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power_n great_a length_n line_n 3,131 5 11.3194 5 false
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A48331 The compleat surveyor containing the whole art of surveying of land by the plain table, theodolite, circumferentor, and peractor ... : together with the taking of all manner of heights and distances, either by William Leybourn. Leybourn, William, 1626-1716. 1653 (1653) Wing L1907; ESTC R20856 115,157 173

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cut off 3 7 parts First draw the line AC making any angle as CAB then from A set off any seven equall parts as 1 2 3 4 5 6 7 and from 7 draw the line 7B Now because 3 7 is to be cut off from the line AB therefore from the point 3 draw the line 3D parallel to 7B cutting the line AB in D so shall AD be 3 7 of the line AB and DB shall be 4 7 of the same line for As A7 is to AB ∷ so is A3 to AD. PROB. XXIV To finde a mean proportionall between two lines given IN the following figure let the two lines given be A and B between which it is required to finde a mean proportionall Let the two given lines A and B be joyned together in the point E making one right line as CD which divide into two equall parts in the point G upon which point G with the distance GC or GD describe the Semicircle CFD then from the point E where the two lines are joyned together raise the perpendicular EF cutting the Periferie of the Semicircle in F so shall the line EF be a mean proportionall between the two given lines A and B for As ED to EF ∷ so EF to CF. As 9 to 12 ∷ so 12 to 16. PROB. XXV How to divide a line in power according to any proportion given IN this figure let CD be a line given to be divided in power as the line A is to the line B. First divide the line CD in the point E in proportion as A to B by the 13 Probleme then divide the line CD into two equall parts in the point G and on G at the distance GC or GD describe the Semicircle CFD and upon the point E raise the perpendicular EF cutting the Semicircle in F Lastly draw the lines CF and DF which together in power shall be equall to the power of the given line CD and yet in power one to the other as A to B. PROB. XXVI How to inlarge a line in power according to any proportion assigned IN the former figure Let CE be a line given to be inlarged in power as the line B to the line G. First by the 13 Probleme finde a line in proportion to the given line CE as B is to G which will be CD upon which line describe the Semicircle CFD and on the point E erect the perpendicular EF then draw the line CF which shall be in power to CE as G to B. PROB. XXVII To inlarge or diminish a Plot given according to any proportion required LEt ABCDE be a Plot given to be diminished in power as L to K. Divide one of the sides as AB in power as L to K in such sort that the power of AF may be to the power of AB as L to K. Then from the angle A draw lines to the points C and D that done by F draw a parallel to BC to cut AC in G as FG. Again from G draw a parallel to CD to cut AD in H. Lastly from H draw a parallel to DE to cut AE in I so shall the plot AFGHI be like ABCDE and in proportion to it as the line L to the line K which was required Also if the lesser Plot were given and it were required to make a greater in proportion to it as K to L. Then from the point A draw the lines AC and AD at length also increase AF and AI that done inlarge AF in power as K to L which set from A to B then by B draw a parallel to FG to cut AC in C as BC. Likewise from C draw a parallel to GH to cut AD in D as CD Lastly a parallel from D to HI as DE to cut AI being increased in E so shall you include the Plot ABCDE like AFGHI and in proportion thereunto as the line K is to the line L which was required PROB. XXVIII How to make a Triangle which shall contain any number of Acres Roods and Perches and whose base shall be equal to any possible number given IF it be required to make a Triangle which shall contain 5 Acres 2 Roods 30 Perches whose base shall contain 50 Perches you must first reduce your 5 Acres 2 Roods 30 Perches all into Perches in this manner First because 4 Roods make one Acre multiply your 5 Acres by 4 which makes 20 to which adde the two odde Roods so have you 22 Roods in your 5 Acres 2 Roods Then because 40 Perches make one Rood multiply your 22 Roods by 40 which makes 880 Perches to which adde the 30 odde Perches and you shall have 910 and so many Perches are contained in 5 Acres 2 Roods 30 Perches Now to make a Triangle which shall contain 910 perches whose base shall be 50 Perches do thus Double the number of perches given namely 910 and they make 1820 then because the base of the triangle must contain 50 Perches divide 1820 by 50 the quotient will be 36⅖ which will be the length of the perpendicular of your Triangle This done From any equall Scale lay down the line AB equall to 50 Perches then upon B raise the perpendicular BD equal to 36⅖ perches and draw the line CD parallel to AB then from any point in the line CD as from E draw the lines EA and EB including the Triangle AEB which shall contain 5 Acres 2 Roods 30 Perches which was required PROB. XXIX How to reduce a Trapezia into a Triangle by a line drawn from any angle thereof THe Trapezia given is ABCD and it is required to reduce the same into a Triangle First Extend the line DC and draw the Diagonall BD then from the point A draw the line AE parallel to BD extending it till it cut the side CD in the point E. Lastly from B draw the line BE constituting the Triangle EBC which shall be equall to the Trapezia ABCD. PROB. XXX How to reduce a Trapezia into a Triangle by lines drawn from any point in any of the sides thereof LEt ABCD be a Trapezia given and let H be a point in one of the sides thereof from which point H let it be required to draw lines which shall reduce the Trapezia into a Triangle First Extend the side which is opposite to the given point namely the side CD both wayes to E and F and then from the point H draw lines to the angles C and D as the lines HC and HD also draw the lines AE and BF parallel to HC and HD cutting the extended line CD in the points E and F. Lastly If from the point H you draw the lines HE and HF you shall constitute the Triangle HEF which shall be equall to the Trapezia ABCD. PROB. XXXI How to reduce an irregular Plot of five sides into a Triangle THe irregular Plot given is ABCDE and it is required to reduce the same into a Triangle First extend the side AE both wayes to F and G and from
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.