in the practise whereof the young practitioner will take much delight and receive no small satisfaction There is also taught how to take the plot of any field or other inclosure severall wayes both by the Plain Table Theodolite and Circumferentor by which will appear what congruity and harmony there is between these severall Instruments for if you take the plot of any field by any one of them and then by another of them and plot your work by the same Scale as both your observations you shall if you be carefull finde that these two Plots will agree together as exactly as if they had been both taken by one and the same Instrument And for this reason I have made one Scheme or figure serve for three severall Chapters which hath much abreviated the number of Diagrams and will I perswade my selfe give better satisfaction to the Learner then variety of figures could have done In the manner of protracting when you have reserved your degrees out by the Needle in the Circumferentor or the Index of the Peractor I have because the practise thereof is very usuall and no lesse difficult in pag 233 inserted a figure so plain and perspicuous that the very sight thereof will be enough if there were no words used to explain the use thereof After the plot of any field is taken and protracted according to any of the former directions I come to shew how the content thereof may be attained severall wayes that is to finde how many Acres Roods and Perches are contained in any field thus plotted Also there is taught how to measure mountanous and uneven grounds and to finde the area or content thereof You are also taught in this fourth Book how to take the time plot of a whole Mannor or of diverse severals both by the Plain Table Theodolite Circumferentor or Peractor with the manner how to keepe account in your Field-book after the most sure and exactest way Also how to reduce your Plot to draw a perfect draught thereof and to deck and beautifie the same And in the last place there is an example of Water-levelling by which you may know whether water may be conveyed from a Spring-head to any determinate place or not Thus have I given you some generall intimation of the principall heads contained in the following Treatise which you may see more aparent in the following Analysis but best of all in the Book it selfe unto which I chiefely refer you wishing that you may take the same delight and pleasure in the practise of those things therein contained as I did in the composing of them so shall I think my labour well bestowed and be the more animated to present thee with some other Mathematical Treatise who am A Friend to all that are Mathematically affected WILLIAM LEYBOURN A GENERAL SURVEY Of the whole WORK The following Treatise is divided into four Books I. Of Geometry which consisteth of 1. Definition page 3. 2. Theorems 10. 3. Problemes concerning 1. Raising and letting fall of Perpendiculars 11. 2. The making of equall angles and drawing of parallel lines 13. 3. The dividing of right lines equally 14. 4. The constituting of right lined figures 16. 5. The working of proportions by lines 17. 6. The dividing of right lines proportionally 18. 7. The dividing of Triangles according to proportion both Arithmetically and Geometrically by a line drawn 1. From any angle 19. 2. From a point in any side 21. 3. Parallel in any side 22. 8. The power of Lines and Superficies 25. 9. The reducing of figures from one form to another as Four Five Six solid figures into Triangles 28. 10. The dividing of any plain Superficies into two or more parts according to any proportion by lines drawn either from any angle or from a point in any side 30. II. Of Instrumēts as 1. In generall 37. 2. Of the Theodolite 39. 3. Of the Circumferentor 40. 4. Of the Plain Table 42. 5. Of Chains and chiefely of Master Rathborns 46. Master Gunters 47. 6. Of the Protractor 50. 7. Of Scales Plain and Diagonall 52. 8. Of a Field-book 53. 9. Of the Parallelogram 54. III. Of Trigonometrie and 1. Of the description and use of the Tables of Sines 57. and Logarithms 63. 2. The application of these Tables as also of the lines of Numbers Sines and Tangents in resolving of Plain Triangles Right angled 74. and Oblique angled 79. IV. The use of Instruments and 1. Of the Scale in taking therefrom laying down lines and angles of any quantity 179. 2. Of the Protractor in laying down finding the quantity of any Angle 182. 3. Of the Plain Table Theodolite Circumferentor to finde an angle in the field therewith 163. 4. Of the Labell thereby to observe an Horizontall line or line of level an angle of Altitude 166. 5. Of taking Distances accessible or inaccessible by the Plain Table 187. Theodolite 189. Circumferent 190. and to protract the same 191. 6. Of the taking of accessible inaccessible altitudes by the Labell and Tangent line 192. and to protract the same 195. 7. Of taking divers distances at once by the Plain Table 196. and Theodolite 198. and to protract the same 199. 8. To take the plot of a Field at one station taken in the middle thereof by the Plain Table 201. Theodolite 203. Circumferentor 205. and to protract the same 206. 9. To take the plot of a Field at one station taken in any angle thereof by the Plain Table 208. Theodolite ibid. Circumferentor 210. and to protract the same 210. 10. To take the plot of a field at two stations taken in any parts thereof by the Plain Table 212. Theodolite 214. Circumferentor 216. and to protract the same 216. 11. To take the plot of a field at two stations taken in any parts thereof only measuring the stationarie distance by the Plain Table 218. Theodolite 220. Circumferentor 220 and to protract the same 222 12. Of Large Champion plains or Woods to take Plots thereof by the Plain Table 223. Theodolite 226. and to protract the same 228. With a way to prove the truth thereof 230. 13. To take the plot of any Field Wood-Park Chase Forrest or other large Champion plain by the Circumferentor 230. And to protract the same 233. With diverse cautions for the exact performance thereof 14. Of the Peractor contrived by Master Rathborn how to make the Plain Table to do the work thereof better then the Peractor it selfe 236. 15. To take the plot of any piece of Land by the Peractor 236. and to protract the same 240. 16. Of finding the Area or superficiall content of any piece of Land the plot thereof being first taken and chiefly of The Geometricall Square 241 The Long Square 242. The Triangle 242. The Trapezia 243. Any irregular plot of a Field 244. The Circle 245. 17. The manner of casting up the content of any piece of Land in Acres c. by Mr. Rathborns Chain 246 Mr. Gunters Chain 249 18.
Vera Effigies Gulielmi Leybourn Philom anno Aetatis 27. THE COMPLEAT SURVEYOR Containing The whole ART of Surveying of Land BY THE Plain Table Theodolite Circumferentor AND PERACTOR After a more easie exact and compendious manner then hath been hitherto published by any the PLAIN TABLE being so contrived that it alone will conveniently perform whatsoever may be done by any of the fore-mentioned Instruments or any other yet invented with the same ease and exactnesse and in many cases much better Together with the taking of all manner of Heights and Distances either accessible or in-accessible the Plotting and Protracting of all manner of Grounds either small Inclosures Champion plains Wood-lands or any other Mountainous and un-even grounds Also How to take the Plot of a whole Mannor to cast up the content and to make a perfect Chart or Map thereof All which particulars are performed three severall wayes and by three severall Instruments Hereunto is added the manner how to know whether Water may be conveyed from a Spring head to any appointed place or not and how to effect the same With whatsoever else is necessary to the Art of SVRVEYING By WILLIAM LEYBOURN LONDON Printed by R. W. LEYBOURN for E. BREWSTER and G. SAWBRIDGE and are to be sold at the signe of the Bible upon Ludgate hill neer Fleet-bridge MDCLIII TO HIS MUCH HONOURED FRIEND EDMVND WINGATE of Grayes Inne Esq SIR THis Treatise being finished and ready to see the light I could not bethink my selfe of a fitter Patron then your selfe to protect it Your knowledge in and affection to the Sciences Mathematicall as also the civill respect which You usually vouchsafe such as affect those Studies arme me with this confidence I foresee that this my presumption in exposing this Work to publique view may meet with some Detractors but Your approbation thereof will both convince them of their Errour and plentifully satisfie me for the pains I have taken therein Howbeit what reception soever it may obtain with the Vulgar my intention I doubt not will give me support and encouragement my ayme therein being nothing else but the publique good and this my Dedication an evidence to let You know how much I am SIR Your most humble and obliged Servant WILLIAM LEYBOURN TO THE READER Courteous Reader ABout three years since there came into the World a little Pamphlet entituled Plaâ—…etria or the whole art of Surveying of Land under the name of Oliver Wâ—…by of which I confesse my selfe to be the Author that name being only the true letters of my own name transposed I was indeed very unwilling the world should know me to be the Author thereof it being so immateriall a Treatise and too particular for a Subject of so large an extent but that was occasioned by over much 〈◊〉 for being urged thereunto it was not above six weeks conceived before it was brought forth and therefore must needs be little lesse then monstrous yet the good acceptance which that pamphlet received occasioned me to prosecute that Subject more at large Now as the opinions of men in the world are various so I knowe this work will be variously censured and therefore it might per chance be expected by some that I should make an apologie for my selfe as to crave pardon 〈◊〉 excusse for whatsoever any man shall be pleased to object against him I mean to make no excuse see I know of nothing that needs it neither did I ever know any Book the more favoured for the Authors bespeaking it besides the subject of the ensuing Treatise being Geometry needeth no such thing for Demonstration the grand supporter thereof is able to with stand all opposers and silently with Lines and Figures to 〈◊〉 the most malevolent tongue or pen that shall 〈◊〉 speak or â—…rice against its But to the judicious Reader I shall say thus much As I dare not think my doings free from all exception so I do not know of any thing herein contained worthily deserving blame Some small oversights which may possibly have crept in by chance I must intreat the friendly Reader to over see or wink at as for the understanding Reader I am sure he will 〈◊〉 to cavill at every â—light mistake or literall fault in the printing as for materiall faults I know of none in the whole Work although I have diligently examined the printed sheets In the following Treatise I have endeavoured to proceed methodically and to insert every particular Chapter as it ought to be read and practised and have omitted nothing that might any way tend to make a man in short time become an exquisite proficient in the Geometricall part of Surveying The first part of this Book consisteth of Geometry only and containeth such Problemes as are meet and necessary to be known and practised by any man that intendeth to exercise himself in this employment by help of these Problemes the plot of any piece of Land may be inlarged or diminished according to any assigned proportion and separation and division thereof made if need be by Rule and Compasse only and also by Arithmetick In the second Book you have a generall description of all the most necessary Instruments used in Surveying as of the Theodolite Circumferentor Plain Table and the like and more particularly of those which I make use of in the prosecuting of this discourse Also I have given such directions for the making of the Plain Table and furnished the Index and other parts thereof with diverse necessary lines for severall occasions so that it being made according to the directions there given it is the most absolute and universall Instrument yet ever invented for by it may be performed whatsoever may be done by the Theodolite Circumferentor or Peractor with the same facility and exactnesse and in many cases better as in the particular uses thereof will plainly appear The third Book is of Trigonometrie or the Doctrine of the dimension of Plain Triangles by Sines Tangents and Logarithms by which the nature and reason of the taking of all manner of Heights and Distances may the better appear and for that reason I have in this third Book added short Tables of Sines and Logarithms namely a Table of Sines to every 10 minutes of the Quadrant and a Table of Logarithms from 1 to 1000 by which more Questions may be resolved in the space of one houre then by the usuall wayes taught by others can be performed in six if the like exactnesse be required And for a further abreviation of these Calculations I have also shewed how to resolve all such Cases in Plain Triangles as may at any time come in use in the practise of Surveying by the lines of Artificiall Numbers Sines and Tangents whereby all such Cases may be resolved without pen ink or paper In the fourth Book is shewn the use of all the fore-mentioned Instruments in the practise of Surveying and first in the taking of all manner of Heights and Distances either accessible or inaccessible
draw the right line GP which shall divide the whole Plot ABCDEF into two parts being in proportion one to the other as the line T is to the line S. PROB. XXXVIII How to divide an irregular Plot according to any proportion by a line drawn from any angle thereof LEt ABCDEFG be an irregular Plot and let it be required to divide the same into two equall parts by a line drawn from the angle A. First draw the line HK dividing the Plot into two parts namely into the five sided figure ABCFG and into the Trapezia FCED then by the 31 Probleme reduce the five sided figure ABCFG into the Triangle HAK the base whereof HK divide into two equall parts in O and draw the line OA which shall divide the five sided figure ABCFG into two equall parts Then by the 30 Probleme reduce the Trapezia FCDE into the Triangle OLM and divide the base thereof LM into two equall parts in the point P and draw the line OP which will divide the Trapezia FCDE into two equall parts and so is the whole Plot divided into two equall parts by the lines AO and OP but to performe the Probleme by one right line only do thus from the point A draw the line AP and parallel thereunto through the point O draw the line ON Lastly if you draw a right line from A to N it shall divide the whole Plot into two equall parts The end of the First Book THE COMPLEAT SURVEYOR The Second Book THE ARGVMENT IN this Book is contained both a generall and particular description of all the most necessary Instruments belonging to Surveying as the Theodolite Circumferentor and Plain Table with all the appurtenances thereunto belonging as the Staffe Sockets Screws Index Label and other necessaries Now whereas these three Instruments are the most convenient for all manner of practises in Surveying I have so ordered the matter that in this Book after the Theodolite and Circumferentor are particularly described as they have usually been made I come to the description of the Plain Table and therein have shewed how that Instrument may be ordered to performe the work of any of the other so that whatsoever may be done by the Theodolite Circumferentor or any other Instrument the same may be effected by the Plain Table onely as it is there contrived with the same ease dispatch and exactnesse and in many respects better as in Chap. 1. doth plainly appear so that this Instrument onely is sufficient for all manner of practises whatsoever And besides the fore-mentioned Instruments for mensuration there is described divers other Instruments belonging thereunto as Chains Scales Protractors and the like all which are described according to the best contrivance yet known A DESCRIPTION OF INSTRVMENTS CHAP. I. Of Instruments in generall THe particular description of the severall Instruments that have from time to time been invented for the practise of Surveying would make a Treatise of it self and in this place is not so necessary to be insisted on every of the inventors in their severall Books of the uses of them having been already large enough in their construction To omit therefore the description of the Topographicall Instrument of Master Leonard Diggs the Familiar Staffe of Master John Blagrave the Geodeticall Staffe and Topographicall Glasse of Master Arthur Hopton with divers other Instruments invented and published by Gemma Frisius Orentius Clavius Stofterus and others I shall immediately begin with the description of those which are the ground and foundation of all the rest and are now the only Instruments in most esteem amongst Surveyors and those are chiefely these three the Theodolite the Circumferentor and the Plain Table Now as I would not confine any man to the use of one particular Instrument for all employments so I would advise any man not to cumber himselfe with multiplicity since these three last named are sufficient for all occasions And if I should confine any man to the use of any one of these Instruments as for a shift any one of them will perform any kinde of work in Surveying yet in that I should do him injury for in many cases one Instrument may make a quicker dispatch and be altogether as exact as another As in laying down of a spacious businesse I would advise him to use the Circumferentor or Theodolite and for Townships and small Inclosure the Plain Table so altering his Instrument according at the nature or quality of the ground he is to measure doth require These three speciall Instruments have been largely described already by divers as namely by Master Diggs Master Hopten Master Rathborne and last of all in Planometria yet in this place it will be very necessary to give a particular description of them again because if any man have a desire to any particular Instrument he may give the better directions for the making thereof For the description which I shall make of these three Instruments in particular it shall be agreeable to those Instruments as they are usually made with some small addition or alteration But when I come to the description of the Plain Table after that I have described it according to the vulgar way I will then shew you a new metamorphosis of that Instrument making it the most absolute and universall Instrument yet ever invented so that having that one Instrument made according to the following directions you shall have need of no other for the due exact and speedy performance of any thing belonging to the Art of Surveying The Plain Table used as the Theodolite For the Frame of the Table being graduated according to that description will be an absolute Theodolite and perform the work thereof with the same facility and exactnesse and whatsoever may be done by the limbe of the Theodolite the same the degrees on the frame of the Table will as well perform The Plain Table used as a Circumferentor Likewise the Index and Sights together with the Box and Needle being taken from the Table and screwed to the Staffe as in the description thereof it is so conveniently ordered will be an absolute Circumferentor and in some respects better then the ordinary one hereafter described because the Sights thereof stand at a greater distance so that thereby the visuall line may be the better directed The plain Table not one but all Instruments And this Instrument as now contrived though it be called the Plain Table only yet you see that it contains both the other and therefore in advising any man to the use thereof chiefely I do not confine him to one but to all Instruments and therefore do not contradict my former expression Besides there is another great convenience which doth ensue by the degrees on the Tables frame for in taking the plot of a field according to the following directions by the Plain Table you may at the same time perform the same work by the degrees on the frame of the Table if at the drawing
from the foot of the farthermost Sight all along the Ruler to the foot of the nethermost Sight and up the side thereof and is numbred from 1 to 90 by 10 20 30 40 50 c. ending at the foot of the furthermost Sight from whence the line proceeded The use of this line of Tangents in taking of Heights is shewed in the fourth Book is used with the Tables of Sines and Logarithms treated of in the third Book without which Tables or something equivalent thereunto this line of Tangents will be of little use therefore it will be convenient to have upon the Index of your Table the lines of Artificiall Numbers Sines and Tangents by which you may work any proportion required very speedily and exactly so that if you be destitute of your Tables these Lines will sufficiently help you There is yet another way by which you may take any altitude or reduce Hypothenusall to Horizontall lines only by Vulgar Arithmetick without the help of Tables by having a line of equall parts divided on the edge of the Index and another line of the same equall parts on the Label by which lines and Vulgar Arithmetick an Altitude may very well be taken Now because I intend only to shew in generall the use of these equall parts I will therefore do it in this place because I shall have occasion to speak no more thereof hereafter The use thereof briefely is thus Now for the reducing of Hypothenusall to Horizontall lines having measured the Hypothenusall line with your Chain the proportion will be As the equall parts cut on the Label Are to the equall parts cut on the Index So is the length of the Hypothenusall line measured To the length of the Horizontall line required I thought good to give the Reader a view of the severall wayes there are to perform these conclusions leaving every man at liberty to use that which he best liketh or all if he please for all the lines may very well be put upon one Instrument without any confusion of lines but the way which I shall chiefly insist upon in the prosecuting of this Work shall be by the line of Tangents as being in my opinion the best of all Now when I come to shew you the use of this line of Tangents with the Tables of Sines and Logarithms in the resolving of Triangles I will also shew you how to perform the same Propositions by the lines of Artificiall Numbers Sines and Tangents and therefore I would advise every man to have these so necessary lines upon his Index Fourthly Unto this Instrument also belongeth a Box and Needle which is to be fastned to the side of the Table by help of two screws so that it may be taken off and put on at pleasure In the bottome of this Box must be placed a Card divided into 360 degrees numbered if you please after the usuall manner from the North Eastward but the Card by which all the Examples in this Book were framed was numbered from the North Westward by 10 20 30 c. to 360 contrary to the common custome There belongeth also to this Instrument a Socket of Brasse to be screwed on the back side of the Table into which must be put the head of the three legg'd Staffe this Staffe ought to be joynted in the middle so that it may be the more portable For the Socket it may be a plain one but a Ball and Socket with an endlesse screw is the best of all for by help thereof you may place the Table or any other Instrument either Horizontall Verticall or in any other position ¶ Note that this Instrument if made according to these directions is the most absolute Instrument for a Surveyor to use CHAP. V. Of Chains the severall sorts thereof OF Chains there are divers sorts as namely Foot Chains each link containing a Foot or 12 Inches and so the whole Pole or Perch will contain 16½ Links or Feet answering to the Statute denomination Some Chains have each Pole divided into 10 equall parts and these are called Decimall Chains and this grosse division may be convenient in some practises The Chains now used and most esteemed amongst Surveyors are especially two namely that generally used by Master Rathborne which hath every Perch divided into 100 Links and that of Master Gunter which hath four Poles divided into 100 Links so that each Link of Master Gunters Chain is as long as four of Master Rathborns Now because these Chains are most esteemed of and used by Surveyors I will therefore make a generall description of them both leaving every man at liberty to take his choise Of Mr. RATHBORNS Chain THe Chain which Master Rathborne ordinarily used as himselfe saith contained in length two Statute Poles or Perches each Pole containing in length 16½ feet which is 198 Inches then each Pole was divided into 10 equall parts called Primes every of which contained in length 19◠Inches again every of those Primes was sub-divided into 10 other equall parts called Seconds so that every of these Seconds contained in length 1 49 50 Inch so that the whole Pole Perch Unite or Commencement as he calleth it was divided into 130 equall parts or Links called Seconds The Chain or one Pole thereof being thus divided at the end of every 50 Links or halfe Pole let a large Curtain ring be fastned so shall you have in a whole Chain of two Perches long three of these Rings the middlemost being the division of the two Poles Then at the end of every Prime that is at the end of every ten Links let a smaller Curtain Ring be fastened By this distinction of Rings the Chain is divided into these three denominations Unites Primes and Seconds whose Characters are these ◯ · · so that if you would expresse 40 Unites 8 Primes and 7 Seconds they are thus to be written 408̇7̇ by which you may perceive that those Figures which have no pricks over them are Unites or Intigers and the figure under the first point Primes and under the next Seconds so also three Unites seven Primes and two Seconds will stand thus 37̇2̇ Besides these divisions Master Rathborn for his own use sewed at the end of every two Primes and a halfe which is a quarter of a Pole a small red cloth and at every seven Primes and a halfe being three quarters of a Pole the like of yellow or other discernable colour which much helped him in the ready reckoning of the several Rings upon the Chain remembring this Rule That if it be the next Ring short of the Red it is two Primes if the next over three if the next short of the yellow seven Primes if the next over eight if the next short of the great halfe Ring it is four the next over six and if the next short of the middle great Ring it is nine and if the next over one ¶ But here is to be noted that if you use this distinction by
of 60 degrees to 400 in the line of Numbers Or extend the Compasses from the Sine of 90 to the Sine of 60 the same extent will reach from 462 to 400 which is the length of the Base BA CASE V. The Perpendicular and angle at the Base being given to finde the Hypothenusall IF the Perpendicular CA be given 231 and the angle at the Base CBA 30 degrees the Hypothenusall BC may be found thus for As the Sine of the angle CBA 30 degrees 9,698970 Is to the Logarithm of the Perpendicular CA 231 12,363612 So is the Sine of the angle CAB 90 degrees 10,000000 To the Logarithm of the Hypothenusall BC 2,664642 ¶ Here because the angle CAB is a right angle or 90 degrees and comes in the third place I therefore only put an unite before the second term and from that second term substract the first term and the remainder is 2,664642 the absolute number answering thereunto is 462 the side BC. By the lines of Sines and Numbers Extend the Compasses from the Sine of 30 degrees to 231 the same extent will reach from the sine of 90 degrees to 462. Or the distance between the Sine of 30 degrees and 90 degrees will be equall to the distance between 231 and 462 which giveth the side required CASE VI. The Hypothenusall and Perpendicular being given to finde the angle at the Base IN the foregoing Triangle there is given the Hypothenusall BC 462 feet and the perpendicular CA 231 feet and it is required to finde the angle CBA the proportion is As the Logarithm of the Hypothenusall BC 462 2,664642 Is to the right angle BAC 90 degrees 10,000000 So is the Logarithm of the perpendicular CA 231 12,363612 To the sine of the angle CBA 30 degrees 9,698970 By the lines of Sines and Numbers Extend the Compasses from 462 to the sine of 90 the same extent will reach from 231 to the sine of 30 degrees Or Extend the Compasses from 462 to 231 the same extent will reach from the sine of 90 degrees to the Sine of 30 degrees which is the quantity of the enquired angle CBA Of Oblique angled plain Triangles CASE VII Having two angles and a side opposite to one of them given to finde the side opposite to the other IN the Triangle QRS there is given the angle QSR 24 degrees 20 minutes and the angle QRS 45 degrees 10 minutes and the side QS 303 feet and it is required to finde the side QR ¶ Here note that in oblique angled plain Triangles as well as in Right angled the sides are in proportion one to the other as the sines of the angles opposite to those sides Therefore As the sine of the angle QRS 45 deg 10 min. 9,850745 Is to the Logarithm of the side QS 303 feet 2,481443 So is the sine of the angle QSR 24 degrees 20 min. 9,614944 the sum of the second and third terms 12,096387 the first term substracted 9,850745 To the Logarithme of the side QR 2,245642 The neerest absolute number answering to this Logarithm is 176 and so many feet is the side QR By the lines of Sines and Numbers The lines of Sines and Numbers will resolve these Triangles by the same manner of work as in the other before For If you extend the Compasses from the sine of 45 degrees 10 min. to 303 the same extent will reach from the sine of 24 degrees 20 minutes to 176 and so much is the side QR Or Extend the Compasses from the Sine of 45 degrees 10 min. to 24 degrees 20 minutes the same extent will reach from 303 to 176 the length of the inquired side In like manner if the angle RQS 110 degrees 30 minutes and the angle QRS 45 degrees 10 min. and the side QS 303 feet had been given and the side RS required the manner of work had been the same for As the sine of the angle QRS 45 degrees 10 min. 9,850745 Is to the Logarithm of the side QS 303 feet 2,481443 So is the sine of RQS 110 deg 30 min. or 69 de 30 m. 9,971588 the sum of the second and third terms 12,453031 the first term substracted 9,850745 To the Logarithm of the side RS 2,602286 The absolute number answering to this Logarithm is 400 and so much is the side RS. ¶ In this case because the angle RQS is more then 90 degrees you must therefore take the complement thereof to 180 degrees so 110 degrees 30 minutes being taken from 180 degrees there remains 69 degrees 30 min. whose Sine is the same with 110 deg 30 min. and being used in stead thereof will effect the same thing By the lines of Sines and Numbers Extend the Compasses from the Sine of 45 degrees 10 min. to 303 the same extent will reach from the sine of 69 deg 30 min. to 400. which is the side RS required Or the Compasses being opened to the distance between the sine of 45 deg 10 min. and 69 deg 30 min. the same distance will reach from 303 to 400 as before CASE VIII Two sides and an angle opposite to one of them being given to finde the angle opposite to the other IN the same Triangle let there be given the side QS 303 and QR 176 together with the angle QSR 24 degrees 20 minutes and let it be required to finde the angle QRS the proportion is As the Logarithm of the side QR 176 2,245513 Is to the sine of the angle QSR 24 deg 20 min. 9,614944 So is the Logarithm of the side QS 303 2,481443 the sum of the second and third numbers 12,096387 the first number substracted from the sum 2,245513 To the sine of the angle QRS 9,850374 The neerest degree answering to this sine is 45 degrees 10 min. which is the quantity of the angle QRS required By the lines of Sines and Numbers Extend the Compasses from 176 to the sine of 24 degrees 20 minutes the same extent will reach from 303 to 45 deg 10 min. the angle QRS Or the distance between 176 and 303 will be equall to the distance between 24 degrees 20 minutes and 45 deg 10 min. CASE IX Having two sides and the angle contained by them given to finde either of the other angles THis Case will seldome come in use in Surveying because the thing required is an angle which are most commonly given they being observed by Instrument and therefore in this place may be omitted partly because the proposition is not wrought by Sines and Logarithms but by Tangents and Logarithms and there is no Tables of Tangents in this Book to work the proportion by Yet those that are desirous to resolve all kinde of Triangles by the proportionall lines may have added to the lines of artificiall sines and Numbers a line of artificiall Tangents and these three lines together will resolve all Cases in Sphericall as well as in plain Triangles For the performance of this Probleme suppose there were given the
on the frame of the Table which supplies the use thereof Thirdly When I mention or make use of the Circumferentor I mean the Index with the Box and Needle screwed to the Staffe ¶ Having thus given you a sufficient description of the severall Instruments and their parts I come now to the use of them shewing how any angle in the field may be measured by any of them And 1. How to observe an angle in the Field by the Plain Table Suppose EK and KG to be two hedges or two sides of a field including the angle EKG and that it were required to draw upon your Table an angle equall thereunto First place your Instrument as neer the angular point K as conveniencie will permit turning it about till the North end of the Needle hang directly over the Flower-de-luce in the Box and then screw the Table fast Then upon your Table with your protracting pin or Compasse point assigne any point at pleasure upon the Table and to that point apply the edge of the Index turning the Index about upon that point till through the sights thereof you espie a mark set up at E or parallel to the line EK and then with your protracting pin or Compasse point or Black-lead draw a line by the side of the Index to the assigned point upon the Table Then the Table remaining immoveable turn the Index about upon the same point and direct the sights to a mark set up at G or parallel thereto that is so far distant from G as your Instrument is placed from K and then by the side of the Index draw another line to the assigned point so shall you have drawn upon your Table two lines which shall represent the two hedges EK and KG and those lines shall include an angle equall to the angle EKG and although you know not the quantity of this angle yet you may by the 1 or 2 Chapters of this Book finde the quantity thereof if there were any need for in working by this Instrument it is sufficient only to give the symetry or proportion of angles and not their quantities as in working by the Theodolite or Circumferentor it is Also in working by the Plain Table there needeth no protraction at all for you shall have upon your Table the true figure of any angle or angles which you observe in the field in their true positions without any farther trouble 2. How to finde the quantity of an angle in the field by the Theodolite Let it be required to finde the quantity of the angle EKG by the Theodolite place your Instrument at K laying the Index on the diameter thereof then turn the whole Instrument about the Index still resting on the Diameter till through the sights you espie the mark at E then screwing the Instrument fast there turn the Index about upon the center till through the sights you espie the mark at G then note what degrees on the frame of the Table are cut by the Index which you will finde to be 114 degrees and that is the quantity of the angle EKG 3. How to finde the quantity of any angle in the field by the Circumferentor If it were required to finde the quantity of the former angle EKG by the Circumferentor First place your Instrument as before at K with the Flower-de-luce in the Card towards you then direct your sights to E and observe what degrees in the Card are cut by the South end of the Needle which let be 296 then turning the Instrument about the staffe the Flower-de-luce alwayes towards you direct the sights to G noting then also what degrees are cut by the South end of the Needle which suppose 182 this done alwayes substract the lesser number of degrees out of the greater as in this Example 182 from 296 and the remainder is 114 degrees which is the true quantity of the angle EKG Again the Instrument standing at K and the sights being directed to E as before suppose that the South end of the Needle had cut 79 degrees and then directing the sights to G the same end of the needle had cut 325 degrees now if from 325 you substract 79 the remainder is 246 but because this remainder 246 is greater then 180 you must therefore substract 246 the remainder from 360 and there will remain 114 the true quantity of the inquired angle and thus you must alwayes do when the remainder exceedeth 180 degrees ¶ This adding and substracting for the finding of angles may seeme tedious to some but here the Reader is desired to take notice that for quick dispatch the Circumferentor is as good an Instrument as the best for in going round a field or in surveying of a whole Mannor you are not to take notice of the quantity of any angle but only to observe what degrees the needle cutteth which in those cases is sufficient as will appear hereafter but in taking of distances by the Circumferentor it is altogether necessary as may appear by the 7 Chap. following and for that reason I have here shewed how to finde an angle by the Circumferentor and also that you might thereby perceive what congruity and harmony there is in all the three Instruments 4. How to set the Index and Labell Horizontall upon the Staffe When you have screwed the Index and sights to the Staffe as a Circumferentor before you put the Labell upon the brasse pin or wier you must hang a line and plummet upon that pin and then put on the Label then move the Index up and down till the thred and plummet hang directly upon a line which is gaged from under the pin all along the Sight and then doth the Instrument stand horizontall or levell which it must alwayes do when you take an altitude therewith 5. How to observe an angle of Altitude The Label which is to be hanged on one of the sights of the Circumferentor as was intimated in the description thereof and the Tangent line on the edge of the Index is only for the finding of angles of Altitude and is therefore only usefull in taking of heights and in surveying of mountanous and uneven grounds The manner how to observe an angle of Altitude by this Label and the Tangent line on the Index is thus Suppose CA to be a Tree Tower or Hill whose height were required Your Instrument being placed at B exactly levell direct the sights thereof towards CA and there fix it hanging the Labell on the farthermost fight upon a pin for that purpose then move the Labell too and fro along the side of the Index till through the sight at the end of the Label and by the Pin on which the Label hangeth you espie the very top of the object to be measured at C then note what degree of the Tangent line is cut by the Labell which suppose 30 and that is the quantity of the angle of Altitude it being equall to the angle CBA Thus by the Rules in this Chapter