length of that trapezium also add your two breadths 60 and 7 together make 67 for every middle breadth of each station-line must be twice added save where you have two severall breadths fall in one place as in the line CD where you have the length 200. twice together the half of 67 is 33½ by which multiply 472 facit 15742 to be set against the latter breadth 7. Then go to the second line BC where the first length is 100 the common breadth 3½ gives 350 and so go on according as the example gives then if you add all those primes or square links into one summe you shall finde it to be 40346 that keep till you have cast up the triangles within the station lines and likewise all the other slabs Therefore I draw a diagonall from A to C which will be the base to both triangles and half the length is 504. the perpendicular falling from B is 514 that from D is 494 the summe of both is 1008. then these multiplied the summe of both perpendiculars by half the base or the whole base by half of them it gives 508032 which added to the summe of the borders 40346 it makes that first close to give 548378 square links in all Now to bring these links into acres you need but onely cut off the five right hand figures the rest to the left hand are acres viz. five acres the reason is there are 25 links in the length of a pole that squared gives 625 square links in a pole and that multiplied by 160 the poles in an acre gives 100000 links by which divide your summe of your links or for the five cyphers cut off five places the rest are acres and the five so cut off are the numerator of a fraction of an acre whose denominator is â—00000 So 548378 gives five acres Now to bring these five figures into poles you may either divide them by 625 the primes in a pole or else multiply those two of the five next the left-hand always by six and set them a place nearer the right-hand and then add those two which you multiplied and the two which are under them together and increasing them so many unites as are sixes in the next two and you shall have 7 pole and 253 links If now that when you have cast up a close you have more then half 625 primes remaining ordinarily it is accounted for a pole if lesse then for nothing But if you have more closes adjoyning you may reckon it with the next close Suppose your ground hath the out-side of this form whose station-line is AD you may set it down in words thus in your note-book At A it is 10 to the brook from the station-line 0 at B where I have gone 20 pole in the station-line there is a square line to a crook stroke with the edge of the table in which at 15 on the left hand is 20 at 28 is 25 on the left hand and 15 on the right hand at 44 is 28 on the right hand at 56 is 33 on the right hand at 70 is 0. on the left and 30 on the right hand then at 30 in the station-line is 10 at which 30 also I strike a station-line forward which when I have stroke it I finde the fore-most acute angle by my scale of chords to be 70 degrees that also I enter in my book by help whereof and a diagonall line from angle to angle I can draw the plot of any ground though many years after without going to it again And after the same manner you may plot and set down single lands in the common-field or a close that is narrow and long CHAP. VI. Of measuring a Wood. THe difference of measuring a wood and pasture is in these two things First in pasture you measure on the in-side but woods on the out-side Secondly in pasture all your trapezia are to be added to that within the station-lines unlesse your station-line be in the close adjoyning but in this to be subtracted CHAP. VII Of dividing or laying out of ground OF this there are three degrees each more difficult then other The first is when the length of a ground is given and a given quantity desired as if you would lay out two acres of grass in a pasture which is 36 pole long and you desire the breath First I turn my two acres into square links it is 200000 which I divide by 900. for 25 times 36 is 900 it gives 224¼ the which if you divide by 25 the links in a pole it gives 8 pole 22¼ links in breadth and this needs no plotting Or if you would do by the foot-chain say two acres is 320 pole that divided by your length 36 gives 8 pole and â—2 36 which abbreviated is 8 9 and to know how many half-feet that is because there are 33 half-feet in a pole therefore I multiply 33 by 8 facit 264 that divide by 9 gives 29 half feet and 3 9 or â…“ that is 8 pole 14 feet 8 inches Secondly In pasture-ground suppose a pasture with crooked hedges is equally to be divided between two men First I plot it and find it 52 acres 2 roods 10 pole that is 26 acres 1 rood 5 pole a peice I ghuess as near as I can to strike a line over the middle of my ploâ— but measuring one end upon the plot I finde it wants 264 pole of his due therefore I measure the length of the dividing line which I finde to be 56 poles Now to work by the decimal chain I multiply 264 my poles wanting by 625 the square links in a pole they make 165000 likewise I multiply 56 pole the length by 25 the links in a poles length they make 1400 by which divide 165000 it quotes 117 6 7 that is 4 poles 17 6 7 links But by the foot-chain if you divide 264 by 56 it quotes 4 poles and 40 56 which to bring into half-feet multiply the numerator 40 by 33 the ½ feet in a pole facit 1320 which divide by 56 it gives 28 half-feet and 16 56 of a half-half-foot in toto 4 pole 14 feet 2 inches almost And so much must you remove your dividing line at both ends and this may be done as well on the out-side as on the in-side Thirdly To divide a standing wood of 200 or 300 acres and to drive a streight line from a mark on one side thereof to any mark on the other though the wood be twenty years growth and a hill in the midst A rare secret Be sure to plot and measure enough or more then you desire to take out of it and where you intend your dividing-line shall come there in your station-line on the first side set a mark keeping also good marks at every station so going on till you be sure you are far enough on the other side also Then draw your dividing-line by ghuess keeping one end thereof still upon the mark in your station-line then
be shown with indico and azure or black-lead for seas a greenish sky-colour of indico azure smalts white-lead and verdigrease CHAP. XVI To measure all manner of ground by the Pandoron or any other graduated Instrument THe Pandoron is an Instrument compounded of First an ordinary foot with three legs for a plain Table Secondly a Table and folding-rulers like it save that it is a true square Thirdly the box and needle Fourthly it hath on one corner a centre in which is a screw-pin on which a moveable ruler with sights turneth Fifthly in the two out-sides furthest from the centre is drawn the Quadrate for terrestrial altitudes and distances Sixthly next to it is the limbe of the Quadrant both for celestial and terrestrial altitudes and distances whether upright flat or aslope Seventhly Gunther's Quadrant for your own latitude for houres both of night and day and Azumeths and divers other problems Eighthly Fale's Quadrant for Planetary houres Ninthly a circle and scale for finding the declination of a plain Tenthly a neck of 14 or 15 inches long to put on the top of the staff the Table being taken off with a pin on the side to hang the Table on to take all manner of altitudes and distances aslope Eleventhly a beam of 6 or 7 foot long about two inches square of deal and a trough on the top gouged all along half an inch deep to fill with water for a water-level having a sight at each end having a lath crossing the beam in the middle above and below 6 foot long fastened with screw-pins and brackets above and below with an hole in the bottom of the middle of the beam in stead of a socket to stand on top of the three-foot staff So that there is nothing that all or any observing Instruments can do but this doth it By this you measure land as by the plain-Table then if the weather be moist or in hilly ground you may uncover the Table and work by the Quadrant whereby you may save the charge of hill-ground sights which are as costly as all the rest of the Instruments Besides which if you know how to work by the Quadrant you cannot be ignorant of working by the Theodelete or semicircles the difference being onely this that they take onely at once which if it be above 90 degrees by the Quadrant you first take some part of it and then the rest of it afterward yet all at the same station and then plot it by your scale of chords Indeed by the Circumferentor you take all the angles by observing the cutting of the South-end of the needle and then either plot the angles by a protractor and the lines by a scale of equal parts or else you may plot the angles either by your scale of chords or by the Circumferentor it self both which I hold better ways then the first So that there being nothing desirable in an observing instrument but this giveth it it so pleased Mr. Hender Roberts the Lord Roberts youngest son a Gentleman every way fitted with a genius for the Mathematicks whom I cannot name without honour who had the first of them to give it the name of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 omne donum So that in shewing the use of it as it is a Quadrant we shall with the same labour shew the use of all graduated Instruments in measuring of land and as for working by it as by the plain-Table we refer you to the ten first chapters of this book Now therefore for working by the Quadrant yet herein we will speak of nothing but what is within the station-lines contenting our selves for the rest with that which hath been spoken before in the use of the plain-Table all the difference consists in three things first the taking of the angles secondly in keeping the field-book thirdly in plotting Now for your plotting it first draw the line AB set out 20 of your scale of equal parts upon it then take always 60 off your scale of chords set one foot at the end of your 20 in B and with the other foot tran always from the last line which here is AB towards the place where you think your next line BC will fall Then take your angle B which is 60 and set it in the said tran from the line AB forward there make a prick and from B through that prick draw the line BC ad infinitum In which line set out â—8 of equal parts there make a prick for your station C. Then take again your 60 of chords set one foot in C and tran from the last line BC toward CD Now because your angle C is more then 90 and that your compass tran at 60 therefore first set out that 60 in the said tran to B and because there wants yet 46 of 106 therefore take those 46 with your compasses and set them on forward from 60 there make a prick and draw your line CD through it and so of the rest So that there are but these things first draw a station-line secondly tran your angle with 60 of chords thirdly prick out the degrees of that angle CHAP. XVII In measuring by graduated Instruments to know if your plot will shut or no. Because in working by graduated Instruments you always plot at home but never in the field and that if any thing be mistaken in the field as oft it comes to pass to be so then will not your plot shut at home therefore either you must look to your needle at every plantation or else you must measure all the angles which by the plain-Table you need not do therefore with such Instruments the needle is more needfull then with the plain-Table and yet the Circumferentor will hardly help you herein neither though you work all by the needle unless you work by taking angles by it which is the slower way Now having measured all the angles if on the inside of a ground because all the three angles of a right line triangle are equall to two right angles or 180 degrees and that there are so many triangles save two as are angles therefore if you reckon so many angles save two for each of them 180 and finde that and the quantities of all your angles to agree there is great hope your plot will shut else not As if there be a triangle they must all make 180 if a quadrangle 360 if a pentangle 540 an hexangle 720 a septangle 800 an octangle 950 but if you measure on the out side as a wood then every outward angle is the complement to 360 of its inner angle therefore to take all those complements is your best way both to prove and plot it by and less labour if you are far from your mark and not to go to it again it oft-times will quit your pains lest you are forced to spend perhaps an whole days-work about that you have done or at least would have done already to prove your angles after this manner CHAP. XVIII
come near to the truth and may indifferently serve in case of letting because it always is a little under the length as will easily appear in this diagram If an hill run streight along a ground if by one side it will be a mere declining level if through the middle it will be two declining levels and that line so running along the top will be a line of level and equal to the line of level under it therefore if you add both ends together as you measured them by the chain and multiply half of them by the length of that line you have the content if it be of equal height at both ends But if it be unequal at both ends though it be a declining level and have more then three angles your best way is to part it in severall triangles whose Hypotenuses and perpendiculars you may finde by either of the two former ways without measuring them by the chain Thirdly If you have no Quadrant nor plain-Table at all save onely the chain and any board of a foot or 14 inches long with one streight edge of ten or eleven inches broad draw a streight line close and parallel to that side and near one end thereof stick a pin in the line with thread and plummet hanging on it then if you are at the bottom of the hill and look upwards turn that end with the plummet from you but if you are at the top turn it towards you and as you espie the mark let a stander by on that side the plummet is on lay his hand gently on the bottom of the board and with his thumb press down the thread there holding it till you have made a prick right under it in a good large tran first drawn with 60 of some large scale of chords whose center shall be the hole where the pin sticketh then take with your compasses the distance between the said prick in the said tran and the beginning of the said tran and apply it to the same scale of chords you drew the tran by it gives the complement of the angle ascending viz. the dâ—grees of the angle descending But if you are at the top and look downward it gives the complement of the top-angle and degrees of the bottom ascending But if you will but erect a perpendicular upon the same center and take the distance between the prick and it it gives the contrary CHAP. X. Of reducing a plot from a greater to a lesser ALthough there are several ways of performing this as likewise of a lesser to a greater whereof there is great use in turning statute-measure into the eighteen-foot pole c. we will lay down onely this one generall rule Or Secondly If you desire a plot equall to another you may oyle a paper drie it well then put it over the other plot that it stir not through which you may see the lines on the neather plot then draw them with your pen on the oyled paper then take it off to prick it then pounch a new paper draw it Or Thirdly Having drawn a line representing AB in your new plot take the line AB off the old either all or ½ or according to your desired proportion set it on the new Also take the proportion of the line AE and set one foot in A and tran where you think E will fall in your new Take also the like proportion of the distance of BE and set in the said tran and so you have E the same 2 distances will set out D also D and B will set out C and so you have all your angles then draw their lines and you have your plot desired CHAP. XI Of measuring pasture-ground by the chain onely and that as speedily and exactly as with any instrument whatsoever and with less help though in misty weather to plot shut and prove the plot thereby also ABout the midst of one of your longest station-lines and some known length in the same as at X in the first or third close chap 3d pag. 2â— set up a mark and mark it in your book both with its proper length letter then having measured round about the ground on the inside or at least all but the last side if you have more then three angles in stead of measuring it from angle to angle viz in the first close from A to C or from B to D you shall measure from C to X and from X to D so making a triangle the more then otherwise which two subtendeâ—ts will easilie be run whilest you can set up the Table once so you shall need less help by one to carry your Table for that is wholly one bodies work and these two subtendents must be set down at the latter end of your notes of that close in your field-book Then if you measure the last side AD having plotted the rest if that AD on the ground and AD on the plot agree all is right neither ever need you divide any more lines then one in the whole ground or close throughout so that at least none of the station lines strike outward for then it must be accounted as another close so much of it till the last line that strook inward being continued streight out do meet with the other plot again See more chap. third Now to plot such a ground measured by the chain onely suppose it be the said first close chap third first I draw the line AXB making a mark at X and another at B secondly you must either take the subâ—endent XC setting one foot of the compasses in X tranning where you think C will falâ— or else take the station-line BC with your compasses and set one foot in B tran at C and then take the other of these two last lines viz. XC setting one foot on its proper mark X and with the other make a prick in the said tran and so have you placed C in his right place then draw the line BC next take CD with your compasses set one foot in C and tran where you think D will fall then take the subtendent DX set one foot in X and make a prick in the said tran and that sets out D then draw the line CD and because D is your last station and that A and D are both set out already therefore draw also the line AD now if AD on the plot and AD in your book agree then all is right else not So that in this kinde of plotting there are onely these three positures First draw a station-line secondly tran with a subtendent thirdly prick with the next station line Nevertheless in great larg plots it will be needfull to use a good larg pair of compasses because you must take the whole length of your lines with them In which case a pair of beam-compasses with a beam of deal willow or sallow or some such soft wood is best of all of 17 or 18 inches long with a piece of an
To take terrestrial distances by the plain-Table or Pandoron aâ— by the Table WE have spoken of taking them by the chain onely in chap. 13. between that and this there is very little difference We will here suppose the same oppositions as there viz. two houses beyond a river between which I desire the distance also between each of them and each of my stations the chiefest difference is this that by this your best way is to have your station-line as near the river as you can which let be as before AB 40 pole long First set your â—nstrument at A and turn the sights to DC and B and draw their lines measure thence to B 40 poles there make a prick but lay down your 40 pole with a very small seale if the distances be long so that the 40 pole be little above an inch long Then set up your Instrument at B laying your index on your station-line of your plot turn it till through the sights you espie A then fasten your Table and one end of your ruler turning upon the center B turn the sights first to C then to D then draw lines whose intersections with the former will give you all your distances desired CHAP. XIX To do the like by the Pandoron as it is a Quadrant or by any graduated Instrument LEt the same example be propounded as afore and let your station-line be AB 40 pole as near the river-side as you can I set up the Quadrant first at A where I find BAD 110 degrees and CAD is 50 degrees likewise set up at B then CBA is 104 whereof CBD 50 this station-line 40 and these angles thus plotted extend you lines till they meet and their intersections will give you the desired distances as afore yet if you will bestow the time and pains to cast it up by the doctrine of Triangles you may come somewhat nearer First for the triangle BAD seeing that BAD is 110 degrees and the angle ABD 54 which make being added 164 which take out of 180 rests the angle ADB 16 degrees Now in the same triangle having all the angle and the line AB to finde the side AD. As the sine ADB 16. Comparithmes 055966 is to AB 40. 160206 So sine 110 degrees that is sine 70 997.99 to 136 3 10 BD. 213.61 Also to finde AD As sine ADB 16. Compar 055966 is to AB 40 pole 160206 so sine DBA 54 degrees 990796 to 117 4 10 AD. 206968 Then in the triangle CBA CBA is 104 and BAC is 60 these added together make 164 which taken out of 180 leaves the angle BCA 16 degrees Now to find BC. As sine 16 d. Compar 055966 to AB 40 p. 160206 so is sine CAD 60 993753 to BC 125 and 7 10 209925 Also to finde AC As sine ACB 16. Compar 055966 to AB 40 160206 so sine 104 that is sine 78 998690 CBA to AC 140 8 10 214862 Lastly having the two sides AC 140 8 10 and AD 117 4 10 and the angle CAD 50 in your triangle CAD to finde CD As the summ of the sides 258 1 10. Compar 658804 to the difference of the same sides 23 4 10 236922 so is the tang of ½ the sum of the angles unknown 65 to the tang of ½ their difference 11 degrees 033133 which add to 65 d. ½ facit 76 the greater 928859 angle D. But subtracted from it makes the angle 54 degrees and then as sine 54. Compar 009205 to 117 4 10 206967 so sine 50 degrees 988425 to CD 111 2 10 204597 CHAP. XX. Of altitudes and distances celestial by the Pandoron or Quadrant FOr taking of altitudes and distances celestial or altitudes terrestrial it is a matter of necessity that besides your Quadrant and three-legg'd foot you get also a neck or piece of close-grain'd wood whose Diameter may be about three inches or somewhat more Let the nether end be turned with a socket that instead of the socket of your Table you may put on that so that it may turn on the top of the staff as the socket doth having also a screw-pin in the side of it to hold it at any situation Also about two or three inches below the top turn it like a bowl in the midst whereof bore an hole with an inch-wimble to which fit a pin of the same wood so hard both driven in and glewed in that it stirs not but let one end thereof be so big and so long as to fit the brass socket that the socket may turn very stiff about it and let the little end of the pin reach past the hole of the bowl almost the depth of the socket and then you may fit that end of the pin either to that or any other Instrument by glewing upon it a piece of its own wood turn'd like a little salve-box then upon this pin put the socket of your Instrument and work as followeth To take the altitude of the Sun Take the string of your plummet in your hand and apply it to the edge of your Instrument and hang it plumb then screw it fast then move the ruler with sights up and down till the Sun shining through the sight next the limb the shadow of the thread run streight along the rule then look how many degrees are between the edge of the rule and the bottom of the limb so many degrees is the height of the Sun and this you may do by setting it on a stool To take the height of a star To do this having hanged your Instrument on the pin of the neck and plumbed one edge by the light of a candle look by the edges of both sights moving the ruler till you see the star desired in a streight line with them both then screw the ruler and take down the Table accounting the degrees from the bottom to the edge of the rule for the height of the star To take the distance of two stars howsoever situate If both be near the Horizon and near of one altitude and within 90 degrees of each other you need not use the neck at all but onely lay your ruler on the beginning of the degrees then screw it and turn the Table till by both sights you see one of the stars then fasten the Table and move the sights to the other star and the degrees on the limb of the fiduciall edge of the rule gives their distance If they be both in one and the same half of a vertical circle take both their heights as afore subtract the lesser altitude from the greater you have your desire If they are in severall halfs of the vertical circle take the complements of both their heights and add them together actum est But if they lie aslope and yet are within 90 degrees one of another then besides the foot and Quadrant or Pandoron get you two round sticks as big as your thumb about six foot long a piece sharpen their little ends
of them without going again into the field I Shewed before in Chap. 2. the manner of keeping your field-book by help of that and this you may readily obtein your desire All the field-books that ever you fill with notes page them all writing at the top of each page the name of the Parishes or Parish wherein the land â—â—th contâ—ined in that page and at every beginning of a new man set down his name and likewise at the beginning of every new field furlong or parcell in a furlong set down the name of the close field furlong or parâ—ell Also write on the cover of your first book A on the second B on the third C c. Then reserve four and twenty pages at the end of your first book A which shall not be paged or else make a little book by it self and on the cover thereof write INDEX and on the top of each page write A B C c. in Alphabetical order Then under each severall letter write first the Towns name beginning with that letter secondly The mans name for whom you measured thirdly The books name in which you wrote it and fourthly The pages either all of them or at least the first and last And whereas you may think this way will not be so beneficial â—o you as to go measure it again for that you may do as you see good you need not finde it unless you will Besides that you deserve pay both for surveying plotting and notes as if you had measured it And if you will measure it again these notes will do you no hurt See an example P. Purton 〈◊〉 Norton lib. C. pag. 31 32 33 34. Panchurch Rob. Audley lib. B. pag. 64. ad 76. Putford Tho. Dennie lib. K. pag. 97. ad finem Refer this following to pag. 85. line 13. But if you would bring water to your house from a conduit where you desire to place a cock as high as you can and that without Instruments First begin at the conduit and dig a trench near a foot deep there but as you go farther off let it be still shallower for five or six pole in length more or less according to the fall of the ground so that the water may but just follow you and when it begins to run over there stay it and begin a new depth as afore but he sure the fall of it be down-right like a stair and so go on till you come where you would be then add the fall at the conduit and all your stairs together and so high may you set your cock above the level of your trench FINIS ¶ An Appendix to my Faithfull Surveyour WE have in the book it self spoken of measuring such things as are measured by observing Instruments as the Pandoron plain-Table Quadrant Quadrat Theodelete Circumferentoâ— viz. of measuring of land taking of Altitudes and Distances taken by the chain here we will speak of such superficies as are done by a two-foot-rule as board glass pavement wainscot and of solid as stone and timber forbearing those things that seldome or never come in question as globes regular bodies and the like First Because land-measure and those seldome meet together in one man Secondly Neither would I have the book to be of two big a price and Thirdly Because my little time I have hath need to be spent to the best advantage for the common good CHAP. I. Of making the Rule FIrst I would have the Rule whether it be of box or of brass whether joynted in the middle or streight out to be just two-foot-long by some standard of brass kept by the Clerk of the Market and not as I have seen some that have been half an inch too long Let it be an inch and an half broad at the least and a third part of an inch thick with a square stroke struck round about it just in the middle of the length thereof Let one edge be besild off which serves that if you have occasion to draw lines with a pen if you turn that side downward you need not fear blotting if your rule chance to be blackt with inke if you rubb it well with sorrel that will fetch it out Through the midst of this besill strike a Gage-stroke an another along the midst of the other edge divide the rest of this side beside the besill into eight equall parts with seaven Gage-strokes In the 4 next coâ—umnes save one to the besill you may place all the under-measure of this Table of board-measure following which will not fall in a scale upon the rule viz. all inches halves and quarters from one inch to six or if you will to ten inches in small spaces the inches of the breadth of the hoard in the column next save one to the besill the feet required to a foot foreward at the breadth in the next the odd inches in the third and the Gentesmes in the fourth And adjoyning to this Table toward the middle of the Rule in the first of those four columnes seâ— one inch divided into ten equall parts and each of those into halves and each of those halves into five or suppose them so divided so is it divided into 100 parts or Centesmes from which inch you shall take off all your Centesmes with your compasses that are to be set in any of your scales For making the scale of board-measure Before you can make this scale you must have one column on the otherside the Rule next the besill parted into three small parts with Gage-strokes and divided in the middle of the length of the rule into two equall parts or feet whereof divide one of them into ten equall parts and each of them into ten more and each of them suppose at least to be divided into ten other so shall that fooâ— be dvided into 1000. and this Gunther calleth foot-measure which must be reckoned both wayes first from the beginning of the rule to the middle thus 1 2 3 c. and backward again and thus 11 12 13 c. and because the other foot makes ten of these inches and these ten make twelve of them therefore divide the other foot into twelve equall parts or inches and each inch into eight parts and number it from the end toward the middle with 1 2 3 4 c. but from the middle to the end with 13 14 15 c. and this he calleth inch-measure By help of this inch-line and the inch aforesaid and by help of your Tables for board and timber-measure are made your scales for board and timber-measure And this Table of board-measure is thus made First for all whole inches divide 144 by the inches of the breadth and you have the inches forward to a foot If any thing remain after division it is the Numerator of a common Fraction whose Denominator is the Divisor to which remain annex two ciphers on the right hand and divide again by the same Divisor and you have the Centesme desired Example
Let a board be seven inches broad I desire to know how many inches forward makes a foot Divide 144. by seven it gives twenty inches or one foot eight inches â— 7. Now to bring â— 7 into centesmes annex two ciphers to the remain four it makes 400 which divide again by seven it gives â—â— 100. But for half-inches reduce the breadth into an improper Fraction as 6½ is 1â— 2 then multiply 144 by the Denominator 2 it gives 288 so that you must always divide 288 by the Numerator or number of half-inches of the breadth of the board which is 13 so have you 22 or one foot ten inches 15 centesmes But if your breadth be an odd quarter or three quarters First reduce it into quarters and divide 576 by it so ◠¼ is 27 quarters therefore divide 576 by 27 it gives 21 inches or one foot nine inches 9 27 or 33 centesmes The Table followeth A Table shewing how many feet inches and centesmes of inches forward are required to make a foot of board measure at all breadths both whole inches half-inches quarters and three-quarters from one inch in breadth to 36 inches Quar. Board feet inch cent Quart feet inch cent Qu. inch cent quar inch cent 1 0 12 0 0 8 0 1 6 0 15 9 60 22 6 55 1 9 7 20 1 1 5 46 1 9 44 1 6 47 2 8 0 0 2 1 4 94 2 9 29 2 6 40 3 6 10 29 3 1 4 46 3 9 14 3 6 33 2 0 6 0 0 9 0 1 4 0 16 9 0 23 6 26 1 5 4 0 1 1 3 56 1 8 87 1 6 19 2 4 9 60 2 1 3 16 2 8 73 2 6 13 3 4 4 36 3 1 2 77 3 8 57 3 6 6 3 0 4 0 0 10 0 1 2 40 17 8 41 24 6 0 1 3 8 31 1 1 2 5 1 8 32 1 5 94 2 3 5 15 2 1 1 76 2 8 22 2 5 88 3 3 2 40 3 1 1 35 3 8 12 3 5 82 4 0 3 0 0 11 0 1 1 9 18 8 0 25 5 76 1 2 9 88 1 1 0 80 1 7 81 1 5 70 2 2 8 0 2 1 0 51 2 7 78 2 5 65 3 2 6 31 3 1 0 25 3 7 68 3 5 59 5 0 2 4 80 12 0 1 0 0 19 7 58 26 5 54 1 2 3 41 1 0 11 76 1 7 48 1 5 48 2 2 2 18 2 0 11 52 2 7 39 2 5 43 3 2 1 4 3 0 11 29 3 7 29 3 5 38     Qu. Inch. Cent.       6 0 2 0 0 13 0 11 8 20 7 20 27 5 33 1 1 11 4  1 10 87 1 7 11 1 5 28 2 1 10 15  2 10 67 2 7 2 2 5 24 3 1 9 33  3 10 46 3 6 94 3 5 19 7 0 1 8 57 14  10 29 21 6 86 28 5 14 1 1 7 86  1 10 11 1 6 78 1 5 11 2 1 7 2â— Â 2 9 93 2 6 69 2 5 5 3 1 6 58  3 9 76 3 6 62 3 5 1 Q. I. C. Q. I. C. Q. I. C. Q. I. C. 29 4 97 31 4 65 33 4 36 35 4 12 1 4 93 1 4 61 1 4 33 1 4 9 2 4 89 2 4 58 2 4 30 2 4 6 3 4 84 3 4 54 3 4 27 3 4 3 30 4 80 32 4 50 34 4 24 36 4 0 1 4 76 1 4 46 1 4 21    2 4 73 2 4 43 2 4 18    3 4 69 3 4 39 3 4 15    Now to place this Table upon the rule divide the second third fourth and fifth columns next to the besill at one end into small squares that may hold two figures a piece in which set over-most the inches of the breadth in the second the feet required in length at each inch half inch and quartern In the next the odd inches and in the next the odd centesmes and this you must do to six inches you may do it to ten inches if you will Then at the end of ten inches set one inch divided into ten equal parts and each of them into halves and suppose each half into five so will it be supposed to be divided into an hundred parts as before Then from six inches to 36 you shall set all in the column next the besill with small strokes after this manner First I begin with six inches and a quarter to which I finde in the Table there belongeth one foot eleven inches four centesmes that is eleven inches four centesmes from the middle cross stroke of the rule But because my compasses will not reach so far I onely take 56 centesmes from the former inch which makes it just two foot from the same end which I set the under measure at Another example let be 9¼ for which I finde in the Table one foot three inches 56 centesmes First I take with my compasses 56 centesmes from my inch of centesmes and prick it down upon a line upon a paper Also with my compasses I take three inches in the foot-line of inch-measure on the other side of the Rule set that distance also on the paper at the end of the 56 Centesme in the same line then take with your compasses the whole length of both set one foot in the middle-cross-line of the Rule and in the said scale and the other toward the beginning of the Rule and it gives the length correspondent to nine inches and â— 4 from the stroke to the end of the Rule Thus do with all the rest marking each whole inch with its proper number to 24 also 30 and 36. And now before we proceed to shew you the making of the Table of timber-measure we will first shew the measure of boards CHAP. II. Of measuring of boards with the Rule THere are divers ways of measuring of boards of which the fundamental way is this 12 inches in length and 12 in breadth that is twelve times twelve or twelve inches square which is 144 inches make a foot of board therefore multiply the inches of the length of the board by the inches of the breadth and divide the product by 144 you have the content in feet If any thing remain divide it by twelve it gives the odd inches or twelve parts of a foot for an inch is the twelfth part of a foot let the foot be what it will Example Let a board be 13 foot five inches long that is 162 inches long and nine and an half broad these multiplied give 1529 and an half which divided by 144 give ten foot 89 square inches and ½ remains which divided by 12 is 7½ ferè inches of board Secondly If you multiply the length in feet 13 feet 5 inches by the breadth in inches 9½ first 9 inches by 13 foot is 9 foot 9 inches half of 13 is 6½ and 6 square inches and 9 times 5
angle upon the same line as if it were the end of it and then making a new plot at home your own reason will direct you better then I can shew it for it is easier perceived upon triall in the field then expressed by word or scheme but then you must lay down none but station-lines and angles 3. The most common help that Surveyours use is to remove the paper nearer one end of the Table and then with a piece of mouth-glue which they usually carry with them they glue on what paper they think they shall need and then fasten it down with the rulers again 4. If your plain-Table be also a Pandoron or have a semi-circle or a Quadrant you may at any time either in this case or case of moist weather take off your paper and help your self thereby as shall be shown hereafter 5. By the chain onely and your field-book whereof also hereafter in its place Eightly Before you begin you must know that both at the beginning and ending of every station-line and every crook of the hedge both inward and outward you must measure the nearest distance between the station-line and the hedge for all breadths must cut the station-line squire-wise and so make two right angles at the station-line and that is the best way and so doing all the pieces on the out-side the station-line will be either rectangle triangles or else compounded of an oblong and a rectangle triangle the area of both which is found by adding the breadth at both ends together and take ½ of it for the common breadth which multiply by the whole length and you have the content And sometime your best way to finde the shortest distance into an angle is to set up the Table right in the station-line if standing at the fore-mark you see by the edge of the Table the backer mark and then standing at the backer end you see the fore-mark then are you right in the line If now withall one or both of your other sides look right into the angle then are you right And all these lines must be entred into your field-book which fall perpendicular upon the station-line every one in their order on the right side of the line and on the left side right against each of them their correspondent lengths how far each of them is off from the last station Or else you may strike a station-line into the angle and so make scalenum triangles but that is not so certain and asks more labour Ninthly Before you go forward you must propound to your self a mark to go upon on the farther side the ground or if it be quite beyond the ground though it be a mile it matters not so that standing at A you may see it clear from the hedge yet as near to the hedge as you can whether it be parallel or no care not If you can see no such mark neither near the further side nor beyond then either you must send one before to stick up a stick with a cloth or paper on it or to stand there till you come with some white before his breast And moreover see if you can see some other mark between him and you right in the same line be it either flower weed grasse dung c to be a guide for the fore-man to keep him right in the line that carrieth the fore-end of the chain Tenthly Whereas you must have ten sticks about a foot long apiece whitled and sharpned at the great end let two take the chain one at one end the other at the other let the former take the sticks and let him be sure to lead streight in the line which for his guide therein he hath these helps First he must always be right in the line with his two marks before him till he comes at the first Secondly after he is come at the first let him every time he sticks down a stick look backward to set himself right in a line with those two And thirdly if there be no middle-man let the hindmost standing at A guide the foremost right in a line to B and after the first chains length let the hindmost guide the foremost and the foremost the hindmost for if the hindmost see the foremost right in a line between him and B and the foremost see the hindmost right in the line between him and A then are they both in the right line between A and B. Then to go forward let the foreman take all the sticks and tell them at the beginning at each change and at the end for the most common mistake is the losing or mis-telling of a stick and carry all save one in his left hand and that one and the chain in his right and let him go on streight in his station-line not looking behinde him till he feel the chain check him then stick down that stick and away as fast you can run and as you go shift an other stick into the right hand ready to stick down again In the mean time the hinder-man first holding the chain in his right hand at A let him look the chain be not tangled and away on till he come to the stick and then clapping his ring of the chain to the foreside of the stick let him take it up with the same hand he carrieth the chain and away after his leader And when the sticks are all run and that they are not yet at the end of that station-line let the fore-man run one chain more holding still the ring in his hand and at the end thereof set his toe there standing still and let the hinder-man take up the tenth stick and hold that still in one hand and the other nine in the other and deliver the nine to the fore-man setting his toe to the fore-mans then let the fore-man tell the nine and if they be right away if not you must measure all that course again and seek the stick for you know not which of you lost it and so going to the end of that station-line or within so much of the end of it that you may have libertie to set up the Table and see to the further end of the next station-line as you did at A without any incumbrances which if you work by a diagonall scale may be in any place but if by a plain scale you had best to have it at some even poles and because by Gunther's chain of an hundred links which is the best way you work not by the diagonall scale by links but by the foot chain by the decimall scale and by poles and parts of poles Set that length in your note-book on the left side of the line close by the line and a Bright under A and on the right side the line write station Then go on still in the said line till you come to the out-side of the ground which in pasture will always be beyond the station but in woods short of it Set down that length
awl-point near one end and a sliding button to be moved pretty and stifly up and down and to be stayed with a screw-pin or wedge at any distance with an other short point in the end thereof Now we will shew you how to continue your plot out of one ground into another that so you may lay all the grounds of a Lordship together in one entire plot by the chain onely and that we will do by severall rules for the understanding thereof we will refer you to the plot in the latter end of the third chapter as also in the end of the book The knowledge whereof consisteth in four rules in the obtaining the first station line in the close which you go unto As for example First Suppose I would go out of the first close at A and would plot the station-line AG now because in plotting these kinds of grounds you must always reduce all into triangles therefore standing at A you may measure two chains length in the line AF or AG likewise two chains back-ward from A towards B in the line AB in the first close then measure the distance between those two lengths and plot them after this manner First your best way is though you have measured but two chains length a piece yet in stead of two take the double if the station-lines be long you may triple that distance setting one foot in A and extending the other towards B there make a prick in that line and tran from thence with that wideness where you think the line AF or AG will fall then look what the distance was between the two lines at the end of your two chains a piece if doubled before then double again that distance upon your scale and set it in the tran from the line AB in the first close to the line AF in the second and draw the line AFG through that prick ad infinitum Thus have you got a line in the second close by help of a part of the line AB which in this kinde you must always take viz. that station-line whereof the whole or part belongs to both the closes But because in this case you must always mete through the hedge from the two chains of one close to the two chains of the other therefore to avoid the trouble of cutting a hole through the hedge if there be ever a gap gate or stile near unto those lengths you may take more or less of those two lines as you please now because here is a gap at two chains and an half from A in the line AB you may measure two chains and an half of either of them or two and an half in that and three in the other as you please and measure the distance upon the ground between those two pricks then you may double all three distances upon your scale as afore and set out the proper distances between those two pricks as afore and then draw your line AG upon your plot in the second close But Thirdly because we have measured the distance between A and X in the first line which is one side of the triangle of that second close and likewise have measured from A to G on the second side and have a gap also at X therefore if you measure GX you will have all the sides of that great triangle which you may use as afore-said First you have the line AX already placed Secondly take the length of AG with your compasses upon your scale and with that wideness set one foot in A and tran where you think G will fall Do likewise with the line GX taken also upon your scale set one foot at X and the other is the foresaid tran and there is your center G. And after the same manner may you go out of that close into the great close from G by help of the line AG. Now having the line AF or AG you may easily set out the triangle AFE as you did AXG. Likewise you set out the triangle that is between the the line XG and the hedge between the two closes onely by the distance of G to the entrance of the great close A second way of going out of one close into another is when I have a station near the middle of a station-line and that there I would go into another close For example Suppose I would go out of the great close into the first close right against the station-line BC in the first from L in the station-line of K then when you come right against BC the station line lengthen that line BC back-ward into the great close from L to M two chains length measure also two chains lengths in the station-line IK and measure two chains lengths from L to I back again and measure the distance between two chains of the one and two chains of the other and that gives you the quantitie of the angle KBC Then from the line LK you may take from your scale four chains length and you may tran from the line KL towards the line LC or BC with one foot set in L and double the distance of the two pricks in the other close and take that with your compasses and set from the line LK to the LC and where it falls draw the line LC ad infinitum After the same manner might you have drawn a line by the South-side of the hedge by BC or LC Also so might you at X in the first close have gone either into the great close or into the little close by drawing a station-line on which side of the hedge you will A third way is by continuation of such a station-line as shoots upon the corner of a close and thus suppose you would go out of the great close into the little close at K if you had but continued your line LK to A and this is the easiest way of all A fourth way If on the West-side of the hedge AK there were a spinny wood of two or three pole broad all along by the sides thereof and that you desire to go out of the first close into that little close but there is no gap save onely you can strike a squire-line from the station-line AB at either end of A K then may you both at A and at X erect a perpendicular into the first close ward and then may you continue those two perpendiculars so far as you shall need them till you are free from the spinny and may draw a line from one to the other by the spinny side and truly plotting out either perpendicular from the last station-line CHAP. XII To measure a wood by the chain onely BEcause a wood cannot be measured on the inside and herefore no subtendents can be taken as they may in pasture-ground we will therefore endeavour how to do it by taking of angles with the chain But in all this that hitherto we have spoken of measuring by the chain onely we would have you to understand that we
and the Sun as afore Then again about three a clock in the afternoon watch where the Suns shaddow falls just on the same circle again and then set up two other sticks so that they may meet in the same centre divide the space between the two furthest sticks into two equall parts and mark that for your meridian-line But lest the Sun should not shine when it comes to that circle you may make severall circles upon the board and stick up marks where the Sun comes at them forenoon and afternoon If both these ways fail this third way is better then either of them In the evening go Southward of the place where you would haue your diall three or four pole turn your face Northward moving Eastward or Westward till you see the North-pole and the place where you will have the meridian of your diall both in a line which by looking over the house you may the better do if you get one to hold a pole a slope with a line tyed to the end thereof and a plummet to it If now the line the meridian-place on the wall and the North-pole are all in a line you are right there stick up a stick till morning another right behinde it for just there is your meridian-line Now to know the pole you may easily ghuess at it near enough for it is a point in the heavens in a right line between the hinder horse of Charles-wain called Alliot and the polar-star so far off fâ—om the pole-star as the pole-star is from the next star to it so that if Alliot be just beyond the polar star then is the polar-star full North è contra A fourth way is this in some plain place near hand where you may see both ways set a mark go South two or three pole then move Eastward or Westward till you see the pole-star right beyond the first staff there set another or rather pitch two good stones like grave stones in Church-yards for so they will not onely serve for this business but also give the hour of the night to a minute by knowing the right ascention of the Sun and stars The use we make of it here is double first it helps us to set out the meridian-line every where near hand for if standing here at the North stone you see the Sun right over a stick or pole holden at the South you run presently set your back against the wall where you would have your diall and set up two sticks between the Sun and you you have a meridian-line desired Having a meridian in some open and plain place to finde the Azumeth set up a stick at the South-end of your meridian-line measure back in it 50 links there make your centre A thence measure 50 forward in the Sun-line measure the distance of those two fifties and plot it then take 60 off your scale of chords and do as in the last rule Having the Azumeth to finde the angle of the wall and Sun by help of the last figure Sometime you are in such a place where you cannot set out a meridian-line yet you may always set out an Azumeth or Sun-line which elswhere I call the angle of the wall and Sun Now finding your Azumeth as in the last rule come presently from thence not staying to cast it up or plot it but presently measure 50 by the wall and 50 in the Sun-line and their distance and then plot both the triangles and finde the degrees of both angles at the centre as afore so have you both the Suns Azumeth and the angle of the wall and Sun Then making a circle with two cross diameters first set out your Azumeth from the South if it was taken in the morning then on the East if in the after-noon on the West Then always reckon backward the angle of the wall and Sun in the course of the Sun and from thence draw a line through the centre representing the wall-line as in the last diagram the distance between that and the East and West line in the circle is the declination of the wall desired And although the Sun be newly gone off the wall or not yet come on by help of the shadow of the end of the wall and these former helps you may finde the declination Onely in stead of setting your Azumeth backward you must set it forward in the course of the Sun if you take it before it shines on the wall And all this may be done by a two-two-foot rule or yard or a boyes cat-stick CHAP. XV. Of colouring and beautifying of plots IN beautifying of plots it is necessary that you draw a square round about the plot the upper-upper-end whereof shall represent the North-side the nether line the South the right-side line the East but you must help your self to these by taking a meridian-line first in the field and drawing a meridian-line through the first plot Secondly Examine your former plot how many chains or poles your plot reacheth from North to South and from East to West and thereby make choise of such a scale that you may lay the whole Lordship within the said square according to the Northing and Southing and distance Or else you may draw your plot first by what scale you will and then draw the square afterward Thirdly Fill the out-borders between the square and the demeans at least such as border next to the demeans with the bordering hedges and names or owners names of the grounds Fourthly Whatsoever you write write it from West to East unless it be the proper name of some river or high-way or such like For if the North be upward the West will be on the left hand Fifthly Describe all houses ways rivers Churches wind-mills arbours great lone-trees gates stiles c. that fall within your plot as also the Lordship-house with other edifices in a corner by it self and the Lords coat in another corner the house being drawn in prospective Sixthly Describe at the bottom the scale that you drew it by adorning it with compasses ovalls squares and compartiments c. Seventhly Having drawn all your severall grounds and distinguished them with their hedges it will not be amiss first to pounce over the paper or parchment with some stanish grain and burnt Allome and a double quantitie of pounced rosen both finely searced and lightly pumiced thereby to preserve the paper or parchment from through-piercing with the colours Then lay on your colours in manner following being first ground and bound with gum-water very thin and bodiless Arable for corn you may wash with pale straw-colour made of yellow-ocre and white-lead For meadows take pink and verdigrease in a light green Pasture in a deep green of pink azure and smalts Fenns a deep green as also heaths of yellow and indico Trees a sadder green of white-lead and verdigrease For mud-walls and ways mix white-lead and rust of iron or with ocres brown of Spain for white-stone take umber and white water or glass may
and nayl their great ends together within five or six inches of the top with one nayl onely that they may open and shut like a pair of tongs also you shall take a joynd-stool and cushion and having put the neck upon the foot and the Pandoron on the pin of the neck close the three feet together with your right-hand and lay them on the cushion and with your left hand under-set the neck with the tongs opening and shutting them as need is or setting them nearer or further from you as need is all with the same hand and turning it aslope with the right hand Then having first placed the sights at the beginning of the degrees turn it till by the edges of both sights you see one of the stars you desire then keep the Table fâ—st there and move the sights till by them you see the other star voti compos â—ris CHAP. XXI Of taking of altitudes terrestrial by the Quadrant THere are divers ways whereby these altitudes may be discovered whether they be perpendicular as properly they signifie or Hypotenuses or bases for all of them are comprehended under the notion of Altitude because the bases may be as well found by the help of the perpendiculars as perpendiculars by the help of bases and any of these may be found severall ways by the Pandoron either as it is a Quadrant or as it is a Geometrical Quadrat of eitâ—er of which we will lay down some Problemes and first as it is a Quadrant Probl. I. A distance being given and the angle of the base to finde an altitude Measure the distance AC â—00 and the angle A 9 degâ—â— 0 min. by your Pandoron the Complement wherof is the angle B 60 d. 20. n. ergò as fine ABC 60 d 20 m. 993898 is to the line AD â—00 230103 so sine BAC 29 d. 40 m. 969496 to CB 114 03. 205700 II. Likewise the height CB given to finde AC the distance As BAC 29 deg 40 min. 969496 is to CB 114 03. 230103 so B 60 deg 20 min. 993898 to AC 200. 230103 To finde either of them by the scale and compasses having the angle A and distance AC First draw the line AC set from A toward C 200 of some scale of equall parts upon C erect a perpendicular and upon A make an angle of 29 deg 40 min. which line will meet CB and you shall finde CB 114 feré So measuring the height CB and the angle B and plotting it you shall have AC 200. III. The height BC and angle A being given to finde the Hypotenuse AB As A 29 deg 40 min. to BC the height 114 03 so ACB 90 deg to AB 230 17. To finde it by the scale Draw the line AC let it be 200 of equall parts upon C erect the perpendicular BC and on A make an angle of 29 deg 40 min. so the Hypotenuse AB wil be 230 17. The part of the distance DA in the same diagram being known to finde DC or AC Let AD or EF be 90 foot and I desire FG or DC but I cannot measure it for impediments therefore first take the angle of altitude B at both stations A and D at AI finde A 29 deg 40 min. so that the angle CBA is 60 deg 20 min. at DI find the same angle D 46 deg and DBC 44 deg subtract 44 deg from 60 deg 20 min. resteth ABD 16 deg 20 min. then say As fine ABD 16 deg 20 min. to AD 90 foot so is BAD 29 deg 40 min. to DB 158 â— 10. Then again As 90 to BD 158 â— 10 so is DBC 44 deg to DC 110 which added to 90 AD makes AC 200 as afore By the scale thus draw the lines AC and AB ad infinitum making the angle 29 deg 40 min. then set 90 feet from A in the line AC to D where you found the angle DBC to be 46 deg because the angle CDB is 44 for they are the complements one of the other therefore plot the angle BDC and it will be 46 deg and the BD 158 4â— then from B let fall a perpendicular upon AC and it cuts it at C making DC 110 and AC â—00 as before To let this perpendicular fall divide either AB or DB into two equall parts and with the compass at that wideness set one foot in the intersection and the other in the line DC at C and there falls the perpendicular BC and the end of the line AC Likewise any part of the altitude being known the rest of it may be found by turning the height into the distance and the distance into the height Secondly As sine DCB 118 deg vel 62 deg Compar 0054164 to BD 600. 2777974 so sine CBD 4 deg 8843588 to CD 47½ the Castles height 1675726 But this will not be found very exactly by plotting by reason of the meeting of the acute angles the lines running so far one in another especially AD and BD that you cannot distinguish their intersection and thus also we have not onely found the height of the Castle 47½ but also the rest of the hill line by measuring AB 200 a part of the same line and up an hill also for if you add BCD 118 deg to CBD 4. deg they make 122 which subtracted from 180 deg rests 58 deg the angle CDB Then say As CBD 4 deg Compar 1156416 to CD 47½ 1675726 so CDB 58 deg 9928420 to BC 776 2. 2760562 which added to AB 200 gives the whole line 976 2. And now if you intend to begin your mine at B. your best way is to go 10 or 12 foot first in BG line as you ghuess half the breadth of the fort to K and thence draw the line KL parallel to BC which two lines are of equal length Elem. 1. prop. 26. and then keep that line up to the top for that must be your line of direction that if by occasion of some rock or other impediment you are forced to raise or sink or go side-ways you may by help of this line drawn on paper with a large scale keeping account stil how far you are gone in the said line and by help of the Quadrant at each station be able to plot how much you are above or below your line of direction and by help of your Needle to finde how far you are gone side-ways but your best way is to draw one line for ascents and descents and another for variations side-ways besides your line of direction and it will not be labour in vain also beside both these lines to set down in a note-book the inches raised by themselves above the line of direction and the fallings by themselves that so you may subtract the summ of the lesser from the summ of the greater just as in conveying of water whereof we shall speak anon Likewise set down the variations on the right-hand by themselves and those on the left by themselves and against
that you would have thought that there had been proclamations set up in all woods groves fields and yards through the whole land to summon them thither or whether that was their beacon when I burnt it or no I know not These for a fortnight together so covered the ground that you could not choose but say it was far blacker then ink for this was of a double die one of black crows and another of black ashes The frost breaking those that they had not eaten they trod into the ground with their feet so that by the later end of the moneth no meadow could be thicker of green grass then that was of green-oats I plowed them in and by Candle-mass it was green again I plowed it again then it lay till the later end of April and was green again then I steeped my seed as I did the year before and sowed it with barley and had a very good crop and so killed the wild-oats The burning of queach also in some ground is exceeding profitable And not onely the steeping seed in dung-hill water helpeth greatly but also in lime and water by reason that which gives it heart lies close to the root Some also wash seed-wheat and rie in lime and water in the seed-leap in the field and then sow it and so no crows nor pigeons will ever touch it CHAP. XXXVI Of planting Willows INstead of beetle and stake or crow of iron make you an augre like a pump-augre make it after this manner Make a plate like a peel of a foot or fourteen inches square well steeled and turn it as an augre is turned let it have a socket like a peel but four-square into which put a stake of good tough ash two foot long and four-square as the socket is with a bar or hoop of iron about it at the top to keep it from cleaving let it be two inches square at the least upward in which near to the top bore an hole or else make a mortes to put in a cross piece to turn it by and to take it out by then enter it a little with your spade as you do a carpenters wimble with a gouch and then bore your holes which in strong clay is an exceeding speedy way Besides that if the sets be not very great you will have room enough to ram the moulds down to the bottom CHAP. XXXVII Of reducing wood-land to statute-measure and statute to wood-land I Have severall times measured ground by statute which should have been done by the eighteen-foot pole but never the contrary One amongst the rest was a close in Hexton in Hartfordshire where three Copy-holders had each of them apart expressed in their severall copies how much by measure but not by what measure thereupon it was taken for granted that it must be statute-measure One of the three had held all in his occupation divers years together and lying in stitches no banks between had plowed one amongst another A and B would have theirs again A must have so much on the East-side B so much on the middle and C the rest for C would neither shew his copie nor yet make known how much he should have So I laid out each man his share accordingly and took a plot of the whole Still it runs in B his minde that his part was not so good as it had been formerly mistrusting that I had done him wrong in laying it forth so that he acquainted the Lord of the Mannour with it who demanded of him by what measure he had measured it he answered by the statute-pole Then quoth the Lord there is the errour the custome is eighteen foot and was the measure taken in Henry the eight his time This being known and reduced C shewed his copie and there was not a pole difference in the whole thing so I gave them direction to alter it without going to the ground To do this there are severall ways First a statute-pole is sixteen feet and an half or 33 half foot long therefore 33 half-feet square is 1089 square half-feet in a statute-pole but in an eighteen-foot pole which is 36 half-feet square are 1296 so then if you multiply your statute-poles by 1089 and divide the product by 1296 you have the number of eighteen-foot poles which divided by 40 gives you the roods and vice versâ And thus six acres of statute which is 960 poles multiply'd by 1089 makes 1045440 and that divided by 1296 gives 806 864 1â—96 or â…” which is five acres six pole â…” of the 18 foot Likewise five acres of 18 foot is 800 pole that multiply'd by 1296 produceth 1036800 which divided by 1089 quotient 952 72 1089 pole that is 5 acres 3 rood 32 pole And this is the best way So that the analogy is thus As 1089 1296 ∷ 800 18 foot pole to 956.1089 id est 5 acres 3 roods 32 pole 72 1089. And as 1296 1089 ∷ 800 statute to 672 2 9 id est 4 acres 3 roods 32 poles 2 9. And this is your best way and thus may you do with all other poles Another way is if upon your scale you have two scales one of 11 in the inch and another of 12 if you lay down staâ—â—â—e-measure by the scale of 12 and then measure the same plot by the scale of 11 it gives you the wood-land measure and likewise on the contrary CHAP. XXXVIII To finde any scale that a plot is made by the content being known SUppose any scale as 10 and measure it by that now if by measuring it by the scale of 10 it comes to but 23 acres 82 parts but it is truely 34 acres 31 parts therefore finde a mean proportional between these two which because the work is somewhat difficult I will therefore shew you the manner of it First multiply 32.82 by 34 31. as here it is set down so you see it produceth 817 ⌊ 264â— And because there are four figures in the Fractions of the two Factours therfore there are also four in the product so the whole number is 817 and 2642 the Fraction the square-root is 28 ⌊ 59. which is the mean proportional desired then say As the lesser of the two numbers viz. 23,82 is to your mean proportional 28 59 so is your supposed scale to 12. the true scale as 23 82. 28 59 ∷ 10 12. See the work But because there is too much difficultie to finde it this way and so little by the line of numbers and so soon done and is exact enough therefore by it divide the distance between 23,82 and 34,31 into two equall parts and the compasses will fall at 28,59 then because 28,59 is more then 23,82 therefore set one foot in 10 and turn the other upward it will fall at 12 the scale desired CHAP. XXXIX Of making an Index or Table whereby readily to finde out any ground that ever you have measured and to tell the quantity of them an hundred years after and draw a plot
inches is 45 square inches and half five inches is two and an half square inches First then add all your inches together 45 6 and 2½ make 53 and ½ which divided by 12 gives 4 board inches and 5½ square inches or half a board inch feré Now add these 4 inches to 9 and 6 inches they make 19 inches that is one foot seven inches to which add 9 foot it gives ten foot seven inches ½ ferè just as afore and both those ways are performed by any common Rule that â—ath no board-measure on it Hence then is discovered this errour that if a board be nine inches broad to take 15 inches forward to make a foot that is so much more then twelve as nine is less whereas our Table saith you must take 16 is a false way for nine times 15 is but 135 which wants nine square inches of 144 and is always the square number of half the difference of nine and 15 equally distant from 12 whose square is 9. So likewise 8 and 16 being multiplied make 124 which wants 16 of 144 and because they are equidistant from 12 and their half difference is 4 therefore their product is less by sixteen the square number of four then the square of twelve 3. A third way of measuring board is by this rule Measure the breadth of the board if it be less then six inches your Table of under-measure will shew you how much forward you must take to a foot forward If it be broader and under 36 inches then the strokes on your scale give it 4. Some measure all the breadths of the boards with a line then stretch the length on a block and so measure the breadths of all the stock at once and then measure the length of a board then multiply the length in feet and parts by the breadth in feet and parts So suppose the breadth of all the boards is ten foot nine inches and the length 154 inches instead of nine inches I take ½ ¼ of a foot and instead of four inches I take â…“ or ¼ one inch and the work will be thus and it makes 164 feet ¾ 1 inch and an half And this is a very good way in case a block be hewn eight-square before it be sawn which if it be fit for boards it is pitty it should be hewn any other way so will it be no loss of timber the boards will be all streight-edged If it be sold in timber and measured as eight square as shall be shewn there will be no loss either to buyer or seller CHAP. III. Of making of a Table of timber-measure for square timber to make the scale of square timber-measure by as also the under-measure FIrst know that a foot of timber is twelve inches every way breadth length and thickness and therefore conteineth 1728 square inches for 12 times 12 is 144 that is a foot of board or a superficies and twelve foot of board make 1728 inches therefore to proceed to the Table First For whole inches square the square of the piece that is multiply the square by it self and by that product divide 1728. Example Suppose the piece be 8 inches square the square of 8 is 64 by which divide 1728 it gives 27 inches or two foot three inches But if you have odd half-inches then you must reduce as before all your inches into half-inches or an improper Fraction by whose Denominator which will always be 4 multiply 1728 it gives 6912 which must always be divided by the Numerator of the Fraction Suppose the square given be 6½ that squared is 42¼ which reduced is 169 quarters by which 169 divide 6912 it gives 46 inches or 3 foot 4 inches ninety Centesmes Again if the square be of odd quarterns or ¾ you must work as before and then your divideâ—t will be 16 times 1728 that is 27648. Example Let your square be 6¾ that squared is 45 9 sixteenths which reduced into 16 parts by multiplying 45 by 16 and adding 9 it gives 7 19 sixteenths Therefore divide 27648 by 729 it gives 7 inches or 3 foot 1 inch 92 Centesmes Here followeth the Table of timber-measure Inch squar feet inch cen inch squar feet inc cent Inc. Inc. C. Inc. Inc. C 1 0 144 0 0 8 0 2 3 0 15 7 68 22 3 57 1 92 1 92 1 2 1 39 1 7 43 1 3 49 2 64 0 0 2 1 11 91 2 7 19 2 3 41 3 47 0 24 3 1 10 57 3 6 97 3 3 34 2 0 36 0 0 9 0 1 9 33 16 6 76 23 3 27 1 8 5 33 1 1 8 19 1 6 54 1 3 20 2 23 0 48 2 1 7 14 2 6 35 2 3 13 3 19 0 60 3 1 6 25 3 6 16 3 3 6 3 0 16 0 0 10 1 5 28 17 5 98 24 3 0 1 13 7 55 1 1 4 44 1 5 81 1 2 94 2 11 9 6 2 1 3 67 2 5 64 2 2 88 3 10 2 88 3 1 2 95 3 5 48 3 2 82 4 0 9 0 0 11 1 2 28 18 5 33 25 2 76 1 7 11 67 1 1 1 65 1 5 19 1 2 71 2 7 1 33 2 1 1 6 2 5 5 2 2 66 3 6 4 75 3 1 0 51 3 4 91 3 2 61     Inch. Inc. C.  In. In. C. In. In. C. 5 0 5 9 12 12 12 0  19 4 78 26 2 56 1 5 2 69 1 11 51  1 4 66 1 2 51 2 4 9 12 2 11 6  2 4 55 2 2 46 3 4 4 26 3 10 63  3 4 43 3 2 41 6 0 4 0 0 13 10 29  20 4 32 27 2 37 1 3 8 23 1 9 82  1 4 21 1 2 33 2 3 4 89 2 9 48  2 4 11 2 2 29 3 3 1 92 3 9 14  3 4 1 3 2 25 7 0 2 11 27 14 8 82  21 3 92 28 2 21 1 2 8 88 1 8 52  1 3 83 1 2 17 2 2 6 72 2 8 22  2 3 74 2 2 13 3 2 4 77 3 7 90  3 3 66 3 2 9 In. In. C. In. In. C. In. In. C. In. In. C. 29 2 6 31 1 80 33 1 59 35 1 41 1 2 2 1 1 77 1 1 56 1 1 39 2 1 99 2 1 75 2 1 54 2 1 37 3 1 95 3 1 72 3 1 52 3 1 35 30 1 92 32 1 69 34 1 49 36 1 33 1 1 89 1 1 66 1 1 47    2 1 86 2 1 64 2 1 45    3 1 83 3 1 61 3 1 43    To place this Table on the Rule Begin at the other end of the Rule taking those 4 columns next the thick edge save one and divide them into little spaces as you did for board-measure setting on them all the under measure to 8 inches and an half square yet you may do it to 12 inches if you will setting the
the basis on which the other are built or scale by which they are made Secondly For the Diameter Archimedes gives this rule Multiply the Circumference by seven and the product divide by 22 so have you the Diameter so on the contrary Thus our circle 88 multiplied by seven gives 616 which divide by 22 quoteth just 28 as afore Thirdly For the square within the circle this is the rule The square without the circle is double in content to the square within Or thus The content of the square within the circle is to the content of the circle as 7 to 11 First therefore by the content of the square without we found the Diameter or side of the square to be 28 that squared or multiplied by it self is 784 the content thereof Therefore the content of the square within is but ½ 784 that is 392. whose square-root is 19 31 39 as afore Secondly by the content of the circle for which Archimedes saith half the Diameter multiplied by half the Circumference gives the content so 44 the half of the Circumference multiplied by half the Diameter 14 gives 616 the content of the circle This therefore multiplied by seven makes 4312 which divided by eleven gives 392 just as afore Fourthly For the square equal to the circle having by this last rule found the content of the circle to be 616 we need but extract the square-root thereof which is 24 40 49 which doth discover a most monstrous and a most gross errour in measuring round timber of which hereafter CHAP. VII Shewing the manner of placing these upon the Rule FIrst To set out the Diameter you may take the nether part of the third column of the besil'd side to set it on from the middle square stroke of the Rule Then Gunther in his Vse of the line of numbers in broad-measure Prop. 11. hath this proportion Having the Circumference of a circle to finde the Diameter As 3143 to 1000 so is the Circumference suppose it 47 ⌊ 13 to the Diameter 15 so that if you take 47 ⌊ 13 in your circle-scale and set in that column from the middle square downward so shall you set out 15 in that distance run that distance as oft as you can to the bottom of the Rule which will be 4 times more divide each of them into 3 equal parts and the uppermost third into 5 equal and number all the other great parts save that with 5 10 15 c. or if you will you may double 47 ⌊ 13 that is 94 26 and take it from the circle-scale set it there they will be 30 then half it and they will be 15 then third it into fives 2. To finde how to proportion the square within the circle by the Diameter Let the Diameter be the Radius 1000 then will the chord of 90 degrees which is the side of the square included be the natural sine of half 90 viz. 45 degrees the sine whereof is 707 therefore then because I would divide my scale into even sines if therefore I take 7 times 5 that is 35 the proportion will be 707 1000 ∷ 35 49 ⌊ 50. or 49½ therefore if you take 49½ on the Diameter and set it on the scale of chords and divide it into 7 equal parts and that part next the end into 5 small parts numbring all but that with 5 10 15 c. you have your scale of chords or square within the circle Or if you think it troublesome to divide it into 7 equal parts you may take 6 times 5 that is 30. and say 707 1000 ∷ 38 42 ⌊ 43 so then you may take 42 ⌊ 43 of the Diameter and set on your scale of chords and then divide each of them into halves and each half in to 3 parts Otherwise thus The content of this circle according to Archimedes is just ½ the content of the square of the Diameter Suppose the Diameter 24 the square thereof is 576 the half whereof is 208 the root whereof is 17 ferè then say If 17 in chords require 24 Diameter what shall 40 in chords or any other even number of fives Answer 56½ therefore take 56½ of the Diameter and set it in the scale of chords which because it gives 8 times 5 first divide it into halves then into quarters then into eight 3. It may also be made by this Rule of his The area of the square within the circle is to the content of the circle as 11 to 7 so that the circle begin known the content is thus found ½ the Diameter multiplied in ½ the Circumference gives the content of the circle which if you multiply by 7 and divide the product by 33 it gives the content of the square within whereof take the square-root and you have the side desired therefore 19 ⌊ 8 88 ∷ 20 88 ⌊ 9 or as Mr. Wingate hath it in Problem 33. of his Appendix to his Rule of Proportion 225 1000 ∷ 20 88 ⌊ 9. So that take 88 ⌊ 9 from the Circumference and set it on this scale and divide it into four fives and this scale may be set on the lower half of the besil'd edge 4. Having the content of the Circumference to find the side of the square equal Take the square-root thereof so we found before that the Circumference being 88 the content is 616 whose square root is 24 ⌊ 40 49 that is more then 24¼ or more easily because as Gunther hath it the Circumference is to the side of a square equal as 1000 the Radius to 282 therefore say 282 1000 ∷ 20 70 ⌊ 9. Therefore take 70 ⌊ 9 of the Circumference and set it in the scale of the square equal it gives 20 of that scale with which distance set out all the twenties the side will bear dividing each 20 into four fives and the last into five little ones and numbring them by five as afore and this scale may be set in the over part of the third column nexthe square edge Errour in round timber to take a quarter of the circumference for the square 5. And here I must acquaint you with that monstrous errour in measuring round timber which I spake of before which is this to gird the piece about and to take the fourth part for the square thereof as suppose the piece be 80 inches about then by this account the square should be but 22 inches whereas in the last section we found it to be above 24¾ whereby the full fifth part of the timber is lost to the seller which notwithstanding the most of them know to be extream false by reason that when they have hewed it they make a great deal more of it then they did before it was hewed But what is their excuse Even this they say That will scarce pay for the hewing and it is but sap and bark I answer The goodness or badness of any thing is considered in the price but neither in the measure nor the manner of measuring
I have seen a sack of fine seed white wheat sold for ten shillings a bushel another of grey wheat at seven sold the same day all to one man yet he had no more measure of the course grey then of the fine wheat Secondly In that they say They had need have that for hewing I say They never hew what they rend to laths pales rails plow-timber cart-timber wheel-timber boles trenchers dishes spoons and infinite other which they rend and sell sap and all Thirdly When they do hew any timber they leave it so wany that in Cambridge-shire especially they leave it nearer round then square and yet allow nothing for the wanes so that in all other things whether sold by weight or measure the buyer is to have the draught though it be but in an ounce of pepper in this he must want of his measure and that no small matter for they seldome hew nigher to square in this Countrey then that the four wanes are as broad as the four flats all which are equal to a square piece of the breadth of one of those wanes although those wanes be less in some places then in other yet will they be of no service so deep as the deepest wane goes And what sense or equity is there that in buying they should desire so much over-measure and yet in selling it hewed sell so much short as in buying Hath not he that buyeth wane-timber that the wanes run not streight as much need and as much reason to have allowance for the wanes and to have the knots and bark left on them for hewing as you to have the fifth part and more and yet never hew a great deal of it at all Besides you have a trick when you buy round-timber with the bark on it be it thick or thin you will cut a notch round about the piece in the middle of the block sometimes deeper then the bark saying That is but a boin now you buying by measure what right have you to the bark which you measure not yet when it is hewed they that buy it must be content with air instead of timber And yet further I have known a Wheel-wright that used to buy all his timber by the foot of fourteen inches every way to the foot and to girdle it and to take the fourth part for the square thus did he over-reach the sellers who thought it to be but a seventh part more then ordinary and that he gave a penny or two pence more in a foot then others gave they thought themselves well enough whereas poor simple fools they sold above two foot for one 6. If you buy round timber that is ordinarily taper little or much then you will be sure to gird it in the middle or nearer the little end whereby you gain no small matter Lastly How common a thing is it with Wood-mongers to have one Rule to buy by another to sell by one a quarter of an inch too long another as much too short And great pity it is that considering there are so many abuses in measuring land and timber it is not a whit looked into whereas in all other things sold by weight or measure the abuses are punished by the Clerk of the market Now for correction of this false measure in round timber committed by this way of taking the fourth part for the square if it be a perfect Cilinder and not taper you may help your self by this Table taken out of Mr. Stirrup's Plain-scale or Carpenters new Rule page 60 which you may draw into a scale as you do for square timber or board-measure all but the first seven inches which are under-measure and set those 7 in four columns between the two Tables of board and timber under-measure Squar Inch. Feet Inch. Cent. Squa. Inch. Inc. Cen. Squa. Inch. Inc. Cen. 1 113 1 71 11 11 22 21 3 11 2 28 3 42 12 9 42 22 2 80 3 12 6 85 13 8 3 23 2 56 4 7 0 85 14 6 92 24 2 35 5 4 6 30 15 6 3 25 2 17 6 3 1 71 16 5 30 26 2 0 7 2 3 70 17 4 69 27 1 86 8 1 9 23 18 4 19 28 1 75 9 1 4 76 19 3 76 29 1 61 10 1 1 57 20 3 39 30 1 51 The use of this Table is thus Girt the piece about and take the fourth part for the square as if it were the true square and therewith enter this Table and it gives the feet inches and Centesmes required forward to make a foot forward at that false square So 44 inches circle gives 11 inches for the fourth part which in the Table gives 11 inches 22 Centesmes forward to a foot-square of timber Or else having taken the Circumference with a nealed wier and there made a twist and measured the number of inches about take off so many with your compasses and apply that wideness to the scale of the square-equal and you have the square you must measure it at And because as I said before that to hew a log for boards the best way is to hew it eight-square both for saving timber and to have all the boards streight-edged so neither shall the sawyers be paid for more then they saw nor he that buieth the boards or the block it self want or have too much we will now therefore give you one rule whereby to measure all equal-sided timber so that it be not taper how many sides soever it hath First finde the centre of your piece and measure the semi-diameter thereof to the middle of one of the equal sides then add all the sides together multiply half thereof by the semi-diameter so have you the content of the base and that multiplied in the length gives the content of the piece So in the figure the 8 sides are ten a piece that is 80 the half whereof is 40 the semi-diameter or perpendicular AB is 1◠that multiplyed by 12 makes 480 which is the content of the base that is one inch sawed off of the end of the piece Then if either you multiply 480 by the inches of the length of the piece and divide the product by 1728 you have the content of the piece Or else you may extract the square-root of 480 which is 22 ferè and then measure it as if it were 22 inches square And thus may you measure all manner of timber not taper by measuring one inch at the end as if it were land then extract the root and measure is as if it were so much square CHAP. VIII Of taper-timber whether Conical or Pyramidal FOr such kinde of timber of either sort measure it as if it were a whole Cilinder or Prisme that is First finde the area of the base and multiply it by the whole length thus Let a Prisme be four-square the side 12 the area of the base is 144 and suppose the length 100 these multiplied make 14400. But by the Corollary of the 7th Prop. 12. lib.
Euclid every Pyramis is the third part of a Prisme having the same base and altitude therefore divide 14400 by 3 it giveth 4800 the content of the Pyramis But suppose it be an imperfect Pyramis that runs not to a point but hath his top cut off you shall then continue out the sides to a perfect Pyramis by plotting it in paper or else finde how much it wants by the Rule of three Example The side of the base being twelve the length of the piece fiftie and the side there is six so that there is six lost in fiftie but the whole side of the base is but twelve whence take six six resteth Then say 6 50 ∷ 6 50. and 50 and 50 make an hundred as before Now then for this little Pyramid the side or Diameter of the base thereof being six whose square is 36 the third part whereof is twelve that multiplied by 50 gives 600 the content of the lesser Pyramid Subtract this perfect Pyramid out of the great perfect Pyramid 4800 rests 4200 the imperfect Pyramis And the reason that holds between the Prisme and Pyramis holdeth also between the Cilinder and Cone Prop. 10.12 Euclid Every Cone is the third part of a Cilinder having the same base and altitude Of the Cone Let us now suppose a Cone also divided in length into 50 and 50 the greater Diameter at the base to be twelve and six in the middle First to finde the Circumference to 12 the Diameter 12 multiplied by 22 is 264 that divided by 7 is 37 5 7 the Circumference Then multiply half 37 5 7 that is 18 6 7 by half the Diameter that is six it gives 115 5 7 the greater area which multiplied by 100 the length it gives 11514 2 7 the Cilinder the third part whereof is 3838 2 21 the greater Cone Now for the lesser the Diameter is six multiply it by 22 it is 132 that divided by seven is 18 6 7 the base which multiply by the length 50 is 942 the third part thereof is 314 2 7 the lesser Cone Now take 314 2 7 out of 3838 2 21 resteth the imperfect Cone 3520 which is almost twelve times as big as the lesser Or if you rather desire 12 and 6 the bases of the Pyramis to be the sides of the square within the circle as there they are and then to see their dimensions then first if twelve be a side of a square within the circle since the content or square thereof is but half the content of the square of the Diameter therefore double the square thereof and out of the double extract the square root and you have the Diameter so 12 squared is 144 that doubled is 288 whose square-root is 17 ferè the Diameter Now to finde the Circumference multiply 17 the Diameter by 22 facit 374. that divide by seven it quoteth 53 ◠7 the Circumference then multiply half the Circumference 26 5 7 by half the Diameter 8½ it gives the area of this base 227 ◠14 which multiplied by 100 the length gives 22707 ◠7 the Cilinder which divided by 3 gives the great Cone 75695½ Likewise for the lesser square within which is six the square is 36 that doubled is 72 the square-root whereof is 8½ ferè the Diameter Multiply 8½ by 22 it gives 187 which divided by 7 gives 26 5 7 the Circumference then multiply half 26 5 7 that is 13 5 14 by half 8 an half that is 4¼ and you have 56 577 879 or ⌊ 72 ferè the content of that area which multiply by 50 the length gives 2835 the third part thereof is 945 the lesser Cone Take this lesser 945 out of the greater 7569 resteth 6624 the imperfect Cone So that the imperfect Cone is more then seven times as big as the little one The discovery of severall errours in measuring the Pyramid and Cone and first of the Pyramid Some hold that to be true To add the areaes at both ends together and multiply the 1 half thereof by the length of the piece as in our example the area of the great end is 144 and the little end nothing therefore half 144 i. e. 72 multiplyed by 100 is 7200 but it should be but 4800 it is too much by 2400. A second errour is to take the area at the third part from the great end as in this figure at C and C but there the square or side is 8 and the square number or area thereof is 64 which multiplied by 100 is 6400 too much by 1600. Secondly in the Cone The common practise is to gird it in the middle and to take the fourth part for the square In measuring the cilinder there was more then the fift part lost to the seller but here that it is taper also is a more intolerable loss For if in the square Pyramid was lost a full quartern onely by reason of tapering what will here be lost where two such errours combine in one to wrong a man The Circumference in the midst of the piece is 26 5 7 the fourth part thereof is 6¾ which squared is 45½ and that multiplied by 100 makes 4556 ◠4 which taken out of 75 9◠there is lost to the seller 3013 which is almost one half thereof Yet this goeth so for currant in all places that he that contradicts it is scorned as a fool and accounted as a knave CHAP. IX Of the making of four other lines on the flat-sides whereof three are Mr. Gunthers lines of numbers sines and tangents and instead of the Meridian-line which is onely usefull for Navigation whereof Carpenters make little or no use we have added a sextant of chords ALthough Mr. Wingate in his book called The Rule of Proportion hath set down the making of them yet for that he hath done them after another manner then here is shown neither will an ordinary Rule bear all those lines we will therefore content our selves with Mr. Gunther's the line of chords onely You shall divide the rest of the Rule beside the columns of feet inch-measure before spoken of into four other great columns and divide each of them into two equal and one of them into two also so the great shall be for figures the other 2 for strokes These two of Mr. Gunthers you may set in the three middle columns and the line of chords on the other outside First for making the line of numbers I told you before that I would have you strike a stroke round about cross the Rule I would also have another at each end of the Rule so close as possibly you can onely to set one point of the compasses on Then first set out your great division in each foot viz. the thousands if your number consist of four figures or howsoever they are to be the left hand figures of any number as 3 in 3 32.346.3654.37046 c. and must be marked with the 9 digits in either foot and the first last and middle-most with
one so that you may understand as many ciphers with it as shall be requisite so that it may signifie 1.10.100.1000 and then if one signifie 10 the next two will naturally signifie 20 but not always Now to take and set the number 2 in his right place take a Table of Logarithmes of absolute numbers and look either the Logarithme of 2.20 or 200. and take the three next figures to the Characteristick which are 301 then with your compasses take 301. viz. three inches no tenth part of an inch and 1 10 of a tenth part or Centesmes of an inch and set one foot in the nether-most cross stroke where you set the first one and turn the other upward in the same column and there set your 2 likewise with the same numbers set one foot in the middle cross stroke where you set the middle one and turn the other upward toward the uppermost one and there set your 2 also likewise do with 3 whose Logarithme is 477 id est 4 inches 7 tenths 7 Centesmes also with 4. And these figures for the making of this line we will call hundreds the next subdivision tens and the least Centesmes But now because we will suppose your compasses will not well reach beyond the figure 4 whose Logarithme is 602 that is above 6 of those inches therefore first let us set on the tens so far on both feet and then the rest of each foot afterward Next set out each fifth tenth so far because you must mark them with longer strokes then each single ten so then you must not account the next of those fifths to 1 as 5. for then you will account the one for nothing but you must account it for 15. or 150. and so take the Logarithme thereof which is 176. Likewise 25 or 250 is 398 which you must take with your compasses and set in their places in in both feet and in like sort shall you do with all your single tens accounting that next â— not for 1 nor 2 but for 11. Or instead of taking them off with your compasses strike out all the first foot with a fine small striking squire of brass laying it upon the Log. in the line of foot-measure and then set out the other foot with your compasses by this Now for the rest of each foot look out the Logar of your numbers and take the distance between it and the middle cross-stroke and with that wideness set one foot in the upper 1 and where the other falls there is the place of that number Example I would set out 70 the Log. is 845 I take the distance between it and the middle-stroke of the Rule or the Arithmetical complement of it 154 and set it both from the upper stroke and middle-stroke downward and you set out seventy But your over-over-foot may bear unites to 20 and from thence to 40 divide each tenth into five and from thence to the end into two To make the line of sines First you must know that neither the line of sines nor tangents enter the Rule till 35 minutes where you see the two next figures to the characteristick 8 are both ciphers there also the characteristick changeth from 7 to 8 for your characteristick shews what foot you are in therefore since we reckon the minutes onely by tens our first number or division upon the Rule will be at 40 minutes of the first foot shewn by the characteristick 8 for 9 is the last and therefore belongs to the last foot so that whereas you see that the Log. of one minute hath 6 the characteristick 463 the three next figures therefore one minute would be above a foot and half before the entrance on the Rule and likewise would the first minute of the tangents be Now the Logar of 40 minutes hath beside the characteristick 8 the three first figures 066 feré therefore take off 0 inch 6 tenths and 6 centesmes or 5 centesmes and 7 millesmes if you caâ— ghuess so near and set them from the nethermost cross-stroke at the beginning of the line of sines forward And thus do for all under two degrees be it sine or tangent but from thence to sine 5 degr 45 min. or tangent 5 degr 43 min. As suppose the sine of 4 degr whos 's Logar beside the characteristick is 843 you shall take the distance between 8 inches 4 tenths 3 cent and ten inches and apply that distance from the middle-stroke down-ward and so of the rest of the quarter But for all both sines and tangents in this first foot you may by their Logarithmes strike them with a square as you did the line of numbers Now for the upper-part shewed by the characteristick for all sines and tangents to 20 degr as suppose the tangent of 20 degr the Logarithmes of 20 degr tangent is 56 set it from the middle-stroke forward but from thence to the sine of 90 and tangent of 45 degr as the sine of 40 whose Logar is 808 take the distance between it and the middle-cross-line and apply it in the line of sines from the upper cross-stroke down-ward then number all the whole degrees to ten with 1 2 3 and after that in the sines with 20 30 40 c. to 90 and the tangents with 10 20 to 45 and back with 50 60 to 80 degrees Lastly for making the sextant of chords Set a pair of beam-compasses with a beam of willow deal or sallow near half an inch thick and â— 4 broad make a little nut of good tough wood with a mortes in it that the beam may slide in it to and fro indifferently stiff and in all places alike with a short prick or little piece of an aule-blade in one end and another longer in one edge of the beam hard by the end so long from the beam as the other point is If it goeth not stiff enough to stand and tran with at any place make the mortes a little the deeper one way to put in a wedge or else help your self with a screw-pin then go to some smooth loft boards opening your compasses to 23½ inches and with that wideness tran an arch that may be two foot long at the least and with each foot of the compasses make a prick in the said arch and set it likewise upon the Rule then divide that space in the arch into two equal parts which will be 30 degr a piece and each of them into three apiece which will be 10 degr apiece and each of them into two which will be five apiece and each of them into five simple ones Then take them off from the floor and set them on the Rule one after another and number them with 10 20 30 40 50 60 and this will be wonderfull beneficial in Dialling and also in many other things as to divide a circle into any number of equal parts or to make an angle of any number of degrees or to finde the quantity of any angle and so by the line of foot-measure you may also divide a streight line into as many parts as you will Now as I have shewed the use of all the lines on the other side of the Rule and also of both the out-side lines on this side so for the other three I must content my self to shew you the use in general for if I should descend to particulars all the paper in Cambridge would be too little to hold them First therefore you see already that as by the line of foot-measure and Table of Logarithms these lines are made so may you by these lines finde the Logarithme of any absolute number tangent or sine as if it were by the Table of Logarithms Secondly By these two lines of numbers and foot-measure may be resolved all questions whatsoever that common Arithmetick can resolve And more for hereby may be resolved all questions of Interest Purchases Annuities c. Thirdly By these three lines of numbers sines and tangents is resolved the whole doctrine of Triangles and whatsoever may be performed by them either in Measuring Dialling Geography Geometry Arithmetick Navigation Cosmography Astronomy c. But because gentle Reader I would have thee learn now to go alone I will commit these to thine own consideration knowing that that chicken that will peck up never a corn but what the hen puts in the mouth will never be a fat chicken Now if the Rule of three is accounted of all men worthy for its excellency of the name of the Golden-Rule which is but the least part of the use of one of the lines of this Ruler then justly may this Ruler be called the Golden-Ruler FINIS Hom. Odyss 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 v. 230. Horat. Ep. 1. lib. 1. * Herbam porrigere Prov. How to set themselves right in a line To plot a plot taken by graduâ—ed Instruments For finding how high you may set your cock in a house see the last page of this Book Street-earth Soot Salt Pigeon-dung Folding of land Rags and Horn-shavings Malt-dust Lime Ashes Chalk c Burning of queach c.
square inches of the block in that column next save one to the edge then the feet required to make a foot forward in the next then the odd inches in the next to that and the Centesmes in the last of the 4. Then from 8 and ½ to 36 you may take off your inches from your line of inch-measure and your Centesmes from your inch of Centesmes as you did in board-measure and place it backward or forward according as it shall be more or less then a foot CHAP. IIII. Of measuring solids as stone timber c. and first of square timber FOr measuring all kind of solids the fundamental or general way is to multiply the inches of the breadth by the inches of the depth and that product by the inches of the length and divide the last product by 1728. This is so plain it needs no example and this is the best way for stone of all other 2. A second way of measuring square timber is by this Ruler Having the square of the piece given look on the Rule and see how often you finde the length required at that square between that and the end of the Rule in the length of the block so many foot of timber is in that block To finde the true square of a piece broader one way then another But to finde the true square of the piece multiply the breadth by the depth and from the product extract the square-root As let the breadth be eight and the depth 14 these multiplyed make 112 whose square root is 10 1â— 21 according to which square you must measure the piece Which disproveth a common errour which is this To add both sides together and to take ½ thereof for the square for so 8 and 14 make 22 the half thereof is 11. And although there seemes but small difference viz. less then ½ an inch between their numbers or roots 10 12 21 and 11 yet between their squares there is no less then 9 inches difference for 11 times 11 is 131 but 8 times 14 is but 112. 3. Now therefore because every Carpenter cannot extract the square-root and to them that can do it it is but a slow way and thirdly we never set any scales of timber-measure upon Rules but for inches halves and quarters take this for the best way of all other where there is such difference of the sides measure it first that false way then take out of it always a square piece of ½ the difference of the sides quite through the block so in our example 8 and 14 their difference is 6 the ½ thereof is 3 therefore take a piece of 3 inches square through the length of the block for that 3 squared gives 9 which is the difference between the square of it and the rectangle of 8 times 14. CHAP. V. Of round tâ—mber BEcause to every circle there belongeth 3 squares first the square without the circle or the square of the diameter secondly the square equal to the circle not in Peripherie but in the area for if the area of a circle of a mile round and a mile about in a square be compared we shall finde the square to contain just 40 acres whereas the circle of the same Peripherie containeth 50 acres 3 roods 25 poles 5 11 and thirdly the side of the square within the circle therefore we will first shew the manner of making these 4 scales and then the measuring of round timber yet before we shew the making of them our best way is to take Virgil's advice and to do as he doth with his Bees Principio sedes apibus statióque petenda So before we shew the making of them we will first finde out a seat for each of them and then the making of them one after each other First in the beginning of the first chapter we shewed that we would have one of the edges on one side besild off and the rest of that side divided length wise into eight equal columns with 7 Gage-strokes upon the besill ½ the length of the Rule you may set a scale of 20 in the inch dividing each inch into halves and quarters Numbring each half-inch with 10 20 30 c. save that half-inch next the beginning which must not be accounted for any of the tens but that must be divided into ten equall parts by it self to take the odd inches above even ones that any round block or circle is about Besides this you have three other scales that are for round measure that shew the three squares belonging to the circle and any of these four being known all the rest are known onely by taking the number thereof upon its proper scale with your compasses and apply that distance to the scale proper to the thing desired and these three scales for these squares are one for the Diameter or side of a square without the circle and that each side thereof toucheth the circle Another is the side of a square within the circle or of the chords of 90 degr and the other is a side of a square whose content is equal to the content of a circle For Example Let a block be girded about with a nealed wyer and then that wyer laid along upon the block being found to be 88 inches I set one foot of the compasses in 80 of the said circle scale and the other foot in 8 of those 10 odd parts next the beginning of the Rule reckoned from ten upward being the contrary way to the other 80. If then you desire to know the Diameter of the circle or side of the square including the circle you shall finde it just 28 inches by setting one foot of the compasses in 25 of the Diameter scale and the other will fall in three odd parts which added make 28 for all these three last scales must be divided into fives and numbred with 5 10 15 c. and five odd ones above at the beginning Likewise if you apply the same wideness of the compasses to the scale of the square within the circle that is to the square that a block being round will be being hewed just to the four edges then set one foot of the compasses in one of those great divisions by fives so that the other may fall amongst the odd small divisions and it gives you 19¾ feré And lastly if you apply the same wideness of the compasses to the scale for the square equal setting one foot in the great divisions so that the other may fall in the five odd small ones it gives 24 and about â…” And in like manner if any of the other three scales be given as if the Diameter 14 be given if you take 14 upon the Diameter and carry that to the circle it gives 44 if to the square equal it gives about 12â…“ and so of the rest CHAP. VI. Of the proof of these scales by Arithmetical calculation FIrst for the circle-scale that needs no proof so that it be truly divided for that is