000 One Deg. is   1 60 3600 One Min. is     1 60 Equation for Time.   Mon. VVeek . Day . Hour . Minute . One Year 13 52 305 8760 52560 Month hath 1 4 28 671 40320 Week hath   1 7 168 10080 Day natural     1 24 1440 Hour hath       1 60 Minute is         1 Equation for Long-measure .   Mile . Furl . Perch . Yards . Feet . Inches . Leag . 3 24 960 5971¼ 1●4● 190080 Mil. 1 8 340 1760 5280 63360 Furlong   1 40 220 660 7920 Per. Rod. P●l .     1 5½ 16½ 198 Acre contains of squa . Perc.   160 4840 43560     Acre is in leng     4 20 660 7020 Acre is in breadth     4 2● 66 792 1 Rood , or ¼ of an acre is in len .     40       1 Rood , or ¼ of an acre is in bread .     1 5½ 16½ 198 One Fathom is         6 72 One Ell English is         3¼ 45 One yard is         3 36 One foot is         1 12 One inch is 1 inch 3 grains .           1 Equation of Liquid-measure .   Gall. Pottl . Quarts . Pints . Tun of sweet Oyl 236 472 944 1888 Tun of Wine is 252 504 1008 2016 But or Pipe is 126 252 504 1008 Tertian of Wine 84 168 336 672 Hogshead is 63 126 252 504 A Barrel of Beer or 2 Runlets of Wine — is 36 72 144 288 Kilderkin or one Runlet . is 18 36 72 144 Barrel of Ale is 32 64 128 256 Kilderkin of Ale 16 32 64 128 Firkin of Beer is 9 18 36 72 Firkin of Ale is 8 16 32 64 Equation of small Dry-measure , and then of great measure .   Peck . Gal. Pottl . Qu. Pi. Bushel of water-m . is 5 10 20 40 80 Bushel of Land-m . is 4 8 16 32 64 One peck is 1 2 4 8 16 One Gallon is   1 2 4 8 One Pottle is     1 2 4 One Quart is       1 2   Last . weig . Chal. Qu. Bush. Peck . Pints , Last of Dry-m . 1 2 2½ 10 80 320 5120 One weight is   1 1¼ 5 40 160 2560 Chaldron of coals     1 4 36 144 2098 Quarter of wheat is       1 8 32 512 One Bushel is         1 4 62 Equation for Averdupoize-Weight .   Hogsh . C. Stons . Lib. Ounces . Drams . Scruples . Grains . Tun-w . gross is 4 20 280 2240 35840 286720 860160 17203200 One Hogshead is 1 5 70 560 8900 71680 215040 4300800 One C. or hund . is   1 14 112 1792 14336 43008 860160 One half C. is     7 56 876 7168 21504 430080 One quarter of C. is     3½ 28 448 4584 10752 213040 One Stone is     1 8 128 1024 3072 61440 One Lib. pound is       1 16 128 384 7680 One Ounce is         1 8 24 480 One Dram is           1 3 9 One Scruple is             1 3 Equation for Troy-weight .   Lib. Ounce . Dp. Carrots . Grains C. w. ●00 ●200 ●400● 2●800 ●●6000 ½ . C is 50 600 12000 14400 288000 ¼ C. is 29 300 6000 7●00 1440●0 ⅛ C. is 12½ 150 3000 ●600 72000 Pound 1 12 340 88 5760 Ounce   1 20 24 480 One penny weight     1 1● 2● One Carrot Troy is       1 20 One Grain is         1 Equation of Money .   Mark. An. Nob Cro. Sh. Groat . Penc . Far. Pound ●● 1½ 2 3 4 20 60 240 960 Mark is 1 1⅓ 2 2⅔ 13⅓ 40 160 640 An Angel is   1 1½ 2 10 30 120 410 A Noble is     1 1⅓ 6⅔ 20 80 320 A Crown is     1 5 15   60 240 A Shilling is         1 3 12 48 A Groat is           1 4 16 A Penny is             1 4 The use of which ( to come to our intended purpose ) may be thus . There you see how many farthings , pence , groats , shillings , and the like , is in one , or any usual piece of Coyn , also , how many ounces , scruples , in any kind of weight ; and the like for measure , both liquid and dry ; and also in Time : now if you would know how many there shall be in any greater number than one ; then say by the Rule ( or Line of Numbers ) thus , If 48 farthings be one shilling , how many shillings is 144 farthings ? facit 3 s. for the extent from 48 to 1 , will reach from 144 to 3. and the contrary . Again , If a mark and a half be 1 pound , how many pound is 12 marks ? the extent from 1.50 to 1 , shall reach from 12 to 8. for reason must help you not to call it 80 l. Again , If 3 nobles be 1 l. what is 312 nobles . facit 104 l. the extent from 3 to 1 will reach from 312 to 104. Further , If a Chaldron of Coals cost 36 s. what shall half a Chaldron cost ? facit 18. ( but more to the matter ) If 36 bushel cost 30 s. what shall 5 bushels cost ? facit 4. 16. that is by Reduction 4 s. 2 d. near the matter , or penny , or 3 farthings , half farthing , and better : or on the contrary ; If 1 bushel cost 8 d. then what cost 36 ? facit 288 d. which being brought to shillings , is just 24. which you may do thus ; If 12 d. be 1 s. how many shall 288 be ? facit 24. for the extent from 12 to 1 shall reach the same way from 288 to 24 , as before . The like may be applyed to all the rest of the Rules of weight and measure ; of which take in fine , some Examples in short , and their Answers . If 14 Stone be 1 C. what is 91 Stone ? facit 6 C. and a half . If 1 ounce be 8 drams , how many drams in 9 ounces ? facit 72. The extent from 1 to 8 , reacheth from 9 to 72. If 1 bushel of water-measure be 5 pecks , how many pecks is 16 bushels ? facit 80 pecks . If 1 barrel hold 288 pints , how much will a Firkin hold ? This being the fourth part of a barrel , work thus , if 1 give 288 , what 25 ? facit 72 , the answer sought . If 1 week be 7 days , how many days is 39 weeks ? As 1 is to 7 , so is 39 to 273. So many days in 39 weeks . If 160 perches be 1 acre , how many acres is 395 perches ? facit 2. 492 , that is near 2½ acres . If 8 furlongs make 1 mile , how much is 60 furlongs ? facit 7½ mile . For the extent from 1 to 8 , gives from 60 to 7. 50. CHAP. XI . To measure any superficies or solid by inches only ( or by foot-measure ) without the help of the line , by multiplication of the two sides . PROB. I. Possibly that this little Book may meet with some that are well skilled in Arithmetick , and being much used to that way , are loath to be weaned from that way , being so artificial and exact , yet tho they can multiply and divide very well , yet perhaps they know not this way to save their division , and yet to take in all the fractions together , as if of one denomination : I shall begin first with foot measure , being the more easie , and I suppose my Two-foot Rule to be divided into 200 parts , and figured with 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100. and then so again to 200 , as in the 3d. Chap. and then the work is only thus : set down the measure of one side of the square or oblong , thus , as for example , 7. 25 and 9. 88 , and multiply them as if they were whole numbers , and from the product cut off 4 figures , and you have the content in feet , and 1000 parts of a foot , or yard , Ell , Perch , or whatsoever else it be . Note the examples following . For any kind of flat superficies , this is sufficient instruction to him that hath read the first part ; but if it be Timber or Stone , you must thus find the base , and then another work will give you the other side , as in Chap. 5. Prob. 2. or multiply the length by the product of the breadth and thickness , and that product shall be the content required . PROB. II. To multiply feet , inches , and 8 parts of an inch together without Reduction , and so to measure superficial ( and solid ) measure . First , multiply all the whole feet , then all the feet and inches across , and right on , then the parts by the feet , and also the inches and parts across and right on ; then add them together , and you shall have the answer in feet , long inches ( that is in pieces of a foot long , and inch broad ) square inches , and 8 parts of a square-inch : as for Example . Let a piece of board be given to be measured that is 3. 3. 5. i. e. 3 foot , 3 inches , and 5 eights one way , and 2. 3. 4. the other way , I set the numbers down in this manner , and then right on , first as the line in the Scheme from 2 to 3 leads . I say thus , 3 times 2 is 6 , set 6 right under 2 and 3 , as in the example in the left page : for 6 foot , as is clear , if you consider the Scheme over the example , viz. the squares noted with f. then for the next , I say cross-wise , 2 times 3 is 6 , viz. long inches , as you may percieve by the 2 long squares marked with 9 L. and 6 L. which 6 I put in the next place to the right hand , as in the example ; then for the next , viz. 3 times 3 is 9 ( cross wise , as the stroke from 3 to ●● shews ) which 9 is also 9 long inches , as the Scheme sheweth , and must be put under 6 in the second place towards the right hand , in the Scheme it is expressed by the 3 long squares marked with L9 . Then lastly for the inches , 3 times 3 is 9 , going right up , as the stroke from 2 threes lead you : but note , this 9 must be set in the next place to the right hand , because they are but 9 square inches , but had the product been above 12 , you must have substracted the 12 out , and set them in the long inches place , and the remainder , where this 9 now standeth , and this 9 is expressed in the Scheme , by the little square in the corner mark't with ( □ 9. ) Then now for the Fractions , or 8 parts of an inch , first say , cross-wise , as the longest prick line doth lead you to , 3 times 4 is 12 , for which 12 you must set down 1.6 , that is 1 long inch , and 6 square inches ; the reason is , a piece 8 half quarters of an inch broad , and 12 inches long , is a long inch , or the twelfth part of a foot superficial ; and if be 12 square inches , then 4 must needs be 6 square inches : therefore instead of 12 I set down 1.6 , as you may see in the example , and in the least long square of the Diagram or Scheme . Then do likewise for the other long square , which is also multiplied across , as 2 times 5 ▪ is 10 , that is , as I said before , 1.3 as the example and Scheme make manifest , considering what I last said , and it is marked by the 2.00 . But if this or the other had come to a greater number , you must have substracted 8 as oft as you could , and set down the remainder in the place of square inches , and the number of 8 in the place of long inches , as here you see . Then for the two shorter long squares next the corner , say cross-wise again , 3 times 5 is 15 , that is 1.7 , because 8 half quarters an inch long , do make 1 square-inch as well as eight half-quarters a foot long made 1 long inch : therefore I set 1 in the place of square inches and 7 in the next place to the right hand , and it is expressed in the diagram by the small long square , and marked with * 1.7 . Then again for the other little long square , say cross-wise , as the shorter prick line leads you , three times 4 is 12 , that is 1.4 ; and do by this as the last : it is noted in the Scheme by 1.4 . Then lastly , for 5 times 4 , as the short prick line sheweth you , is 20 : out of which 20 take the 8 s. and set them down in the last place , and the 4 remaining you may either neglect ( or set it down a place further ) for you cannot see it on the Rule ; therefore I thus advise , if it be under 4 , neglect it quite , but if above , increase the next a figure more , if 4 , then it is a half , and so may be added ; for note , 64 of these parts make but 1 square inch ; of which parts the little square in the right hand lower corner of the Scheme is 20 , for which I fet down 2.4 , that is two half quarters and 4 of 64 , which is the last work , as you may see by the Scheme and example . Now to add them together , say thus , 4 is 4 , which I put furthest to the right hand , as it were useless , because not to be expres●● ; then 472 are 13 , from which take 8 , and for it carry 1 on to the next place , or as many times 1 as you find 8 , and set down the remainder , which here is 5 , then 1 I carried , and 13619 is 21 , from which I take 12 and set down 9 , because 12 square inches is 1 long inch ; then 1 I carried ( or more , had there been more 12 s. ) and 1169 is ▪ 18 , from which take 12 , as before , there remains 6 , that is 6 long inches ▪ and so had there been more 12. so many you must carry to the next place , because 12 long inches is one foot , lastly , 1 I carried and 6 is 7 foot , so that the work stands thus , F. Lo. In. Sq. In. 8. 64 7 — — 6 — 9 — — 5 — 4. And so for any other measure Superficial or Solid . To multiply feet , inches , and 12 parts of an inch , by feet , inches , and 12 parts of an inch . Although the usual way of dividing the line of inches , on ordinary Two-foot Rules , is into 8 parts , according to which counting the former R●●● is worded , into feet , inches , and 8 parts of an inch . Yet if every inch be divided into 12 parts , o● conceived so to be , which you may easily count by calling every quarter of an inch 3 , every half inch 6 , every three quarters 9. Then the parts between 3 , 6 , and 91 may be easily estimated by help of the halfquarter cut . Then I say , the Arithmetical multiplication will be much easier , being brought to one denomination after the way by decimals , and somewhat more exact , as will appear by this following example . Suppose a Cedar ▪ board be 2 foot 3 inches , and 7 twelve parts of an inch broad , and 9 foot 5 inches and 8 twelve parts of an inch long , how many superficial feet is it ? First , set down the numbers as is usually done in their three denominations of feet , inches , and 12 parts , as in the example is made apparent by the black lines and the pricked lines down right and sloping from figure to figure , done so only for ease and plainness , in wording of it to a Learner . Then proceed in the multiplication thus ; 1. First , 2 times 9 is 18 , as the down right black line from 2 to 9 sheweth , for which set down 18 foot under the 2 and 9 , as in the Example . 2. Secondly , say 9 times 3 is 27 long inches , as the sloping black line from 9 to 3 sheweth , for which 27 you must set down 2 foot in the place of feet , and 3 in the place of long inches , because 27 long inches is 2 foot 3 inches . 3. Thirdly , Say 2 times 5 is 10 long inches , as the sloping black line from 2 to 5 sheweth , for which you set down only 10 long inches under 5 and 3 as in the the example , because it is under 12 long inches , which make 1 foot , as in the second just before . 4. Fourthly , Say 3 times 5 is 15 square inches , as the short black line from 3 to 5 sheweth ; every 12 whereof makes 1 long inch . Therefore I set down 1 long inch in the place of long inches , and the 3 square inches over in the place of square inches right under 7 and 8. 5. Fifthly , Say 9 times 7 is 63 square inches , as the long pricked line from 9 to 7 sheweth , every 12 whereof makes 1 long inch , and every 144 one foot ; therefore I set down 5 long inches in that place and 3 in the place of square inches . 6. Sixthly , Say 2 times 8 is 16 square inches , as the other long sloping pricked line from 2 to 8 sheweth , for which set down one long inch and 4 square inches , each in their proper places , as in the fifth last mentioned . 7. Seventhly , Say 5 times 7 is 35 , 144 or 12 parts of 1 square inch , as the shorter sloping pricked line from 5 to 7 sheweth , every 12 whereof is 1 square inch ; therefore set down 2 in the place of square inches under 7 and 8 and 11 , the parts over in a space beyond , under 144. 8. Eighthly , Say 3 times 8 is 24 , 144 or 12 parts of one square inch , as before , as the other short sloping pricked line sheweth , every 12 whereof is one square inch ; therefore I set down 2 in the place of square inches under 8 and 7 , and no more , because nothing is over 2 halves . Lastly , Say 7 times 8 is 56 , 1728s . as the short down right pricked line from 7 to 8 sheweth , every 12 whereof makes one 144 , or the 12 part of 1 square inch ; therefore I set down 4 in the place of 144 & the 8 parts over in the place of 1728 s. being a place further , as in the example you see done . Feet Lon. inc . Sq. inc . 1●4 . 1728. 18 3 3 11 8 2 10 3 4     1 4       5 2       1 2     21 9 3 3 8 1. Then add them together , saying , 8 under 1728 is 8 and no more . 2. Then 11 and 4 is 15 , set down ● under 144 , and carry 1 to the next place . 3. One I carried and 2. 2. 4. 3 , 3 , under the square inches , or 12 is 15 , for which I set down 3 and carry 1 to the next place . 4. One I carried and 1. 5. 1. 10. 3 is 21 , for which I set down 9 under the long inches , and carry 1 to the next place . 5. One I carried and 2 and 8 is 11 , for which set down 1 and carry 1 to the next place . 6. One I carried and 1 is 2 , in all 21 foot 9 long inches , or 12 parts of a foot ; 3 square inches , or 12 parts of a long inch ; 3 144 or 12 parts of a square inch , and 8. 1728 parts of 1 long inch . If it be a piece of Timber o● Stone ; then having thus gotten the Area of the Base , then multiply that Area by the length in feet , inches , and 12 parts , and the product shall be the solid content required . As in this Example , 1. 2. 4 thick ▪ 2. 1. 6 broad , and 9 foot 7 inches 6 twelves long . The Use of the LINE of NUMBERS ON A SLIDING ( or GLASIERS ) RULE , In Arithmatick & Geometry . AS ALSO A most excellent contrivance of the Line of Numbers , for the measuring of Timber , either Round or Square , being the most easie , speedy , and exact as ever was used . WHEREBY At one setting to the length , all ordinary pieces of Timber , from 1 inch to 100 Foot , is with a glance of the eye resolved , without Pen or Compasses . First drawn by Mr. White , and since much inlarged , and made easie and useful by John Brown. London , Printed in the Year , 1688. The Use of the Line of Numbers on a Sliding-Rule , for the measuring of Superficial or Solid Measures . CHAP. I. A Sliding-Rule is only two Rules , or Rule-pieces fitted together , with a brass socked at each end , that they slip not out of the grove ; and the Line of Numbers thereon , is cut across the moving Joynt on each piece , the same divisions on both sides , only the placing of the Lines differ ; for on one side of the Rule you have 1 set at the beginning , and 10 at the end on each piece ; but on the other side , 1 is set in the middle , and the rest of the figures answerably both ways , on purpose to make it large , and to take in all numbers ; and the reading of this is the very same with the other ; for if you pull out the Rule , and set 10 at the end , right against 1 at the beginning , then on both pieces you have the former Line of Numbers complealy ; therefore I shall say nothing as to description , or reading of it , but come streight to the use . On the edges of the Rule is usually set Foot-measure , being the Foot or 12 inches parted into 100 parts ; and on the flat sides next to the foot-measure , inches in 8 parts , and on the other flat edges on the other side , the Line of Board-measure , and sometimes Timber-measure , whose use is shewed in the first Chapter of the Book ; but note , if the Rule be a just foot when it is shut , as Glasiers commonly is , then the inches are set alike on both sides , and the foot-measure alike on both edges ; and being pulled out as far as the brasses will suffer , it wants about one inch of 2 foot just when pulled out , as it is made for Carpenters use , then the inches on one side , and Foot-measure on the same reciprocal edge , must be figured otherwise , as 13. 14. 15. 16. 17. &c. to 25 inches , and the Foot-measure with 110 , near the end 120 , 130 , 140 , 150 , &c. with 210 at the very end , shewing the measure from end to ▪ end , being drawn out to any distance ; as is very easie to conceive of , and need no example to illustrate it withal . Note also in using the Line of Numbers , that that side or part of it , on which you find the first part or term in the question , shall always call the first side , and then the other must needs be the second , that the Rules and Examples may be shortned and made easie . 1. Multiplication by the Sliding-rule . Set the 1 on any side ( which being found , I call the first side ) to the Multiplicator on the other ( or second ) side ; then seek the Multiplicand on the first side where 1 was ▪ and right against it on the second , is the Product required . Examp. If I would multiply 25 by 28 , set 1 on the first side to 25 on the second , then just against 28 on the first side on the second is 700 : for the right naming the last figure , and the true number of figures , you have a Rule in the 2d . Chapter , and 2d . Problem of the Carpenters-Rule , as in pag. 28. 2. Division by the Rule . Set the divisor found always on the first side , to 1 on the second side then right against the Dividend found out on the first side , on the second is the Quotient required . Examp. If I divide 156 by 12 , the Quotient is 13 ; note , to find how many figures shall be in the Quotient , do thus , if the two first figures of the Divisor be greater than the two first figures of the Dividend , then the quotient hath so many places or figures as there is more in the Dividend than in the Divisor ; but if it be less , that is to say , the Dividends two first figures greater than the Divisors , then the quotient shall have one place or figure more : then the Dividend exceeds the Divisor . Examp. 2964 divided by 39 , makes a quotient 76 , of 2 figures , but if you divide the same number by 18 , you shall have the figures in the quotient , viz. 164 , and 12 remaining , or by the Rule , 2 third parts of one more , for the reason above said , the two first figures of the dividend being greater than the Divisor , it must have one place more than the difference of the number of figures in the Multiplicator and Multiplicand . 3. The Rule of 3 Direct . Set the first term of the question sought out on the first side , to the second term of the question on the second ( or other ) side : then right against the 3d. term found out on the 1st . side , on the 2d . side is the 4th . proportional term required . Examp. If 2 yards of Cl●th cost 8 s. what cost 11 yards● ? The answer is 46 s. for if you s●t 2 on any one side to 8 on the other , th●n look for 11½ on the first side where 2 was , and right against it on the second you shall find 46 , the number required . Note , that all your fractions on the Line of Numbers are decimal fractions , and to work them , you must reduce your proper fractions to them , which for ordinary fractions , you may do it by inches and foot-measure ; but this general Rule by the Numbers will reduce any kind whatsoever , as thus ; suppose I would have the decimal fraction of 9 foot , 7 inch . ¾ , first note that 9 are integers , for the rest , say thus , as 48 the number of quarters in ( 12 inches or ) 1 foot is 1000 , so is 31 the number of quarters in 7 inches 3 quarters to 645 the decimal fraction required , for 9645 is equal to 9 foot 7 inches ¾ ; & so for any other whatsoever . 4. To work the Rule of 3 Reverse . Set the first term sought out on the first side , to the second , being of the same denomination on the 2d . line or side , then seek the third term on the 2d . side , and on the 1st . you shall have the answer requir'd . Ex. If 48 men perform a piece of work in 24 hours , how many men may there be to do the like in 4 hours ? Set 24 on the first side , to 4 on the second , then right against 48 found out on the second , on the first is 288 , the number of men required ▪ 5. To work the double Rule of 3 direct . To perform this , you must have two workings ; as thus , for an Examp. If the increase of 3 bushels of Wheat in one year , be 36 bushels , what shall the increase of 8 bushels be for 7 years ? First , set 3 on the first side , to 36 on the second , then against 8 on the first , on the second you find 96 , then set ●on the first side to 96 , then against 7 on the first side on the second you have 612 , the increase in 7 years , the answer required . CHAP. II. To measure Board or Glass by the Sliding ▪ rule , the length and breadth being given . PROB. I. The breadth given , to find how much makes a foot . If the breadth be given in inches then set 12 on the first side , to the inches on the second ; then right against 12 on the second , on the first is the number of inches required . Examp. At 6 inches broad set 12 to 6 , then against 12 on the second on the first you have 24 But if it be given in foot-measure , then instead of 12 use 1 , and do in like manner as before . Examp. At 0 ▪ 50 broad , set 1 to 0.50 , then right against the other 1 is 2.00 the answer required . But to find how much is in a foot long at any breadth , do thus : First , for foot-measure , just as the Rule stands even look for the breadth on one side , and the quantity in a foot is on the other side ▪ but for inches set 1 to 12 , the● right against the inches broad i● the feet and tenths in a foot long . Ex. At 6 inches broad is 50 o● half a foot in a foot long . Again a● 30 inches broad is 2 foot and a ha●● in a foot long . PROB. II. The length and breadth given , ●● find the content . First , the breadth given in inch●● and the length in feet and inches set 12 on the 1st . side to the breadt● on the second , then right again●● the length on the first , on the s●cond is the content required . Ex. At 16 inches broad and ●● foot long : Set 12 to 16 , then right against 20 you have 26 foot 7 10th . look for the 7 10th . on the foot-measure , and right against it on the inches you have 8 inches and ¼ and ½ quarter , the answer desired . But if the breadth be given in foot-measure , then set 1 to the breadth ; then right against the length on the first side , on the second you shall have the content required . Ex. At 1.20 broad , 20.00 foot long , you shall find 24 foot . For if you set 1 to 1.20 , then right against 20 foot , you have on the 2d . 24 as before . PROB. III. The breadth given in feet and inches and the length also in the same parts , to find the content . Set 1 on the first side to the feet and inches brought to a decimal fraction , or as near as you can guess ( for 6 inches is half , 3 inches is 1 quarter , 9 inches is 3 quarters , 4 inches is one third , 8 inches is two thirds , and 1 inch is somewhat less than 1 tenth on Rule ) on the other or second side ; then right against the length found on the first , on the second is the content required . Ex. At 3 foot 3 inches broad , and 9 foot 9 inches long , you shall have 31 foot 8 inches ½ near , the very same is for foot-measure ( only much easier ) because the divisions on the Line of Numbers , and on the Line of foot-measure on the edge , do agree together . This being premised as to the using of it , you may apply all the former precepts and examples to this Rule as well as the other . CHAP. III. To measure Timber by the Sliding-Rule . PROB. I. To measure Timber by this Rule , is nothing else but to work the Double Rule of Three . Examp. At 8 inches square & 20 foot long , I would know the content . Set 12 , if the side of the square be given in inches ( or 1 , if in foot me●sure ) on the first side , to 8 the inches square on the second : then right against 12 on the second side on the first is 18 , the fourth proportional part : then for the second work , set 18 , the fourth proportional last found to 8 the inches square on the second , then right against 20 the length , is 9 , the content required . Or rather thus ; Set 12 against 8 , then right against 20 on the same side 12 was , is 13. 5 near on , then look for 13. 5 fere on the first side , and right against it on the second is 9 foot the content required . PROB. II. To measure a piece that is not square Set 12 if you use the inches ( or 1 if you use foot-measure ) on the first side , to the inches thick on the second ; then right against the inches broad on the first side , on the second is a fourth proportional : then in the second operation , set 2 on the first side to the fourth proportional on the second , then right against the length on the first side , on the second is the content required . Ex. At 8 inches thick and 16 broad , and 20 foot long , you shall find 18 foot fere . PROB. III. The square given , to find how much makes a foot . Set the inches square on the first side , to 12 on the second ; then right against 12 on the first , on the second is a fourth proportional number : then in the second work , as the inches square to the fourth proportional , so is 12 to the number of inches required , to make a foot of Timber . Ex. At 6 inches square , set 6 to 12 , then against the other 12 is 24 : then set 6 to 24 , then right against 12 you shall have 48 , the length in inches required . After the same manner are other questions wrought , but the Compasses are easier and more ready ; therefore I shall say no more to this but only refer you to the former Rules in the third , fourth and fifth Chapters . Only note , that in those Sliding-rules made for Glasiers use , the one half of the Line of Numbers is on one side of the Rule , and the other on the other side ; and whatsoever leg or piece of the rule is the first on the one side , the same leg or piece is the first , when the Rule is turned , on the other side , which must well be observed ; but note , that for measuring of Timber , those that use it may have one side fitted for that , as I shall more plainly and fully shew in the next chapter , being the easiest , speediest and nearest way that ever yet was used by any man , resolving any Contents by having the length and the diameter , the circumference or square given . A Table of the true size of Glasiers Quarries , both long and square , calculated by J. B. Square-Quarries 77. 19 gr . Quarries . Rang. Sides bread . leng . content in feet content in inc 8 in 100 i. 100 I. p i. pts . F. p ●ts Inc. p. 8 4 20 - 4 30 5 36 6 70 0. 1250 1. 50 10 3 76 3 84 4 80 6 00 0. 1000 1. 20 12 3 43 3 51 4 38 5 47 0. 0833 1. 00 15 3 07 3 13 3 92 - 4 90 0. 0667 0. 80 18 2 80 2 86 3 57 4 47 0. 0555 0. 666 20 2 66 2 72 3 39 4 24 0 , 5000 0. 60 Long Quarries 67. 22 Quarries Rang. Sides bread . leng . content content   in pts . I. pts . ● . 100 ● pts . F. 100 ● . pts . 8 4 09 4 41. 4. 90 7. 34 0. 1250 1. 50 10 3 65 3. 95 4. 38 - 6. 57 0. 1000 1. 20 12 3 34 3 61 4. 00 6. 00 0. 0833 1. 00 15 2 98. 3 23 3. 58 5. 37 0. 0667 0. 80 18 2 58 2 79 ▪ 3. 10 4. 90 0. 0555 0. 666 20 2 72 - 2 94 - 3. 27 4. 65 0. 0500 0. 60 Note , that a prick after the 100 parts of an inch , notes a quarter , and a stroke ( - ) a half of 100 parts of an inch ; to make this Table , work thus by the Line of Numbers . Divide the distance between the content of some known size , as square 10s . or long 12s . and the content of the inquired size , into two equal parts , for that distance laid the right way ( increasing for a bigger , or decreasing for a less ) from the sides of the known size , shall give the reciprocal sides of the inquired size . Example for square 12s . The half distance on the Line of Numbers , between 1000 the content of square 10s . and 0.833 the content of square 12s . shall reach from 6 the length of square 10s . to 5.47 the length of square 12s . and from 4. 80 the breadth of square 10s . to 4 38 - the breadth of square 12s . and from 3 84 - to 3. 51 , and from 3 76 to 3 43 ; and so for all the rest . CHAP. IV. The description of the Line of Numbers on a Sliding-rule , to measure Solid measure only , according to Mr. White 's first contrivance , but much augmented by J. B. First , when the figures on the Timber side ) stand right towards you , fit to read , then that half or piece next to your right hand , I call the right side , the other is of necessity the left . Secondly , the figures on the right side are , first at the lower end , ( where the brass is pin'd fast ) either 3 , or 4 , or 5 , it matters not much which , yet to have 3 there is best ; then upwards , 4. 5. 6. 7. 8. 9. 10. 11. for so many inches then 1. 2 3. 4. 5. 6. 7. 8 9. 10. 11. 12. under the brass at the top , for so many feet , the divisions between to 1 foot , are quarters of inches , the next above 1 foot , are only whole inches , as you may plainly see . Thirdly , at 1 foot you have the word square , at 1 foot , 1 inch ½ is a mark , and right against it is set TD , noting the true diameter of a ●ound Cylinder ; at an inch further ●s 12 set , which I call small 12 , being in small figures . Again at 1 foot 3 inches better , is another mark , and right against it the word diameter , for the diameter of a piece of timber according to the usual english allowance . Then again at 3 foot 6 inches ½ near , is TR , for the true circumference of a round cillander . Lastly , at 4 foot is the word round , noting the circumference according to the usual allowance , whose use followeth . Also at 13 foot 7 inches is TD , and at 3 inches and ½ is TR. Note also , if you put on the Gage-points for Ale or Wine , with the mean diameter , and length , you may gage any Wine or Beer-vessel , the Wine at 17 inches 15. the Ale or Beer at 18. 95. Also the Gage-points for a Beer-barrel at 35 inches and 9525 parts ; and the gagepoint for a barrel of Ale at 33 inches and 89645 parts . Fourthly , the figures on the left side are not much unlike the right , for 1 at the beginning is one inch , and so it proceeds by quarters of inches to 1 foot ; then by figures at the feet , and the divisions all whole inches to 10 foot , then every whole foot , and half , and quarter , or 10th . to 100 , or 140 , or 150 foot ; and this I call the left side , the other the right side ; so that from 1 inch at the lower end to one foot , every inch hath a figure ; from 1 foot to 10 foot , every foot hath a figure , and from 10 foot to 100 , every 10th . foot only is figured . I have been very plain in explaining this , because I would avoid vain repetitions in the following uses , wherein you shall have first the most ordinary and easie questions , and then the more hard and critical , and less useful . The Uses follow . PROB. I. A piece of Timber being not square , to make it square , or to find the Square-equal . Set the breadth on the left side , to the breadth on the right , then right against the inch and quarters thick found on the left side , on the right is the inches square required . Examp. At 18 broad and 6 thick you shall find 10 inches ⅜ , the side of the square required . For if you set 6 inches against 6 inches , on the right and left side : then right against 18 inches , or 1 foot 6 inches on the left , on the right you have 10 inches 1 quarter , and half a quarter ; for the side of the square-equal to 18 one way , and 6 the other way . PROB. II. The side of the square given , to find how much makes a foot . For all pieces between 3 or 4 inches and 42 inches square , which are the most useful : this the best way , set the inches or feet , and inches square , found out on the right side , to one foot on the left , then right against 1 foot on the right , on the left is the inches or feet , and inches required to make a foot of Timber . But when the piece is small , count 1 foot on the right for 1 inch , and call 12 on the right for 1 foot . Example . At 8 inches square set 8 on the right , to 1 foot on the left , then right against 1 foot on the right , on the left is 2 foot 3 inches , the length required . To find how much is in a foot long . Just as the Rule stands even look for the inches the piece is square on the right , and on the left is the inches or feet , and inches required . Example . At 17 inches square , there is 2 foot of timber in 1 foot long , which if you multiply by the length , you shall have the true content . A very good way for large pieces , and very exact . PROB. III. The side of the square , and length given , to find the content . For all pieces between 1 inch or ●● part of a foot , and 100 foot , this is the easiest way . Set the word square or 1 foot to the length on the left , then right against the inches or feet and inches square on the right , on the left you have the content . Examp. At 9 inches square and 20 foot long ; set the ( long stroke by the ) word square , to 20 foot on the left , then right against 9 inches on the right side , on the left side you have 11 foot and a quarter , the content required . But if it be a very great piece , as above 100 foot , then call 1 foot on the left side 10 foot , and 2 foot 20 , &c. then 10 shall be 100 , and 100 a 1000 , that will supply to 1500 foot in a piece . But for all small pieces under 3 inches square , and above 1 quarter of an inch , do thus : Set 12 on the top ( or the small 12 when it is most convenient to use ) to the length on the left side , then right against the inches ( or 12s . of 1 inch ) squares found on the right side , on the left is the true content required . Example . At 2 inches ( 3 twelves or ) 1 quarter square , and 10 foot long , you shall find 4 inches and a quarter ferè . But note , when you use the small 12 , the answer is given in decimals of a foot , therefore the top 12 is best . ROB. IV. The square of a small piecè of Timber given , to find how much makes a Foot. For all pieces from 12 inches to 1 inch square , do thus ; set the inches and ( 12s . or ) quarters square , counting 1 foot on the right side for 1 inch , and 2 foot for 2 inches , &c. found out on the right side to 100 on the left ; then right against the upper , or small 12 on the right , on the left is the length required to make a foot of Timber . Examp. At 2 inches ¼ square , you must have 28 foot 4 inches , to make a foot . PROB. V. Under 1 inch square , to find the length of a foot . Set 1 foot 9 inches , 6 inches o● 3 inches , found on the right side , for 1 inch ¼ , ½ , or ¼ of an inch , against 10 on the left side , counted for 100 : then right against the small 12 you have the feet in length required . Examp. At 1 inch square you find 144 feet , at ½ square 256 feet , at ½ an inch square 576 feet , at ¼ or an inch square , you find 2034 feet in length , to make 1 foot of Timber . Or , if you set the former numbers 12 , 9 , 6 , 1 , against 1 inch on the left , then right against the upper 12 is a number , which multiplied by 12 , is the number of feet required . PROB. VI. A great piece above 3 foot ¼ square , to find the length of a foot . Set the feet and inches on th● right , to 100 on the left ; the● right against small 12 is the inche● and 12s . or 12s . of a 12th . tha● goes to make a foot . Examp. At 4 foot square , yo● have 9. 12ths . or ¾ of an inch t● make a foot of Timber ; at 5 foo● square , 5. 12ths . and 10. 12ths . ●● a 12th . to make a foot . Thus you see the Rule as no● contrived , resolves from 1 quarte●● square , to 12 foot square , the content or quantity of a foot of Timber in length at any squarenes● without Pen or Compasses . CHAP. V. For round Timber . PROB. I. The number of inches that a piece ●● Timber is about , being given , find how much makes a foot . First , for all ordinary pieces , s● 1 foot . on the left , to the inches ●● feet , and inches above on the righ● then right against TR for true measure , or round for the usual measure , is the feet , or feet and inches required to make a foot of Timber at that circumference about . Examp. At 4 inches about , 113 foot 2 inches is for true measure , but for the usual measure , 142 foot goes to make a foot of Timber . At 12 foot 3 inches about , 1 inch is a true foot , but for the usual allowance , as the fourth part of a fine girt about gives : it must be 1 inch ¼ long , to make a foot of Timber at that circumference . But for very large pieces , count 1 foot on the right for 12 foot , 2. 24 , &c. and set 1 foot on the left , as before ; then in the answer , 1 foot on the left is 1. 12th . of an inch , and 1 inch 1. 14th . of an inch . Example . At 144 foot about , 1. 144th . part of an inch , is a foot of Timber . PROB. II. For very small wood , to find a foot in length . But for very small pieces of under 4 inches about , set 1 foot , 2 foot , &c. on the right ( counted for 1 inch , 2 inches , 3 inches , or 4 inches ) to 1 foot on the left , then right against TR or round , you have a number , which multiplied by 12 , is the number of feet required . Example . At 1 inch round true measure is 151 foot ferè , but for the usual allowance 196 , which numbers multiplied by 12 , is the number of feet required , viz. 1809 , and 2352. But note , you must read the 196 , and 151 right , as thus ; 1 foot on the left is 12 , 2 is 24 , &c. so that 12 foot is 144 , and our number by the same account is 151 near . To find how much is in a foot in length , set round or TR to 1 foot on the left , then right against the inches , or feet and inches about , found on the right , on the left is the answer required . PROB. III. The inches , or feet and inches about and length given , to find the content . Set the word round , or TR for the usual or true measure , to the length on the left ; then right against the inches about on the right , on the left is the content required . Example . At 2 foot 3 inches about , and 20 foot long , it is 6 foot 2 inches of the usual allowance , or 8 foot of true measure . But if it be a great Tree , then set TR , or round to 1 called 10 , or to 10 called 100 , then is the content augmented to 1000 foot , as you did in the Rules for square Timber . But if you would have it measure bigger still , then set the 4 inches or a TR set close by the brass on the right side , to the length on the left , either as it is , or augmented , counting at last according ; ( the note 1 foot on the right is 12 foot , and 12 at the top is 144 foot ) then right against the feet about on the right , on the left is the content required . Examp. A Brewers Tun 3 foot long or deep , and 72 foot about , set the TR by the brass to 36 inch . ( which is thus counted ) on the left side ; ( 1 inch is 10 inches , 2 is 20 , 3 is 30 , 3½ is 35 , somewhat more is 36 ; so then 1 foot is 120 inches , o● 10 foot ) then right against 6 times 12 foot on the right ( which is at ● foot ) on the left you have 12 30 foot , as near as the Rule will give it , which counting 6 foot to a barrel , is 205 barrels , the content required . PROB. IV. To find the content of a very smal● piece . Set the word round , or TR to the length on the left , as in the third Problem of this Chapter ; the● right against the inches about o● the right , ( calling 1 foot 1 inch and 6 inches ½ an inch ) on the lef● is the 13s . of 1 inch ) or 12s . of a 12th . required . Example . At half an inch about , and 10 foot long , it is 2 12s . and a half of 1 12th . of an inch , or 2 square inches and ½ true measure . Again , 2 inches ¼ about , and 10 foot long , is half an inch of true measure , 12 inches to a foot solid , or ½ a foot superficial of one inch thick . CHAP. VI. To measure Timber , having the Diameter and the length given . PROB. I. The Diameter given in inches , to find the length of a foot . Set 1 foot on the left to the inches diameter on the right : then right against TD for true diameter or the word diameter for the usual allowance ( of a string girt about and doubled 4 times for the side of the square ) you have the feet and inches required . Example . At 10 inches diameter , 1 foot 10 inches makes a foot . But for very great pieces , set 1 foot as before , but look for TD beyond the upper 12 , & right against it on the left you have the 12 of 1 inch , or the 12s . of a 12 , that makes a foot . But for very small sticks , set 1 , 2 , or 3 foot on the right ( for 1 , 2 , or 3 inches ) to 1 foot on the left , then right against TD or Diameter , you have a number , which multiplied by 12 , is the number of feet required to make a foot of Timber . Examp. At 1 inch Diameter you shall have 15 foot 3 inches and better , which multiplied by 12 is 123 foot 3 inches . Note , that 1 on the left is reckoned 10 foot , and 2. 20 foot , as before in the same Rule , for the circumference , and then note 1 inch is 10 inches . At any diameter , to find how much is in 1 foot long , do thus ; set Diam . or TD to 1 foot , then just against the inches , or feet and inches diameter found on the right , on the left is the answer . Example . At 2 foot diameter is 3 foot 2 inches in 1 foot of length , which multiplied by the length , gives the true content of any round piece , and very exactly . PROB. II. The diameter and length given , to find the true content . For all ordinary pieces , set the word diameter for the usual measure , or TD for true measure , always to the length on the left ; then right against the inches or feet and inches diameter on the right , on the left is the content required . Examp. At 5 inches diameter , and 30 foot long , you shall find 4 foot and ½ an inch true measure for the content required . But for very small pieces set TD or Diam . to the length , as before ; then counting 1 foot on the right for 1 inch , and 6 inches for ⅓ an inch ; on the left you shall have the answer or content required . But note , as the right side is diminished , so is the left , for 1 foot on the left is a 12th . of an inch of Timber , whereof 12 makes a foot , or 1 long inch , a foot long , and 1 inch square , and every ineh on the left is 1 square inch ; thus , at 2 foot long , and half an inch diameter , it is 4 □ inches , ¾ in content . But for a great piece under 1000 foot , set TD or diameter to 1 , 2 , or 3 foot , called 10 , 20 , or 30 foot : then right against the feet and inches diameter , you have a content augmented accordingly , as at 30 foot long , and 7 foot diameter , you have 1140 foot , for the true content using TD . Note , that in large Taper-timber , whether square or round , when i● is measured by the usual way ; that is , by the middle square or girt , or the 2 squares or girts put together , and the half counted for the equal square , or girt ; I say , a square of half the difference of the squares or girts , and one part of the length is to be added to the former measure , as is proved in the circles of proportion , pag. 50. As thus for Example . Suppose a Taper piece be at one end 16 inches square , at the other 30 inches square , and 30 foot long , the square in the middle is like to be 23 inches , the content then is 110 foot ; now half the difference of the two ends square is 7 inches , and 1 third part of the length is 10 foot ; a piece 7 inches square , and 10 foot long , is 3 foot 5 inches which added , is 113 foot 5 inches , the true content of that taper piece abovesaid . The general way of Gauging by this Rule , is thus . Set the W. or the A. for Wine or Ale-measure , always to the length of the Vessel found out on the left . Then right against the mean Diameter found out on the right side , on the left is the answer required . Examp. At 30 inches Diameter and 36 long , you shall find about 90 gallons and a half Ale-measure The Gage-point for a Beer-barrel i● near 3 foot , and the Ale-barre● near 34 inches , which use thus . Set 3 to the depth of the Tun then right against the mean Diam is the content in Barrels . Example . Set 3 to 36 inches , and then right against 5 foot Diam . is 10 barrel of Beer-measure , a very good an● spedy way . FINIS . Notes, typically marginal, from the original text Notes for div A29760-e29120 3. 3. 5. 2. 3. 4.