THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES JOHN S. Civil & Mechanical Engineer. SAN FRANCISCO, CAL. Cyclopedia of Civil Engineering A General Reference Work ON SURVEYING, RAILROAD ENGINEERING, STRUCTURAL ENGINEERING, ROOFS AND BRIDGES, MASONRY AND REINFORCED CONCRETE, HIGHWAY CONSTRUCTION, HYDRAULIC ENGINEERING, IRRIGATION. RIVER AND HARBOR IMPROVEMENT, MUNICIPAL ENGINEERING, COST ANALYSIS, ETC. Editor-in- Chief FREDERICK E. TURNEAURE, C. E., Dr. Eng. DEAN, COLLEGE OF ENGINEERING, UNIVERSITY OF WISCONSIN Assisted by a Corps of CIVIL AND CONSULTING ENGINEERS AND TECHNICAL EXPERTS OF THK HIGHEST PROFESSIONAL STANDING Illustrated with over Three Thousand Engravings EIGHT VOLUMES CHICAGO AMERICAN TECHNICAL SOCIETY 1908 COPYRIGHT, AMERICAN SCHOOL OF CORRESPONDENCE COPYRIGHT, 1908 BY AMERICAN TECHNICAL SOCIETY Entered at Stationers' Hall, London. All Rights Reserved. Uhrary n i+t . Editor-in-Chief FREDERICK E. TURNEAURE, C. E., Dr. Eng. Dean, College of Engineering, University of Wisconsin Authors and Collaborators WALTER LORING WEBB, C. E. Consulting Civil Engineer American Society of Civil Engineers Author of "Railroad Construction, ' "Economics of Railroad Construction," etc. FRANK O. DUFOUR, C. E. Assistant Professor of Structural Engineering, University of Illinoi American Society of Civil Engineers American Society for Testing Materials HALBERT P. GILLETTE, C. E. Consulting Engineer American Society of Civil Engineers Managing Editor "Engineering-Contracting" Author of "Handbook of Cost Data for Contractors and Engineers," "Earthwork and its Cost," "Rock Excavation Methods and Cost" ADOLPH BLACK, C. E. Adjunct Professor of Civil Engineering, Columbia University, N. Y. EDWARD R. MAURER, B. C. E. Professor of Mechanics, University of Wisconsin Joint Author of "Principles of Reinforced Concrete Construction" W. HERBERT GIBSON, B. S., C. E. Civil Engineer Designer of Reinforced Concrete V AUSTIN T. BYRNE Civil Engineer Author of "Highway Construction," "Materials and Workmanship Authors and Collaborators Continued FREDERICK E. TURNEAURE, C. E., Dr. Eng. Dean of the College of Engineering, and Professor of Engineering, University of Wisconsin American Society of Civil Engineers Joint Author of "Principles of Reinforced Concrete Construction," "Public Water Supplies," etc. THOMAS E. DIAL, B. S. Instructor in Civil Engineering, American School of Correspondence Formerly with Engineering Department, Atchison, Topeka & Santa Fe Railroad ALFRED E. PHILLIPS, C. E., Ph. D. Head of Department of Civil Engineering, Armour Institute of Technology ^ DARWIN S. HATCH, B. S. Instructor in Mechanical Engineering, American School of Correspondence CHARLES E. MORRISON, C. E., A. M. Instructor in Civil Engineering, Columbia University, N. Y. Author of "Highway Engineering." ERVIN KENISON, S. B. Instructor in Mechanical Drawing-, Massachusetts Institute of Technology *f EDWARD B. WAITE Head of Instruction Department, American School of Correspondence American Society of Mechanical Engineers Western Society of Engineers ^ EDWARD A. TUCKER, S. B. Architectural Engineer American Society of Civil Engineers sides. Whatever the number of sides, a sufficient number of diagonals or tie-lines should be measured to divide the area into triangles from which the area of the entire field may be calculated, If N represents the number of sides of a field, there will be required N-3 diagonals or tie- lines, form- ing N-2 triangles. To simplify calculations when tie-lines are used in place of the long diagonals, the following method may be adopted: Measure off Am any fractional por- tion of AE, and An the same fractional portion of AB and measure mn. Then mn will be to EB as A;/?, is to AE or as An is to AB. Suppose for example that Am is T L of AE and An is J^- of AB. There- pj g 15 fore EB is 10 times the length of mn. EXAMPLES FOR PRACTICE. 1. Given the three sides of a field as 5.25, 6.50 and 4.60 chains. Find the area of the field in acres, and square rods. Ans. 1 acre, 31.52 square rods. 2. Given CB=3.65 chains, CD=2.85 chains, C*=C?=0.50 chains and t?=0.65 chains.* Calculate the area of the triangle BCD. Ans. 5.14 square chains area. Off=sets and Tie=Iines. To find the area of a field which is bounded in part by a stream, it is necessary to use off-sets, as follows: Measure the sides of the field in the usual manner and for the irregular boundary run a straight line, as ED, Fig. 16, and calculate the area of the field included between these boundary lines. To this area must be added the area included between the line ED and the irregular boundary. *NOTE; In all problems involving the measurement of land, the chain referred to is the Gunter's chain of 66 feet, unless otherwise noted. 27 PLANE SURVEYING To find this area, at points along ED, erect perpendiculars to the irregular shore line at such distances that the lines 1' 2', 2' 3' etc., may be considered straight. The desired area will evidently equal the sum of the areas of the trapezoid? thTi3 formed. The distance from E to any point 1, 2 or 3 on ED is called the abscissa of that point and the perpendicular distances from ED to 1' 2' 3' etc., are called the ordi nates of the point. Pig. 16. Instead of summing the trapezoid as above, the desired aroa may be found by the following rule: Multiply the difference between each ordinate and the second succeeding one by the abscissa of the intervening ordinate. Multiply also the sum of the last two ordinates by the last abscissa; one-half of the algebraic sum of these several products will be the area required. To lind the area of an inaccessible swamp, a lake or other area, run a series of straight lines entirely enclosing the given area, and since the diagonals cannot be measured, measure tie-lines either inside or outside of the area. As already stated, calculate the area included between the straight boundary lines and from this area substract the area included between off-sets let fall from PLANE SURVEYING 21 points upon these boundary lines. Reference to Fig. 17 will make the method of procedure plain. Surround the inaccessible area by straight lines, AB, BC, CD, etc., and calculate the enclosed area. At proper intervals along these straight lines, erect and measure perpendiculars ex- tending to the edge of the inac- cessible area. Compute the area between these perpendiculars by the rule on page 20 and for the required area, subtract it from the area previously found. Since the long diagonals are not accessible, measure the area by measuring the interior tie-lines ; remembering that the required number of tie-lines will be less by 3 than the number of sides enclosing the area. Example. Given the dis- tances measured along the straight line AB(Eig. 18) with the corresponding off-sets measured to the broken line ACDE. It is required to compute the area between A B and the broken line ACDE. Fig. 18. Difference of 1st and 3rd ordinates = 0' 55'= 55' etc. " 2nd " 4th =40' 35'=+ 5' " 3rd " 5th =55' 18'=+37' " 4th " 6th =35' 40'= 5' " 5th " 7th " =18' 60'=^2' Sum of last two ordinates = 40 / +60'= lOCX PLANE SURVEYING Abscissa of intermediate ordinates between 1st and 3rd= 40* X 55'=-2200 2nd " 4th= 90' X+ 5'= 450 3rd " 5th=132'X+37'-= 4884 4th " 6th=172'X 5'=- 860 5th " 7th=217'X 42'=-9114 last ordinate = 267' XlOO'=26700 One-half the algebraic sum of the products as given above will give the required area. 32034 12174 Area = ~ = 9930 square feet. EXAHPLE FOR PRACTICE. 1. Given the distances measured along the straight line AB Fig. A with the corresponding off-sets measured to the broken line ACDEFB, to find the area between AB and the broken line ACDEP^B. Check the result by calculating the areas of the trapezoids and triangles of the figure. Ans. 11,875 square feet. 38- Keeping the Field Notes. In keeping field notes, clearness and fullness should be constantly kept in mind. As field notes often pass into the hands of a second party, they should admit of but one interpretation to a person at all acquainted with the nature of the work. Extra time spent in the field in acquiring data will avoid confusion and vexatious delays when the notes are worked up in the office. Avoid the habit of keeping notes upon scraps of paper or in vest-pocket note books. Provide note books especially adapted to the keeping of field records and number and index them so that the contents may be understood at a glance. Remember that sketches made upon the ground aid materially in interpreting field notes that otherwise might be unintelligible. There are three principal methods of keeping field notes; first, by sketches alone; second, by notes alone; and third by full notes RODMAN OF A PLANE-TABLE PARTY AT A STATION The divisions on the telemeter rod. read between the cross-hairs of the surveyor's telescope, indicate the distance of the rod from the instrument. PLANE SURVEYING 23 850 750' supplemented by sketches. The third method is without doubt the best, but examples of the others will be given. For keeping the notes of the chain survey there should be provided what is known as a field book, a pencil (preferably 411), rubber eraser and and a short rule for drawing straight lines. First* by Sketches Alone. Either page of the note book may be used for sketching but it will be more convenient to use the right- hand page, as it is ruled into squares, thus permitting sketching to scale. Always sketch in the direction of the sur- ,.. vey, beginning at the bottom of the page and making the center line of the page correspond approximately with the North and South lines. Second, by Notes Alone. Use the left- hand page of the note book beginning at the bottom as before. Do not crowd the notes, and if necessary use two or more pages. See Fig. 19. Fig. 19 shows the method of keeping the notes of the survey shown in Fig. 20 400' 330' -B- 500' 470' 420' 695' 1065' 470' 400' 805' 740' 070' 600' 475' 415' Fig. 19. Third, by Notes and Sketches. It is apparent that in this method both the first and second methods are embodied in the notes. THE VERNIER. The vernier is an auxiliary scale for measuring with greater precision the spaces into which the principal scale is divided. The smallest reading of the vernier, or the least count, is the difference in length between one division on the main scale and one on the vernier. A vernier is said to be direct when the divisions on the 31 24 PLANE SURVEYING vernier are smaller than those on the main scale Fig. 21 A; retro- grade, when the divisons on the vernier are greater than those on the main scale. See Fig. 2 IB. In Fig. 22 let MM represent a scale divided into tenths; then since ten spaces on the vernier W are equal to nine spaces upon the scale, it is evident that each space upon W is short by one- tenth of a space of MM. The least count is there- fore, T V of T V or T J . The vernier and slow motion screw of the ver- tical arc of the engi- neer's transit are attach- ed to the left hand standard of the instru- ment. Fig. 23 represents a vernier as applied to an engineer's transit. It will be noticed that the main scale is divided so A Fig. 20. as to read directly to 30 minutes. The vernier is so divided that 29 spaces upon the main scale equal 30 spaces upon the vernier, therefore the least count of the vernier is -J^ of 30 minutes or 1 minute. A Mill 1 1 1 1 1 1 1 1 1 1 1 ^ Fig. 21. It will be apparent, therefore, that the readings are taken in the direction of the increasing graduations of the main scale. Thus, for example, in Fig. 23, it will be noted that the zero a. PLANE SURVEYING 25 has passed the 156th space on the main scale, and is near the 30 minute (half degree) division &, therefore the coinciding lines of the vernier and main scale must be between and 30', and we 123456789 M M 234567 89 10 Fig. 22. find them, by looking along the scale of the vernier, at 17 minutes hence, the reading is 156 00' + 17'= 156 17'. Fig. 23. Fig. 24 represents another method of division of the circle of the transit. The vernier is double, and the figures on the vernier are inclined in the same direction as the figures on the scale to which they belong. *,0 Fig. 24. It will be noticed that the main scale reads directly to 20 minutes and that the vernier is so divided that 39 spaces upon the scale correspond to 40 spaces upon the vernier. The least count of the vernier is therefore -^ of 20 minutes or |^ of 1 minute equals 30 seconds. To read the inside scale, it will be noticed the zero of the PLANE SURVEYING vernier is beyond the 138 mark and about half way between the first and second 20' divisions. The reading so far is then 138'20'. Now look along the vernier to the right until a line upon the ver- nier is found that seems to be a prolongation of a line upon the scale. This occurs at the division marked 10 upon the vernier so that the reading is 138 20' + 10' or 138 30'. For the outside scale, the zero of the vernier is beyond the 221 mark and about half way between the first and second 20' divisions. The reading so far is therefore 221 20'. Now look along the vernier to the left as before, and the divisions coin- cide at the division marked 10 upon the vernier, so that the read- ing of the outside scale is 221 20' -f 10' or 221 30'. The sum of the readings of the two scales equals 300 as it should. EXAMPLES FOR PRACTICE. 1. Determine the least count of the vernier in Fig. A, 39 spaces upon the scale, being equal to 40 spaces upon the vernier. 2. Determine the least count of the vernier in Fig. B, 59 spaces upon the scale being equal to 60 spaces upon the vernier. The figure represents what is called 2k folding vernier. To read it follow along the vernier in the usual way until the division marked 10 is reached. If there are no corresponding lines, then go back to the other end of the vernier beginning with the other 10 mark and follow it back toward the center of the vernier. 3. Determine the least count of the vernier of Fig. C, which represents the usual method of dividing the vertical circle of the transit. The Level Bubble is one of the most important attachments of an engineering instrument, and an instrument otherwise good may be rendered useless by imperfect level tubes. The spirit level is a glass tube nearly tilled with a mixture of ether and alcohol, the remaining space being occupied with the vapor of ether. Alcohol alone has not proved satisfactory as it is too sluggish in its movements, thereby rendering an instrument lacking in sensitiveness. If the tube were perfectly cylindrical, the bubble would occupy the entire length of the tube, when hor- izontal, or when slightly inclined to the horizon, thus rendering it impossible to tell when the tube is in a truly horizontal position. 34 PLANE SURVEYING The tube is, therefore, ground on the inside so that a longitudinal section is a segment of a circle. If the tube is not ground to an an even curvature the bubble will not travel the same distance for every minute of arc to the extreme ends of the tube, and an other- wise perfect instrument will not work well. **, B Fig. C. A line tangent to the circular arc at its highest point, as indi- cated by the middle of the bubble, or a line parallel to this tangent, is called the axis of the bubble tube. This axis will be horizontal when the bubble is in the center of the tube. Should the axis be slightly inclined to the horizontal, the bubble will move toward the higher end of the tube, and the movement of the bubble should be proportional to the angle made by the axis with the horizontal. Therefore if the tube is graduated, being a portion of the circum- ference of a circle, with a radius so large that the arc of a few sec- 35 28 PLANE SURVEYING onds is of an appreciable length, it will be possible to determine the angle that the axis may make at any time with the horizontal, provided the angular value of one of the divisions of the tube is known. This is done by noting by how many divisions the center of the bubble has moved from the center of the tube. Since divisions of uniform length will cover arcs of less angu- lar value as the radius of the tube increases, and since a bubble with a given bubble space will become more elongated as the radius is increased, the sensitiveness of the bubble is proportional to the radius of curvature of the tube and the length of the bubble. The length of the bubble, however, will change with changes in tem- perature, becoming longer in cold weather and shorter in warm weather. This is due to the fact that the liquid in the tube expands and contracts more rapidly than the glass. If the bubble contracts excessively, the sensitiveness is thereby impaired, and it should be possible to regulate the amount of liquid in the tube. This is done by means of a partition at one end, having a small hole in it at the bottom. A bubble should come to rest quickly, but should respond easily and quickly to the slightest change of inclination of the tube. To determine the radius of curvature of the tube, proceed as follows: Let S = length of the arc over which the bubble moves for an inclination of 1 second. Let R = its radius of curvature. Then S: 27rR :: 1" : 360. From which R = 206265x8 Or = S = 20b2bo S may be found by trial, the level being attached to a finely divided circle. Or, bring the bubble to the center and sight to a divided rod; raise or lower one end of the level and again sight upon the rod. Call the difference of the readings />, the distance of the rod fl, and the space which the bubble moved S. Then from approx- imately similar triangles d S EXAMPLE FOR PRACTICE. 1. At 100 feet distant, the difference of readings was 0.02 foot, and the bubble moved 0,01 foot, What is the radius of the bubble tube ? Ans . 50 f eetf PLANE SURVEYING 29 Locke's Hand Level. This instrument consists of a brass tube six inches long with a small level mounted on its top at one side of the center near the object end. See Fig. 25. Underneath the level is an aperture across which is stretched a horizontal wire attached to a frame. This frame is made adjustable by a screw and a spring working against each other, or by two opposing screws placed at the ends of the level mounting. In the tube, directly below the level, and at 45 to the line of sight, is placed a totally reflecting prism acting as a mirror. The images of the bubble and wire are thus reflected to the eye. The prism divides the section of the tube into two halves, in one of which is seen the bubble and wire focussed sharply by a convex lens placed in the draw tube at the eye end of the instrument, while the other permits of an open view. Putting the instrument to the eye and raising and lowering the object end until the bubble is bisected by the horizontal wire, natural objects in the field of view can be seen through the open half at the same time, and approximate levels can then be taken. To prevent dust and dampness from entering the main tube, both the object and the eye ends are closed with plain glass. There are two adjustments necessary in this instrument: First, the bubble tube: it should be so adjusted that the bubble will be in the center of the tube when the instrument is horizontal. Second, the horizontal wire; it should bisect the bubble when the latter is in the center of the tube. The methods of executing these adjust- ments are so apparent it will be unnecessary to dwell upon them here. The instrument is intended to be carried in the pocket and is of especial value upon reconnaissance surveys, and for sketching in topography upon preliminary surveys. For topographical purposes, the topographer should provide a rod about eight feet long, divided into foot lengths, the divisions 30 PLANE SURVEYING painted alternately red and white. Upon this rod, the topographer should mark by a notch or other means, the height of his eye above the ground. Standing then upon a station of the line of survey, the topographer directs his assistant to carry the rod out upon either side of the line and in a direction at right angles thereto, until a point having the proper elevation above or below the center line is found as determined by the topographer holding the instru- ment in a horizontal position at the eye. The topographer then paces the distance, while the assistant carries the rod to the next point. It is evident that if the line of sight from the instrument coincides with the mark upon the rod, the two points upon the ground are at the same level. If the line of sight strikes the rod, say one foot below the mark upon the rod, it is evident that the ground where the rod is held is one foot higher than \vhere the instrument is held. These operations can be repeated indefinitely and made to extend as far as necessary upon either side of the line. The points of proper and equal elevation are then connected form- ing contour lines, but the topographer should fill in details by the eye. The methods of keeping the field notes will be illustrated and described later. Let BCXDEFG and H, Fig. 26, represent the successive rod readings on the right of the center line A, and B' C' D' E' F' G' the readings on the left. Now suppose the leveler stands with a Locke- level at zero and the rod is held vertically at B. The line of sight ab bisects the rod at 8.6 feet. The distance from the ground to the observer's eye is 5.5 feet. Thus it is apparent the elevation at B will be 3.1 feet lower than at A. The observer now paces the dis- tance between A and B, and finds it to be 50 feet. The reading is now taken at C on the line of sight cd which reads 6.2 feet, hence the elevation of C is .7 foot lower than B, and the distance be- tween 20 feet. Suppose an attempt is made to take a reading near D. Since the horizontal plane from the observer's eye to the ground does not strike the rod, it is apparent that the rod is too faraway, therefore it should be moved back^to a point X where the horizontal plane ?f will bisect the rod at some division. The ele- vation of X having been ascertained, pace the distance CX, as in the former cases. This method is continued until II is reached, taking the rod readings at g A, ij, k I, m u, and s y and pacing 38 PLANE SURVEYIKG 31 the distance between each. The same method is used on the left-hand side of the center line. However, where the surface of the ground has an abrupt change between stations, it is customary to take cross sections at such changes and ascertain the distances between the sta- tions by pacing; the center line at such points is accepted as zero ; in the same manner perform the operation as if at a station. Where a cross road inter- sects the center line or any portion of the cross section, take readings at places that show an abrupt change, as the top of a bank, side of the road, or gutter, center ^ \ of the road and on the other side in the same way and place as before. This rule holds good in places where small streams are situated. It is not necessary to find the depth of the water, because the pur- pose of the cross section deals solely with J CENTER LINE the surface. Where obstacles prevent the section being run at right angles to the center line, use the method of off -sets and secure the desired elevation as closely ap- proximate as circumstances will permit. The Abney Hand- Level and Clino- meter. This instrument is similar to the Locke hand-level, see Fig. 27, but the small spirit level mounted on top can be moved in the vertical plane and is clamped to a dial graduated upon one side into single degrees and upon the other into slope ratios, so that it is possible to meas- ure angles of slope. The adjustments of the instrument are the same as for the Locke hand-level. The instrument can be used in the field PLANE SURVEYING in the same manner as the Locke hand-level, but is of more universal application. It is of especial value upon steep slopes when the effi- ciency of the Locke level would be limited by the length of rod. In using the Abney instrument it is only necessary to mark the height of the eyes upon the rod. In sighting upon the rod, with the horizontal line coinciding with the mark upon the rod, move the vertical circle until the bubble is in the center of the tube. Read the vertical angle, and the tangent of this angle multiplied by the horizontal distance to the rod will give the difference of elevation. If the distance to the rod is measured along the slope of the ground, multiply this distance by the sine of the vertical ano-le to get the difference of elevation. The most satisfactory method of using this instrument is in connection with a straight edge from 8 to 10 feet in length. The straight edge is laid upon the ground parallel to the direction of slope and the clinometer is then applied to it, the vertical circle being turned till the bubble is in the center. The angle of slope is then read, or better still, the slope ratio is read from the vertical circle. This operation is repeated at every change of slope, the distances being either paced or measured with a tape. For in- stance, suppose the slope is found to be GO feet in length and the slope ratio as given by the clinometer is ^. It is evident then that at the end of the slope the difference in elevation will be 6 40 PLANE SURVEYING 33 feet. The instrument is sometimes fitted with a small compass and a socket for use upon a tripod or Jacob staff. The Leveling Rod is an important part of the leveling outfit; it is used in measuring the vertical distance between the horizontal plane through the line of sight and the point upon which the rod is held. There are three forms in common use known as the New York, Philadelphia and Boston. They are made of hard wood 6^ feet long, sliding out to 12 feet and provided with target, vernier and clamps. Leveling rods are of two kinds, the target and the self -reading. Of the target rods, the New York and Boston are generally used for precise work. Of the self-reading rods, the Philadelphia shown in Fig. 28 is in more common use. The self-reading rods are used only in connection with that class of work where approximate accuracy only is required; this form is generally read to hundredths of a foot and can be read directly from the instru- ment by the observer without the aid of the tar- get, as is suggested by the name. However, with the aid of the target this rod can be read to thousandths of a foot approximately. The target is used when greater accuracy is re- quired and when the rod is so far from the instrument that it cannot be distinctly read. The rod consists of a graduated scale di- vided into feet, tenths and hundredths of a foot, and when properly made, readings to thousandths of a foot can be easily taken. Fig 28 The numbers making the tenths should be 0.06 foot long and so placed that one-half the length is above and one-half below the line. The numbers marking the feet are 0.10 foot long and each is bisected by the foot mark. 41 34 PLANE SUEVEYING This class of rod is painted white, the foot graduations are red and the tenths and hundred ths are black horizontal lines. No attempt will be made to describe the reading of the vernier of either the New York or Boston rod, but the Philadelphia rod is so divided as to make its reading easily understood. With this rod each side of the black horizontal line indicates lOOths, that is, the lower side of the first black space is called "one," and the upper side of the same space is called "two," the lower side of the third space is called "three" and so on until the tenth is read. The reading is taken without the aid of the target, in feet, tenths and hun- dredths as the case may be. The mov- able target has a vernier which reads to thousandths of a foot and is read from zero to ten. To read this rod, move the target to any convenient place on the scale of the rod and note where the vernier at zero coincides with a black horizontal line ; then note where a line of the vernier coincides with a line of the scale. For example, if the zero of the vernier is just above one foot, four- tenths and five hundredths, as shown in Fig. 29, and a line of the graduation of the vernier coincides at 7 with a hori- zontal black line on the rod, the reading will be 1.457 as is shown in Fig. 29. If reading to the nearest 100th, the reading will be 1.46. This is because the 7 naturally brings the zero .002 above the line of graduation on the rod, therefore, the zero of the vernier is .002 nearer the 6 than the 5, hence, the reading is as above. Should the vernier read .002 instead of .007 the reading would be 1.45. It is apparent that .002 now brings the zero of the vernier below the line, hence it will be nearer to the 5 than the 6, thus the Fig, 29. LEVELER AND RODMAN ENGAGED IN BASCULE BRIDGE CONSTRUCTION OVER CHICAGO RIVER, CHICAGO, ILLINOIS PLANE SURVEYING 35 rod reading is 1.45. Therefore, in all readings with the Phila- delphia rod, read the thousandths to the nearest half hundredth. This is true whether or not the lines coincide. These readings apply only to the face of the rod or to 6^ feet. When the rod is extended to 12 feet, or any fractional part thereof, the reading is a little different, both as to its graduation and vernier. The scale, of course, is the same on the face of the rod when extended, except as to the vernier, which is placed on the back at 6^ feet and the scale of graduation on the ex- tended part of the rod is also on the back of the extension which runs through the ver- nier, as shown in Fig. 80. The scale of hundredths is the only part to be particularly A> observed, together with the vernier in the former. For example, the first horizontal black space equals "one," which is the top line of the foot mark. The lower side of the first black space is "two," and the upper side of Fig. 30. Fig. 31. the same space is "three" hundredths, and so on until the tenth is reached. The tenth and feet are placed the same as on the face of the rod. The vernier, as already stated, is a little different in point of reading and is graduated from ten to zero, instead of zero to ten, as on the movable target. However, with some recently- made rods of this type, the scale and vernier reading is the same throughout. See Fig. 31. The graduation at ten is taken as the zero in determining thousandths. The vernier in question is firmly attached to the upper end of the rod 6^ feet, (and the extension of the rod runs through this vernier). The differences in graduation of the two sides should be carefully noted. The rod has two clamp screws, one attached to the movable target and the other near the vernier on the back of the rod. In running the rod, it is customary, where a target or rod reading exceeding 6i feet is 43 PLANE SURVEYING desired, to set the target at 6| feet and run the rod to its full length, then move down as signalled; where no target reading is required, run the rod to its full extent (12 feet) and as the face of the rod has a scale throughout, the reading can be taken from the instrument. Should the instrument not be near enough to enable the leveler to see the rod distinctly without the aid of the target, he should first read the rod through the telescope of the in- strument and then notify the rod- man at what distance the intersection of the cross-hairs in the instrument approximately bisects the rod, such as 3.21, which means three feet, two tenths and one hundredth. The rod- man then sets his rod to read this distance and another sight is taken, being careful to have the rod plumb. Should the intersection of the target fail to coincide with the cross -hairs in the instrument, the leveler then signals, or calls out if sufficiently near to do so, the true rod reading, as up a tenth, down two hundredth^, as the case may be, and the target is placed at this distance. When pre- cision is required, this method is relied upon only for the approx- imate placing of the target; the method used in this case is to slowly move the target by standing behind the rod and holding it between the thumb and fingers of one hand, while the target is moved with the other. Then the target is slowly moved by the signals of the observer. When a slow motion with the hand above the shoulder or below the hip is made by the observer, it means that the rod is to be moved in that direction a fractional part, as one tenth, but when a quick motion is made and the hand drawn back in the same manner it implies that the target is to be moved just a trifle. In this way and by proper attention to the signals of the observer, the rodman can become an efficient and helpful assistant, Fig. 32. PLANE SURVEYING 37 thereby saving mncli time. "When the target is finally set, the rodman reads the rod and calls out the reading to the observer, when within reasonable distance. The target rods are read en- tirely by the rodman, and the readings are kept by him in a note book for that purpose ; these notes should be given to the observer at every opportunity and results checked. To obtain correct re- suits when leveling, it is absolutely essential that the rod be ver- tical and the rodman should remember to hold the rod in this position. 45 38 PLANE SURVEYING The observer or leveler, by means of the vertical wire upon the target of an ordinary leveling rod, can tell whether or not a rod is vertical and in a position at right angles to the line of sight, but he is not able to determine whether the top of the rod is in- clined towards the instrument or in the opposite direction; because when looking through the telescope of a level he can see only a fractional part of the rod. Therefore the necessity of overcoming this difficulty led to the invention of the bent target which obviates this latter trouble as can readily be seen from Fig. 32. The American target fulfills the same requirements, but differs from the ordinary target in having two discs, one behind the other, as in Fig. 33. The principle of construction of this target is ex- tremely simple, and may be best explained in the figures above Suppose a target of the old kind, which in its front view looks exactly like the front view of the new target in A, to be cut along the vertical lines an, ll>, thus dividing it into three parts; that is, one center-piece and two wings. Suppose furthermore, the centerpiece to remain in its former place at the front of the rod, while the two wings are removed to the rear of the rod. Then the result evidently will be that the horizontal line ec, d. Fig. 46. of the vertical axis of the instrument. Turn the screws until the bubble appears along the graduations of the bub- ble tube and bring it to the middle, then turn the telescope at right angles to these two screws and over the other two. In like manner perform the same operation as before. This will cause the bubble to run away from its former position; bring the bubble ac- curately in the center of the tube over these screws, that is, hav- ing equal spaces on each side of the zero of the scale, then turn the telescope over the former screws and bring the bubble in the center. Do this several times until the bubble remains stationary at any angle the telescope may turn; to test this, turn the telescope half way around and see if the bubble moves; should it remain stationary, the instrument is level. The level is, with few excep- tions, never placed in line (except when being adjusted under the PLANE SURVEYING 55 peg method). It is usually placed in some convenient spot where the greatest number of horizontal sights can be secured. As al- ready stated, the tripod legs must be so placed as to make the plates horizontal. This will save time in bringing the bubble in its proper position. Should it be required to set up the instru- ment on the side of a hill, place one leg at an altitude and the other two in apparent line with each other (see Figs. 45 and 46), but where the tripod is adjustable the proper method is apparent. After the instrument is set up and leveled, focus the eye- piece upon the wires and focus the object-glass on the rod by means of the screw placed for that purpose on the top or side of the telescope Care should be taken not to take a reading until the bubble has been carefully observed and brought in the exact center of the bubble tube. When this is completed, sight through the telescope and note the rod reading or set the target rod; again look at the bubble and see if it has moved away from its former position; if not, again -sight on the rod and see if the first observa- tion was correct. Should the intersection of the cross-hairs fail to coincide with the horizontal and vertical lines of the target or the center of the rod, the rodman is to incline the rod by the sig- nals of the observer, until it coincides or is in line of collimation. Care of the Instrument. This duty properly belongs to the instrument man or leveler, and the requirements should be thor- oughly understood. While in the field, the instrument remains on the tripod and is carried from place to place as the work re- quires, but when taken any distance, such as on railway trains, street cars, etc., it should be carefully placed in the box and car- ried by one who is capable of giving it proper care and attention. The instrument man being responsible for the instrument, it is natural, and perhaps best, that he should always carry the in- strument. In fact, the greatest amount of precaution should be exercised in the care of the instrument, both in the field and while conveying it. Instruments in general, the level in particular, should never be unduly exposed to the rays of the sun, as this will have a tend- ency to throw its various sensitive parts out of adjustment, there- fore, whenever possible, place the instrument, whether it is on the tripod or not, in the shade. 63 56 PLANE SURVEYING The leveler should always exercise great care not to disturb the instrument after it is set up and should avoid, as much as possible, walking around it unreasonably, especially if the ground is soft, or the position of the instrument not very firm. This ap- plies to all persons whether in the active performance of duty or not. It is frequently necessary to set up the instrument in places such as loose timber, rocks, etc., thus the importance of this care is apparent. If disturbed to any great extent it will be necessary to relevel it, and if the position of the legs of the tripod is disturbed, the entire work must be done over, because the height of the instru- ment will not be the same as in its former position. Should the in- strument be disturbed after a turning point has been established and its elevation ascertained, it will only be necessary to take a' reading on the last turning point to determine the new height of the instrument. After leveling, the instrument man should keep his hands off the instrument except for the purpose of leveling and adjusting the telescope. He should not make a practice of leaning his weight on the tripod. It is often necessary to send instruments great distances, and in so doing, in no case should it be sent by express or freight without first being properly packed and secured against breakage; because of its- fine construction and sensitive- ness it may get out of adjustment to such an extent as to render it impossible to readjust it for good work by any method known to the engineer, and may become worthless and beyond repair even to an instrument maker. The student should appreciate that the care of the instrument is just as important to good work as its original excellence. Leveling. To determine the difference in elevation between two points, both of which are visible from a single position of the instrument, set up the instrument in such a position that the rod held upon either point will be visible. Now send the rod to one of the points as at A in Fig. 48; direct the telescope upon it and take the rod reading; now direct the telescope to the rod held at B and again note the reading. Evidently the difference of the rod will give the difference in elevation of the two points. If the points are too far apart or if the difference of elevation is too great to be determined from one setting of the instrument, intermediate points must be taken. For instance, suppose it is 64 PLANE SURVEYING 57 desired to find the difference of elevation of A and C in Fig 47, C being too far below A to permit of being read upon both points from a single position of the instrument. Set up the instrument (not necessarily on line from A to C) in some position such that the line of sight will strike the rod as near its foot as it is pos- sible to take a reading: send the rod to some point B such that the line of sight will strike the rod near the top when extended. The difference of these rod readings will give the difference of level of A and B. Now carry the instrument to some point such that rod readings can be taken upon B and C. The difference of the rod readings upon B and C added to the difference of rod read- Fig. 47. ings upon A and B will give the difference of level of A and C, proper attention being given to signs. If the line of levels is very extended the above method is awkward, as some of the differences will be positive and some negative. Choose some plane called a datum plane, such that all of the points in the line of levels will lie above it. Beginning at the point A, assume the elevation of the point above the datum plane. Read the rod held upon A, and the read- ing added to the assumed elevation will give the height of the cross-hairs above the datum plane, called the "height of instru- ment" (II. I.). Now, turn the instrument upon the point B and read the rod and it is evident that this last rod reading subtracted from the height of instrument will give the elevation of B above the datum plane. Next move the instrument beyond B, or at least where it can command a view of B and C and again sight to the rod held upon B. This last rod reading added to the elevation of B will give the new height of instrument from which if the rod 65 58 PLANE SURVEYING reading at C is subtracted will give the elevation of C above the datum plane. Fig. 48 will make the method of procedure apparent. Eeferring now to that figure, the first rod reading taken upon the point A is ordinarily called a "back-sight" and the first read- ing taken upon B is called a "fore-sight". There seems to be no crood reason for adhering to this method of distinguishing between the rod readings and it is illogical and misleading. A back sight is not necessarily taken behind the instrument, that is, in a direc- tion contrary to the progress of the survey, neither is a fore-sight ~5 \Datum Plan& Fig. 48. necessarily taken in front of the instrument. It is more logical and less misleading to designate these rod readings by the terms -plus-sight" and "minus-sight". A plus-sight, therefore, is one taken upon a point of known or assumed elevation, to determine the height of instrument. A minus-sight is one taken upon a point of unknown elevation and which, subtracted from the height of instrument, will give the required elevation. A "bench-mark" (B.M.) is some object of a permanent character, the elevation of which, together with its location, is accurately determined for future reference and for checking the levels. A "peg", "plug", or turning point" (T.P.), is a point used for the purpose of changing the position of the instrument. This PLANE SURVEYING 59 turning point may be taken upon a bench-mark, but is oftener taken upon the top of a spike or stake driven into the ground. If a self-reading rod is used, the instrument man will carry the notebook and record the rod readings as they are observed. The leveler should cultivate the practice of calculating the ele- vations of his stations as the work progresses, thereby enabling him to discern errors when they occur. If a target rod is used upon the work, the rodman should also carry a notebook in which he should at least enter all readings upon turning points and bench-marks and check up with the in- strument man at every opportunity. Under the circumstances, the instrument man is more or less dependent upon his rodman for the correct reading of the rod and when an inexperienced rod- man must be employed, the self-reading rod will give the better results. The limit of range of an ordinary leveling instrument is about 400 feet, and sights should not be taken at a greater distance. The method of keeping the field notes for the work above out- lined is given below. A level notebook especially adapted to the purpose should be procured, the notes entered on the left-hand pages, the right-hand pages being reserved for remarks, sketches, etc. STA. + S 11. 1. S ELEV. A 0.650 1000.650 1000.00 N. E. cor. B 1.250 993.140 8.760 991.890 of abutment 2.380 987.670 7.850 985.290 Main street D 9.570 978.100 bridge. It will be noticed that the algebraic sum of the plus and minus readings equals the difference of elevation of the first and last stations, and these quantities should be checked as often as possible to discover errors in addition or subtraction. Profile Leveling. The method of profile leveling is the same in principle as above outlined, but the details of field work are a little different. In this sort of work it is intended to determine a vertical section of the ground above a datum plane. To this end, rod read- ings are taken sufficiently close together that when the elevations 67 60 PLANE SURVEYING are plotted and the points connected, the resulting irregular line will closely approximate the actual line of the surface. Profile levels are usually run in connection with a transit or chain survey of the line, the positions of the points being first established upon railroad surveys. These points are usually 100 feet apart unless the ground is very irregular, when they may be 50 or 25 feet apart or even less, the points being indicated by stakes. Upon sewer or street work they should seldom be more than 50 feet apart and the readings should be taken with the rod held upon the ground. Fig. 49 will illustrate the difference between profile leveling and the first system outlined, sometimes called differential leveling or "peg" leveling. Referring now to that figure A Bis the datum Fig. 49. plane and the full lines at C,D, and E represent positions of the rod for turning points. Assuming the elevation of the point C, the rod is held upon it and the reading added to the elevation for the height of instrument. The rod is then carried successively to the points a, J, c, <7, and each reading is in turn subtracted from the height of instrument at C, to get the elevations of these points. The rod is then held upon the point D, the instrument moved and the plus-sight upon D added to the elevation for the new height of instrument. The rod readings upon are taken 68 PLANK SURVEYING 01 behind the instrument, but the rod reading at C is the only plus-sight. The method of keeping the field notes is illustrated below. STA. O. -f 50 1 + 50 T. P. + 62 2 + 50 3 + 50 4 + S 3.25 2.64 H.I. 585.70 581.49 S 3.78 4.18 5.06 6.85 3.10 3.18 3.90 4.60 ELEV. 582.45 581.92 581.52 580.64 578.85 578.39 578.31 577.59 576.89 576.24 CROSS=SECTIONING. One of the most important problems that confronts the leveler is the setting of "slope stakes," called cross-sectioning, from. which may be determined the amount of earthwork in cut or fill, and which mark the extreme limits of the operations of the construction corps in building railways, highways, sewers, canals, irrigation ditches, etc. The problem is as follows: Given the required width of finished roadbed or channel, with proper side slopes (depending upon the kind of material), it is required to determine where these side slopes will intersect the natural surface of the ground with reference to the center line of the survey. The center line is defined by stake, carefully aligned and leveled, and a profile of it is prepared upon which the grade line is laid down, showing the 62 PLANE SURVEYING elevation of the finished roadbed or channel with reference to the natural surface of the ground. Let us assume the ground to be level transverse to the center line. Depth of cut at center = 12 feet; side slopes 1| feet hori- zontal to 1 foot vertical; width of cut at bottom 20 feet. See Fig. 50. Set up the instrument in some convenient position that will command a view of as much ground as possible. Hold the rod upon the ground at the center stake and note the reading. Sup- pose it to be 3.5 feet. Now if the ground is level, the distance from C to B is evidently 10 + (12xH) =28 feet and the rod should again read 3.5 feet when held at B. The point A would be found in the same way. The notes would be kept as shown below. Sta. Dis. Left Center Right' Area C.Yds. 175 170 176 50 50 + 12.0 28~ + 3.0 14~5 + 2.5 + 12.0 + 6.0 + 5.0 + 7 9 + 12.0 28 + 9.2 2OO + 8.0 22~ 13.751 The preceding example illustrates one of the simplest cases that occur in practice. Let us now take the case of a line located upon the side of a hill. See Fig. 51. Depth to grade at center 6 feet; width at bottom 20 feet; side slopes 1| to 1. As before, hold the rod upon the ground at C and determine the height of instrument abov.e C. Suppose this to be 5.5 feet. Now, if the ground were level through C it would be necessary to measure to the right 10 + (0 X 1|) = 19 feet to the point D and the rod should read 5.5 feet. Instead it reads, say 2.8 feet. We know therefore that we have not gone out far enough by (5.5 2.8) H 4.05 feet, if the ground were level through the point D, bringing us to the point E where the rod should read 2.8 feet. Suppose it reads 2.3 feet. We must then go out 0.75 foot farther, each move bringing us closer and closer to the point B. This operation may be repeated as often as is considered necessary, but with a little experience in this sort of 70 PLANE SURVEYING 63 work the instrument man can direct the rod closely enough to the point B for all practical purposes. We then enter the notes in the second line of the record shown above. Upon the left of the center, these operations are reversed. That is to say, we measure out 19 feet and instead of the rod read- ing 5.5 feet, it reads, say, 8.5 feet. We know then that we are out too far by 4.5 feet. We then move in toward the center the Fig. 51. required distance and read the rod again, noting how much it dif- fers from 8.5 feet, if any, and enter the final results in the notes. A third case is shown in Fig. 52, in which the transverse slope is not uniform. The method of procedure is the same as in the other cases, but the rod should be held at the point where the slope changes in order to find its height above grade. Enter this and the distance out in the third line of the notes. The transverse section may be very irregular, in which case it may be necessary to take readings at several points in order to Pig. 52. calculate the area of the sections with more exactness. At times a section will be cut partly in rock and partly in earth, forming a 71 64 PLANE SURVEYING compound section. Each material will, of course, have its own proper side slope, and the depth and extent of each must be deter- mined by soundings. In case the section is in fill instead of in cut, the method is the same as in the preceding cases, as will be illustrated in the fol- lowing examples. Let us first take a section level transversely. See Fig. 53. In this case the finished grade is to be 9 feet above the point Fig. 53. C. Hold the rod at C and suppose it reads 3.25 feet. Now since the ground is level we go out to the right and left 9-|-(9xl4) = 22.5 feet and set the stakes at A and B entering the record in the notebook as before, except that now the numerator of the fraction will be marked instead of + . We will next take the case where the surface of the ground has a transverse slope. See Fig. 54. Now hold the rod at the point C, and suppose it reads 9.25 feet. Now if the ground were i is'- 1 svajy- Fig. 54. level through C we would have to go out to the right 9 +(6. 25 Xl.5) = 18.4 feet to some point D. But there the rod reads, say, 1.5 feet, hence we know we are out too far by 7.75x1.5 = 11.63 feet, bringing us back to some point as E and the rod now reads, PLANE SURVEYING 65 say, 3.5 feet and we move out again 2.0 X 1.5 = 3 feet. Therefore we move back and forth until we find the point B where the com- puted rod reading and the actual reading agree. Sometimes it will be found that a part of the section is in cut and a part in fill, but methods outlined will serve in any case. The distance between the sections longitudinally will depend upon the nature of the ground. On uniformly sloping or level ground they may be taken 100 feet apart. Over uneven ground it may be necessary to take them as closely together as 25 feet or even less. In the sections themselves, a sufficient number of rod readings should be taken that the area of the sections may be determined with reasonable accuracy. After the field work is completed, the notes are plotted, usually upon cross-section paper, and the areas determined either with a planimeter, by Simpson's rule or some other method. These sections then divide the earthwork into a system of prismoids of which the volume must be calculated. The formula for calculat- ing volumes is known as the Prismoidal Formula and is as follows: in which I = length between consecutive sections, A = one end sec- tion, B = the other end section and M = the section midway be- tween the two. The result is given in cubic yards. The mistake must not be made of assuming that M is a mean between A and B; but a theoretical section must be plotted whose dimensions are a mean between those of A and B. This often results in quite a complicated problem, and various other formulas have been devised to give sufficiently close results without the labor and time involved in the preceding. This will be treated in detail in Railroad Engineering. 73 COAST SURVEY PARTY STARTING TO WORK FROM A TRIANGULATION STATION The topographer and his assistant are adjusting the instrument under the "signal;" the instrument man is at his never-ending task of putting needle-points on pencils: one of the rodmen (with telemeter rod) Is waiting for instructions to set out, and the other is pit-king up other "signals" with the glasses. PLANE SURVEYING. PART II. The meridian plane of any place upon the earth's surface is a great circle passing through the zenith of the place and the poles of the earth. A true meridian is, therefore, a line lying in this plane, and would, if produced, pass through the poles. The magnetic meridian plane would in the same way be defined by the zenith and the magnetic pole of the earth ; but since this pole is not fixed in position, the magnetic meridian is defined as the direction of the line indicated by the position of the magnetic needle. At a few places upon the surface of the earth, the true meridian and the magnetic meridian coincide at times, but for the most part they differ in direction by an ever varying quantity. The angle at any place between the true meridian and the meridian as defined by the magnetic needle, is called the magnetic declination for that place. If the direction of the magnetic meridian were constant, or if the changes followed any particular law, it would be a comparatively simple matter to determine the declination for any time or place. The variations occurring are of three principal kinds diurnal, annual, and secular, the last being the most important. Diurnal Variation. On continuing observations of the direction of the needle throughout the day, it will be found that the north end of the needle will move in one direction from about 8 A. M. until shortly after noon, and then gradually return to its former position. Annual Variation. If observations be continued throughout the year, it will be found that the diurnal changes vary with the seasons, being greater in summer than in winter. Secular Variation. If accurate observations on the declination of the needle, in the same place, are continued over a number of years, it will be found that there is a continual and comparatively constant increase or decrease of the declination, continuing in the same direction over a long period of years. Copyright, 19O8, by American School of Correspondence. 68 PLANE SURVEYING Besides the above, the declination is subject to variations more or less irregular, due to local conditions, lunar perturbations, sun spots, magnetic storms, etc. The declination in any part of the United States may be approximately determined by consulting the chart issued from time to time by the United States Coast and Geodetic Survey. (See chart, page 132.) Upon this chart all points at which the needle points to the true north are connected by lines, called agonic lines or lines of no declination. Lines are also drawn connecting points of the same declination, called isogonic lines. The isogonic curves or lines of equal magnetic declination (variation of compass) are drawn for each degree, a + sign indicating West declination, a sign indicating East declination. The magnetic needle will point due North at all places through which the agonic or zero line passes, as indicated on the chart. Before undertaking an extensive or important survey, it is the first duty of the surveyor to determine accurately his declina- tion. This is best done by laying out a true meridian upon the ground and comparing its direction with that indicated by the needle. Before describing the methods of laying out a true meridian, it will be best to describe the compass. THE COMPASS. Construction. The surveyor's compass consists primarily of a circular brass box, carrying, upon a pivot in its center, a strongly magnetized needle (see Fig. 56). The inside edge of the box on a level with the needle, is usually graduated to half degrees, and smaller intervals may be " estimated." Two points diametrically opposite each other are marked 0, and form the north and south ends of the box, the south end being indicated by the letter S, and the north end either by the letter N or by a fleur-de-lis or other striking figure. The divisions extend through 90 upon both sides of these points, to the east and west points marked respectively E and W. The east side of the box, however, is on the left' as the observer faces the north end; this is because the needle remains stationary while the box revolves around it. The divided circle is sometimes movable, being fitted with a clamp and tangent-screw for setting off the declination of the needle. 76 PLANE SURVEYING The magnetic needle is the most essential part of the compass. It consists of a slender bar of steel, usually five or six inches long, strongly magnetized, and balanced on a pivot so that it may turn freely and thus continue to point in the same direction however much the box carrying the pivot may be tunied around. To this end the pivot should be of the hardest steel, ground to a very fine point, or, better still, of iridium; and the center of the needle resting upon the pivot should be fitted with a cap of agate or other hard substance. To distinguish the ends of the needle, the north end is usually Fig. 56. cut into a more ornamental form than the south end, or the latter end may be recognized by its carrying a coil of wire to balance the "dip." Intensity of directive force and sensitiveness are the chief requisites in a magnetic needle, and nothing is gained by making a needle over five inches in length. Indeed, longer needles are liable to have their magnetic properties impaired by polarization. The needle should not come to rest too quickly. Its sensitiveness is indicated by the number of vibrations that it makes in a small space before coming to rest. Should it come to rest quickly or be sluggish in movement, it indicates either that the magnetization 77 PLANE SURVEYING is weak or that there is undue friction between needle and pivot. The under side of the box should be fitted with a screw which, engaging a lever upon the inside of the box, will serve to lift the needle off the pivot when the instrument is carried about. The sights form the next most important feature of the compass. They consist of two brass uprights, with a narrow slit in each, terminated at intervals by circular apertures. They are mounted directly upon the compass-box ; or the bottom of the box may be extended at each end in the form of a plate, and the sights attached at the ends of the plates. How- ever mounted, the sights should have their slits in line directly over the north and south points of the divided circle. The right and left edges, respectively, of the sights, may have an eye-piece and a series of graduations, by which angles of elevation and depression for a range of about twenty degrees each way can be taken with considerable accuracy. This device is called a tangent scale, so called because of the distance of the engraved lines from the O line being tangents (with a radius equal to the Pig. 57. distance between the sights) of the angles corresponding to the numbers of the lines. The spirit levels may be placed at right angles to each other in the bottom of the compass-box, or mounted in the same way upon the plate. The compass is usually fitted to a spindle made slightly conical, which has on its lower end a ball turned perfectly spherical, confined in a socket by a pressure so light that the ball can be moved in any direction in leveling the instrument. The ball is placed either in the brass head of a Jacob staff, or, better, in the top casting of a tripod. A plumb-bob should be provided with the instrument to center it over a stake. A telescope is sometimes provided, to be attached to one of the vertical sights, for the purpose of more clearly defining the 78 PLANE SURVEYING 71 line of sight, The compass is, however, so inaccurate that it would seem to be an unnecessary refinement. Prismatic Compass. This is a form of compass used in general where merely ordinary work is required. It is about 3 inches in diameter with a floating metal dial (see Fig. 57), and is provided with folding sights and prism. By means of the latter it may be read while being pointed. This is especially useful when the instrument is held in the hand. Although it can be mounted on a Jacob staff, it is usually held in the hand and carried in the observer's pocket. Adjustment. To Adjust the Levels. First bring the bubbles to the middle of the tube by the pressure of the hand on different parts of the plate, and then turn the instrument half-way round. If the bubbles remain in the middle of the tubes, the tubes are in adjustment. If the bubbles do not remain in the middle, raise or lower one end of the tube to correct one-half the error. Relevel the instrument, again test, and apply the correction as before. Continue the operation until the levels are in perfect adjustment. To Adjust the Needle to the "J9>." While the compass is still in a perfectly level condition, see if the needle is in a horizontal plane. Should this not be the case, move the small coil of wire towards the high end until the needle swings horizontally. To Adjust the Sight- Vanes. Observe through the slits a fine hair or thread made exactly vertical by a plummet. Should the hair appear on the side of the slit, the sight-vane must be adjusted by filing its under surface on the side that seems the higher. To Adjust the Needle. Having the eye nearly in the same plane with the graduated rim of the compass-box, bring one end of the needle in line with any prominent graduation mark in the circle, as, for instance, the zero or the 90-degree mark, and notice if the other end corresponds with the same degree upon the opposite side; if it does, the needle is said to "cut" opposite degrees; if not, bend the center pin, until the ends of the needle are brought into line with the opposite degrees. Then, holding the needle in the same position, turn the instrument half-way round, and note whether the needle now cuts opposite degrees; if not, correct one-half the error by bending the 79 PLANE SURVEYING needle, and the other half by bending the center pin. The operation of testing and correcting should be repeated until perfect reversion is secured in the first position. This being obtained, the operation should be tried on another quarter of the circle; if any error is found, the correction must be made in the center pin only, the needle being already straightened by the previous operation. When the needle is again made to cut, the test should be tried in the other quarters of the circle, and the correction made in the same manner, until the error is entirely removed and the needle will reverse at every point of the graduated circle. Use. In the operation of locating points, and therefore lines, by angle-measuring instruments, two operations are necessary: (1) to measure the angle at the instrument between some given line and the line passing through the given point; (2) to measure the distance from the instrument to the given point. For the first operation two types of instrument are in general use the compass and the transit. For the compass, the line of reference from which all angles are measured is a meridian, and the angular deviation from this line is called the bearing. The bearing and length of a line are collectively named the course. The compass, therefore, measures bearings directly and angles indirectly. To Determine the Bearing of One Point from Another. "Set up" the compass over one of the points, and level carefully. Turn the sight-vanes in the direction of the second point, with the north end of the plate ahead. Hold a rod upon the second point, and cover it with the slits in the sight-vanes. Now lower the needle upon the pivot, being sure that the instrument is still level; allow it to come to rest, and read the bearing. To Survey a Series of Lines with the Compass. "Set up" the compass over the point A, with the north end of the plate ahead (Fig. 58); and after leveling, turn the sight-vanes to cover a rod held upon the point B. Now send out the tape in the direction of B, and, sighting through the slits, signal the head tapeman into line. Continue this until the point B is reached. Now read and record the bearing and the length of the line. Take up the instrument, and carry it to B. Set it up over B, with the north end ahead, that is, pointing in the direction of the survey. Level, and turn the south end so as to cover a rod held upon the point A. PLANE SURVEYING 73 Read the bearing as a check upon the former one, but reversed in direction; i. e,, if the bearing from A to B was north by east, the bearing from B to A will be south by west. If the direct and reversed bearings check, turn the north end of the compass to cover a rod held upon C. Read the bearing, measure B C, take the instrument to C, and proceed as before. If at any station, such as C, the direct and reversed bearings do not agree, take the instrument back to B and again take the bearing of B C. If they still disagree; it indicates local attraction at C. Take the instrument to D and take the bearing of D C, Fig. 58. comparing it with the bearing of C D. If these disagree, record the bearings of B C and D C as well as those of C B and C D. The latter should check the former, since the local attraction at C will affect both lines equally; and the correct angle between the lines can be computed. Locating a series of lines with certain lengths and bearings is essentially the same as above, except that after the compass has been turned in the proper direction, the stations must be brought into proper line. Here it may be well to remark upon the proper method of reading and recording bearings. Always read the north or south end of the plate first; i. e., if a line has a bearing 35 east of north, it should be read and recorded N 35 E. If the bearing is 90 east or west of north or south, record the bearing as E or W. The Gunter's chain is always used in land surveys made with the compass, and deeds and records of such surveys are based upon the Gunter's chain as the unit. Hints Regarding the Use of the Compass. Sometimes, as when the line of which the bearing is required consists of a fence, 74 LANE SURVEYING etc., the compass cannot be set upon the line. In such a case measure off equal distances at right angles to the line, and find the bearing of the parallel line; the length should be measured upon the line itself. In other cases it may be more convenient to set the compass or rod "in line" upon the line produced, or upon some intermediate point of the line. It is more important to have the compass level, crosswise of the sights, than parallel with them. Avoid reading the bearing from the wrong number of the two between which the needle points, as for instance 35 for 25. Check the vibrations of the needle by gently raising it off the pivot and lowering it again by means of the screw on the under side of the box. If the needle is slow in starting, smartly tap the compass to destroy the effect of any possible adhesion to the pivot or friction of dust upon it. Avoid holding the pins, axe, or any other body of iron, in close proximity to the needle. Should the needle adhere to the glass after the latter has been dusted with a handkerchief or has been carried so as to rub against the clothes, the trouble is due to the glass being thereby charged with electricity and may be obviated by moistening the finger and applying it to the~glass. RELOCATION. Suppose it is required to relocate a line, no trace of the old survey being at hand except the given line. Now, between the date of the old survey and the present, the declination of the needle has changed several degrees. The first duty of the surveyor is to consider this question very carefully, and to ascertain the probable amount of change in the magnetic needle. Suppose the result of his inquiry leads to N 38 15' E as the bearing. Starting at corner A, Fig. 59, the surveyor runs a random line AS on the bearing N 38 15' E, and measures along this line a distance of 32 chains, or 2,112 feet, to point S. On arriving at S, the surveyor proceeds to look over the ground on both sides of this point for a lost corner, which is described in the old record as a monument, stump, or some other well-defined mark. If, after diligent search, PLANE SUKVEYINQ no trace of this mark can be found, nothing further can be done from the data at hand. However, should the mark be found at m, a perpendicular is dropped upon the line AS, arid its length is measured, as is also the distance nS. It is now evident that the distance An becomes known. From the right triangle, the angle nAm can be com- puted, and the present magnetic bear- Fig. 59. ing of Am can be determined. For example, suppose that mn is found to be 37.4 feet, while An is 2,110.5 feet, then tan. nAm=-^- 0.01772, whence nAm An = 1 01', and the present magnetic bearing of Am is X 39 16' E- The distance Am = ^f^^ = 2,110.84 feet. This indicates that the present work is correct, and that the old survey was in error by 1.16 feet. As there is a principle of law that establishes corners and monuments, resurveys must control; therefore the new record of the line Am is N 39 1C' E, 2,110.84 feet. Intermediate points of the line Am may now be established from the starting point A, running it out with the new bearing. EXAMPLE FOR PRACTICE. Compute the distance and bearing of two points which are not intervisible. Call the line GH. A line is run approximately near H, from the known corner G to a point A which is visible to H; the bearing and length of this line being N 42 15' E, 714.5 feet: AH being N 1 08' E, 210.5 feet. Ans. N 33 14' E, 883.24 feet. To Find the Bearing of One Line to Another. Suppose, in Fig. 60, that of the tract of land therein described there has been prepared a rough plot upon which the angles, bearings, and distances as taken from the field book are figured. In order to find the bearing of one line to another, add together the interior angles formed at all the corners; call their sums a; multiply the number of the sides by 180; from the product subtract 360. If the remainder is equal to , this is proof that the angles -have been accurately measured, This, however, will rarely if ever occur; 83 76 PLANE SURVEYING there will always be some discrepancy, but if the field work has been performed with reasonable care the discrepancy will not exceed two minutes for each angle. In this case divide it, in equal parts, among all the angles, adding or subtracting, as the case may be, until it amounts to less than one minute for .each angle, when it may be entirely disregarded in common farm surveys. The corrected angles may now be marked on the plot in ink, and the penciled figures erased. We shall suppose the corrected Fig. 60. ones to be as shown in Fig. 60. Next, by means of these corrected angles, correct the bearings also. Select some side, the longer the better, from two ends of which the bearing and the reverse bearing agree, thus showing that the bearing was probably not influenced by local attraction. Let side 2 be the one so selected; assume its bearing N 75 32' E, as taken on the ground, to be correct; through either end of it, say at its farther end 2, draw a short meridian line, parallel to which draw others through every corner. Now, having the bearing of side 2, N 75 32' E, and requiring that of side 3, it is plain that the reverse bearing from corner 2 is S 75 32' W, and that therefore the angle 1 2 ra is 75 32', Therefore, if we take 75 32' from the entire corrected angle 123, or 144 57', the remainder 69 25' will be the angle m 2 3; conse- quently the bearing of side 3 must be S 69 25' E. For finding the bearing of side 4, we now have the angle 2 3 a of the reverse PLANE SURVEYING 77 bearing of side 3, also equal to 69 25', and if we add this to the entire corrected angle 2 3 4, or to 69 32' we have the angle a 3 4 = 69 25' + 69 32' = 138 57', which, taken from 180, leaves the angle I 3 4 = 41 3' ; consequently the bearing of side 4 must be S 41 3' W. For the bearing of side 5, we now have the angle 3 4 c = 41 3', which, taken from the corrected c angle 3 4 5, or 120 43', leaves the angle c 4 5 = 79 40 ' , consequently the bearing of side 5 must be N 79 40' W. At corner 5, for the bearing of side 6, we have the angle 4 5 d = D 79 40', which, taken from 133 10', leaves the angle d 5 6 53 30'; consequently the bearing of side 6 Fig. 61. must be S 53 30' W: and so with each of the sides. Nothing but careful observation is necessary to see how the several angles are to be employed at each corner. FARM SURVEYING. Method of Progression. Farm surveying with the compass does not differ in any essential particular from the methods outlined for surveying a series of lines. If the boundary lines are irregular, it will be necessary to measure offsets at proper intervals, that the included area may be calculated. The method above described is known as the method of progression. Method of Radiation. The method of radiation consists in setting up the instrument at some point inside or outside the field, from which all the corners are visible and accessible, and then measuring the bearing and lengths of the lines to these corners. Fig. 61 illustrates the method. Set up the compass at the point O, and take the bearings and lengths of the lines O A. O B, O C, O D, and O E. Method of Intersections. Lay off a base-line of convenient length inside or outside the field, from which all the corners are visible. Set up the compass at one end of the base-line, and take the bearings from it to each corner in succession. Remove the 85 78 PLANE SURVEYING compass to the other end of the base-line, and take the bearings from it to each corner in succession. Take the bearing and length of the base-line. Now, when these bearings and lengths are plotted, the intersections of the lines will define the corners. Proofs of Accuracy. When the survey of a field is plotted, if the end of the last course meets the starting point, it proves the work, and the survey is then said to "close." Errors of closure may be due either to incorrect lengths of lines or to incorrect bearings, or to both. Diagonal lines running from corner to corner of a field may be measured and their bearings taken. When these are plotted, their meeting the points to which they were measured proves the accuracy of the work. Finally, the accuracy of the work may be tested by calculating the "latitudes and departures" of all the courses. If their algebraic sum is equal to zero, the work is correct. A check upon the bearings may be had by calculating the "deflection angles" between the courses. If their sum is equal to 360 degrees, the bearings are correct. This, however, will seldom be the case. A certain amount of error is permissible, depending upon the nature and importance of the work. Field Notes. The field notes may be recorded in various ways, the object being to make them clear and full. 1. The surveyor may make, in the field book, a rough sketch of the survey by eye, and note on the lines their bearings and lengths. If a protractor and scale are available, the actual bearings and lengths of the lines may be plotted in the notebooks, as well as offsets, etc. 2. Draw a straight line up the page of the notebook, and record on it the bearings and lengths of the lines. Offsets, tie-lines, etc., can be plotted in their proper positions. 3. Write the stations, bearings, and distances in three columns. This method has the advantage, when applied to farm surveying, of being convenient for use in the subsequent calcula- tion of contents, but does not give facilities for noting effects. It is illustrated as follows: PLANE SURVEYING 79 STATIONS. BEARINGS. DISTANCES. N. 32 E. 16.82 1 S. 36 E. 18.90 3 4 S. 27^ W. S. 16 W. 7.85 15.30 Notice that distances are given in Gunter's chains, and in calculating content the result will be given in square chains, which can be reduced to acres by pointing off one decimal place. To Change Bearings. In certain kinds of work with the compass, it is convenient to assume one of the lines as a meridian, and it then becomes necessary to change the bearings of all of the other lines to conform with the assumed meridian. This case is best illustrated by an example. The bearings of the sides of a field are here shown: Suppose now that the first course is assumed as a meridian, that is, that its STATIONS. BEARINGS. DISTANCES. 1 N. 35 E. 2.70 2 N. 83} E. 1.29 3 S. 57 E. 2.22 4 S. 34^ W. 3.55 5 N. 56^ W. 3.23 bearing is due north and south. Required the bearings of the remaining courses. Since the courses are changed to the west by 35, the new bearing of course 2 will be N 48| E. Of course 3 it will be 57 + 35 = 92, or the new bearing will be N 88 E. Of course 4 it will be 34^ 35, or | in the next quadrant, or the bearing will be S | E. Of course 5 it will be 56| + 35 = 91, or the bearing will be S 88| W. EXAMPLE FOR PRACTICE. The bearings of a series of courses are given as follows: The bearing of the first course is changed to due north and south. 87 80 PLANE SURVEYING It is required to determine the bearings of all the courses, due to this change. Find bearings and plot the lines. Ans. Course 2 = N 62 E; 3 = N 9 W; 4 = N 68 W. STATIONS. BEARINGS. DISTANCES. 1 S. 21 W. . 12.41 2 3 N. 83*4 E. N. 12 E. 5.86 8.25 4 N. 47 W. 4.24 Latitudes and Departures. The latitude of a point is its distance north or south of some line taken as a parallel of latitude, or line running east and west. The longitude of a point is its distance east or west of some line taken as a meridian, or line running north and south. The distance that one end of a line is north or south of the other end is the " Difference of Latitude" of the two ends of the line, and is called its northing or southing, or its latitude. The distance that one end of a line is east or west of the other end is the " Difference of Longitude " of the two ends of the line, and is called its easting or westing, or its departure. The terms Latitude Difference and Longitude Difference have of late come into quite general favor; but while they are perhaps more explicit, they are certainly cumbersome, and the older terms will be adhered to in what follows. In Fig. 62, N S represents a meridian, and E W a parallel of latitude. If we take the line O A, its bearing as given by the compass is the angle NOA. The latitude or northing of the point A is therefore A B = O A cos NOA. Its departure or easting is O B = O A sin NOA. To find the latitude of a course, multiply the length of the course by the natural cosine of the bearing; and to find the departure of any course, multiply the length of the course by the natural sine of the bearing. PLANE SURVEYING 81 If the course be northerly, the latitude will be north, and will be designated by the sign +, or plus; if the course be southerly, the latitude will be south, and will be designated by the sign , or minus. If the course be easterly, the departure will be east, and will be designated by the sign -+-, or plus; if the course be westerly, the departure will be west, and will be designated by the sign , minus. Thus in the figure, OA is of plus latitude and plus departure; OP is of plus latitude and minus departure; OD is of minus latitude and minus departure; and OC is of minus latitude and plus departure. For calculating latitudes and departures, a set of traverse tables may be procured; but a table of natural functions will be satisfactory', though possibly less convenient. Testing a Survey by Latitudes and Departures. It is evident that after the surveyor has gone completely round a field or farm, measuring all the lengths and bearings, returning to the starting point, he has gone as far north as south, and as far east as west. In other words, if the work has been done correctly, the algebraic sum of the latitudes must equal zero, and the algebraic sum of the departures must equal zero. This condition, however, will seldom be attained, and it becomes necessary to decide how much error may be permitted without necessitating another survey. This will depend upon the nature of the work and its importance, and a surveyor will soon determine for himself his factor of error, depending partly upon his instrument, partly upon personal skill, for ordinary cases. If it is necessary to depend upon a "green" hand to carry the tape or chain, this may prove a fruitful source of error. We shall now proceed to calculate the latitudes and departures of the survey as given below. Arrange the diagram as below with seven columns: STATIONS. BEARINGS. DISTANCES. LATITUDES. DEPARTURES. N. s. E. w. 1 S. 21 W. 12.41 11.591 4.443 2 3 N. 83^ E. N. 12 E. 5.86 8.25 0.691 8.069 5.819 1.716 4 N. 47 W. 4.24 2.81)2 3.104 30.76 11.652 11.591 7.535 7.547 89 82 PLANE SURVEYING The cosine of the bearing of course 1 is 0.934.12.41=11.591 Latitude. The sine of the bearing of course 1 is 0.358.12.41= 4.443 Departure. The cosine of the bearing of course 2 is 0.118.5.86 = 0.691 + Latitude. The sine of the bearing of course 2 is 0.993.5.86 = 5.819 + Departure. The cosine of the bearing of course 3 is 0.978.8.25 = 8.069 + Latitude. The sine of the bearing of course 3 is 0.208.8.25 = 1.716 + Departure. The cosine of the bearing of course 4 is 0.682.4.24 = 2.892 + Latitude. The sine of the bearing of course 4 is 0.732.4.24 = 3.104 Departure. The latitudes fail to balance by 0.061 chains, and the departures by 0.012 chains. The error of "closure" of the survey is therefore I 2 E ==^.061 + .012 = 0.062 + chains, or approximately 4.09 feet. This sum may be divided up among the courses in proportion to the length, or the bearings may be corrected, or partly one and partly the other, as will hereafter be explained. Balancing the Survey. Before proceeding to the calculation of the content of a field or farm, the survey must be balanced; that is, the latitudes and departures must be corrected so that their sums shall be equal, or shall balance. As to whether the bearings or lengths shall be corrected, will depend somewhat upon the conditions tinder which the survey was made. If the surveyor has reason to think that the error is entirely in the bearing of one or more, or even of all of the courses, the corrections may be made accordingly. If, on the other hand, one or more of the courses were measured over difficult ground, it may be presumed that the error occurred in those lines. If, however, there is no reason to believe that one course is in error more than another, the differences may be distributed among the courses in proportion to their length, according to the following proportions: As the length of any course is to the sum of the lengths of all the courses, so is the correction of the latitude of that course to the total error in latitude of all the courses. As the length of any course is to the sum of the lengths of all the courses, so is the correction of the departure of that course to the total error in departure of all the courses. The practical application of these proportions to balancing a survey will be illustrated from the preceding problem: For course 1 12.41 : 30.76 : ; x : 0.061 x = .0246, correction for latitude. For course 2. ... 5.86 : 30.76 :: x : 0.061. . . .x = .0116, correction for latitude. 90 PLANE SURVEYING 83 For course 3. ... 8.25 : 30.76 : : x : 0.061. . . .x = .0164, correction for latitude. For course 4. ... 4.24 : 30.76 :: x : 0.061. . . .x = .0084, correction for latitude. Since the sum of the north latitudes is the greater, the corrections will be subracted from them and added to the south latitudes. That is to say, the correction for course 1 will be added to 11.591, the result being 11.6156. The correction for course 2 will be subtracted from 0.691: that for course 3 will be subtracted from 8.069; and so on. For course 1. . .12.41 : 30.76 :: x : 0.012. . .x = .0048, correction for departure. For course 2. . . 5.86 : 30.76 : : x : 0.012. . .x = .0023, correction for departure. For course 3.. .. 8.25 : 30.76 :: x : 0.012.. ..x = .0032, correction for departure. For course 4.. .. 4.24 : 30.76 :: x : 0.012.. ..x .0017, correction for departure. The corrections are to be subtracted in this case from the west departure and added to the east departure. In this example, the errors are small, but often they will be so large as to raise doubt as to the accuracy of the survey. In such a case, go carefully over all the computations, and, if the error is still too large, check the exterior angles of the figure (their sums should equal 360), and if necessary repeat the survey. Having corrected the latitudes and departures, the corrected bearings of the courses may be deduced from the trigonometric ratio: corrected departure Tan. bearing = - corrected latitude Calculating the Content. After a field has been surveyed, its content may be calculated by dividing it up into triangles, trape- zoids, etc., calculating the various contents, and adding them together. This, however, is at best a cumbersome method, involv- ing much work of calculation and great chance of error. The method of latitudes and departures is at once simple, easily applied, and easily checked. Before proceeding to develop a formula for this method, it will be necessary to illustrate and define certain terms. Draw a line, as N S (Fig. 63), through the extreme east or west corner of the field for a meridian. From the definitions previously given, the difference of longitude of the two ends of a line is the departure of the line. I B is therefore the departure of the line A B. The departure of the line B C is L C; that of E F is S F; and that of A F is O Q. The perpendicular distance of each station from the given meridian is the longitude of that station, plus if east, minus if fri 84 PLANE SURVEYING west. Thus the longitude of A is zero; that of B is I B; that of C is I B + L C; that of E is O Q + F S; and that of F is O Q = ZS FS. The difference of latitude of the two ends of a line is called the latitude of the line. Thus the latitude of A B is A I; that of B C is B L; that of E F isES. The distance of the middle of any side of a field from the meridian is called the longi- tude of that side. Thus the longitude of the side A B is GH;thatofBCisJX = GH + K M + MX; and that of A F is WV = OR QR Fig. 63. QP, the minus signs being used in this instance because the lines E F and A F bear to the west. An analysis of W V will show that it equals O R (longitude of preceding course) + [ R Q (one-half departure of preceding course)] + [ QP (one-half departure of the course itself)]. To avoid fractional quantities, double the preceding expres- sions and then deduce a general rule for finding double longitudes. The double longitude of the first course equals the departure of that course. The double longitude of the second course equals the double longitude of the first course, plus the departure of the first course, plus the departure of the second course. The double longitude of any course equals the double longi- tude of the preceding course, plus the departure of the preceding course, plus the departure of the course itself. We shall now proceed to deduce a rule for determining areas by double longitudes and departures, and shall first take a three- sided field, as in Fig. 64. Drawing a line through the most westerly corner A, we see that the area of the field will be the difference between the area of the trapezoid D B C M and the combined area of the triangles D B A and A C M. The double area of the triangle D B A is the PLAttE SURVEYING 85 product of D B by D A, or the double longitude of A B by the latitude of A B. The resulting product will be north or plus. The double area of the trapezoid D B C M is the product of (D B + M C) = 2 G H, by D M, that is, the double longitude of B C by its latitude. The resulting product will be south or minus. The double area of the triangle ACM will be the product of M C by Fig. 64. Fig. 65. A M, or the double longitude of the course A C by its latitude. The resulting product will be north or plus. Adding together, then, the plus products, and subtracting from the minus product, gives as the result the double area of the field. We shall next take a four-sided field, as in Fig. 65. It is evident that the area of the field A B C D is the difference between the sum of the areas of the two trapezoids T B C R and RODE and the sum of the areas of the triangles A B T and A D E. The double area of the triangle A B T is the product of B T by A T, or the double longitude of the course A B by its latitude. The result will be a north product or plus. The double area of the trapezoid T B C R will be the product of (T B + C R) = 2 L P by T R that is, the double longitude of the course B C by its latitude. The result will be a south product or minus. The double area of the trapezoid RODE will be the product of (R C + D E) = 2 F K, by R E, or, the double longitude of the course C D by its latitude. The result will be a south product or minus. The PLANE SURVEYING STA- TIONS BEARINGS DIS- TANCES LATITUDES N. S. DEPARTURES E. W. DOUBLE LONGI- TUDES DOUBLE AREAS 1 S. 21W. 12.41 11.616 4.438 7.204 83.682 2 5.86 0.679 5.821 -5.821 8. 062 3 X. 12 E. 8.25 8.053 1.719 + 1.719 13.843 4 X. 47 W. 4.24 2884 3.102 +0.336 0.969 98 494 3.952 47.271 AREA=47.271 SQ. Cns.=4 ACHES, 2 HOODS, 37 SQ. PERCHES. double area of the triangle A D E will be the product of E D by A E, or the double longitude of the course A D by its latitude. The result will be a north product or plus. Finally, adding together the north products, adding together the south products, and taking the difference of their sums, gives as the result the double area of the field A B C D. The same principle will apply to any enclosed area, however great the number of the sides. The area will always be one-half the difference of the sums of the north and the south products arising from multiplying the double longitude of each course Inj its latitude. For systematic computation arrange the work as follows: Arrange the columns as in the problem on page 83. Balance the latitudes and departures, putting the corrected quantities above the others in red ink: or else arrange four additional columns, and enter them in their proper places. Compute the double longitude of each course with reference to a meridian passing through the extreme east or west station, and place the results in another column. Multiply the double longitude of each course by the corrected latitude of that course, and place north products in one column and south products in another. Add together the north products and also the south products, ana take the difference of their sums. Divide the difference by two, and the result will be the area desired. If the survey has been made with a Gunter's chain, the result will be in square chains. Divide by ten to reduce to acres. \\\ PLANE SURVEYING 87 To test the correctness of the calculation, assume the meridian as passing through the extreme station upon the other side of the field, and carry out the work in detail as before. We shall now proceed to calculate the content of the field given by the notes on page 81. The corrections to the latitudes will be found on page 82, and the corrected departures on page 83. The arrangement of the columns for convenient calculation is as described on page 86. Upon making a rough sketch of the courses, it is found that station 3 is the farthest east; and therefore the double longitudes will be calculated beginning with course 3. From the definition previously given, the double longitude of course 3 is equal to its departure = + 1.719. The double longitude of course 4 equals the double longitude of course 3, plus the departure of course 3, plus the departure of course 4 - 1.719 + 1.719 + (- 3.102) = + 0.336. The double longitude of course 1 equals the double lon- gitude of course 4, plus the departure of course 4, plus the departure of course 1 = 0.336 + (- 3.102) + (- 4.438) = - 7.204. The double longitude of course 2 equals the double longitude of course 1, plus the departure of course 1, plus the departure of course 2 = + (- 7.204) + (- 4.438) + 5.821 - - 5.821. Multiplying these double longitudes by their respective latitudes, gives the quantities in the last two columns, the first, third, and fourth being positive, and the second negative. Taking the difference of the sums of the quantities in these columns, and dividing the result by 2, gives the content of the field, 47.271 square chains. Dividing by 10 gives 4.7271 acres. Reduce to roods and perches by multiplying the decimal part by 4 and 40 successively. The result may now be checked by beginning with the most westerly station, and it will be necessary to recalculate the quantities in the last three columns. The following problems are taken from "Gillespie's Surveying" (Staley): EXAMPLES FOR PRACTICE. Calculate the content of the fields from the data tabulated below. The result, where found in square metres, should be reduced to acres: 1 sq. metre = .000247 acre. PLANE SURVEYING (1) STATIONS. BEARINGS. DIST \NCES. 1 N.34M" E. 2.73 2 N. 85 E. 1.28 3 S. 56% E. 2.20 4 S. 34)4 W. 3.53 5 N. 56^ W. 3.20 Ans. 1 acre, roods, 14 perches. STATIONS. IJKARINOS. DISTANCES. 1 N. 35 E. 2.70 2 N. 83i E. 1 29 3 S. 57> E. 2.22 1 s. 3414 w. 3.55 5 N. 56^ " W. 3.23 Ans. 1 acre, roods, 15 perches. (3) STATIONS. BEARINGS. DISTANCES. 1 S. 5 35' W. 2,388.88 meters 2 S.3935' W. 1,060.27 meters 3 S. 50 25' E. 3,078.31 meters 4 S. 79 5' E. 325.00 meters 5 S. 53 50' E. 275.00 meters 6 S. 48 15' W. 200.00 meters 7 N.82 45' E. 450.00 meters 8 S. 87 40' E. 186.72 meters 9 N. 8,768.12 meters 10 N.84 25' W. 1,898.54 meters 11 S. 5 35' W. 3,530.60 meters 12 N. 84 25 ' W. 257 . 50 meters i 4,999 acres 3 roods. 39^ sq. rods. Supplying Omissions. The method of latitudes and depar- I'lres may be applied to supplying any two omissions in the field PLANE SURVEYING 89 notes, as will be explained in connection with the "Use of the Transit." Azimuth. The azimuth of a line is the horizontal angle which the line makes with some other line taken as a meridian. It differs from bearing in that it is measured continuously from to 360. All descriptions of property must be given in terms of bearings, but line surveys with either the compass or the transit had better be given in terms of the azimuth. In astronomical and geodetic work it is customary to reckon azimuth from the south point around through the west, through Fig. 66. 360. For the ordinary operations of surveying, however, it is better to measure the azimuth from the north point to the right through 360. RESURVEYS. Where the boundary lines of a farm or town have been obliter- ated and the corners lost, it is often necessary to make resurveys in order to re-establish them. If the corners can be found by reliable evidence, they must be accepted as corners even though the second bearings and lengths of the lines indicate different points. It sometimes happens that some corners can be found while others cannot. In such cases a series of random lines is to be run with the old bearings, or with the old bearings corrected for a change in declination of the needle between the two dates. As an example, let the records in an old deed give the length and bearings of three lines as follows: (See Fig. 60.) 90 PLANE SURVEYING A6 N 60 E 10 chains. be N 45 E 4 chains. cd S 45 E 8 chains. There being no definite data at baud to determine the change in the magnetic declination between the dates of the two surveys, the lines AB, BC and CD are run with the given bearings and distances from the known corner A. The old corners I and c cannot be found; but on arriving at D the old comer d is discov- ered at a point 20.4 links S and 12 "W from D It is required to locate the old corners J and c. By the method of latitudes and departures explained before, the lengths of the lines DA and (IA. may be computed. They are: for DA, south 82 47', west 17.29 chains; for dA, south 83 26', west 17.22 chains. Now the error Dd between the two corners is due to two causes: (1) the continued variation in the magnetic bearings of the old surveys, (2) the difference in the length of the chains used. The first cause alters the? polygon AbcdA. around the point A by a small angle. The second cause alters the length of the sides in a constant ratio. The difference between the bearings DA and t?A is the constant angle, while the ratio of the length of the old lines is the constant ratio. To find the bearings of the old line, there- fore, each of the given bearings is to be corrected by the amount 83 26' minus 82 47' = 39'. To find the length of the old 17 22 line each of the given lengths is to be multiplied by - '" = 0.99(5. Suppose now that the work of computation has been done with such precision that the error in chaining must be regarded as lying in the old survey. Applying these results, we find the adjusted bearings and lengths of the old line to be, Ah = N 60 39' E 9.% chains. he = N 45 39' E 3.99 chains. cd = S 44 21' E 7.97 chains. With the new data the line may be rerun and the corners I and c located, a check on the field work being that the lost line 4 should end exactly at (L It is, however, not difficult to compute the length and bearings of B h and C r, so that I and c may be located from the points B and C. PLANE SURVEYING 91 Since the angle B A b is small, the triangle B A b may be con- sidered similar to the triangle D A d. We will then have the pro- portion : Bb:Dd::AB:AD. A similar proportion may be written for the side B c, and the result added to the value of B 6. The same principle .may be used to determine the bearings of B b and C c, so that, with the lengths and bearings of these lines determined, the most probable location of the old corners b and c can be fixed. EXAMPLE FOR PRACTICE. The records of an old survey read as follows: "Commencing at a point marked No. 5 and running X 62 E 14 chains to a stake marked A, thence running N 432 E 8.00 chains to a stake marked B, thence N. 5 W. 12.00 chains to a stake C, thence N 72 E. 10.25 chains to a stake D, thence S 12 W 6.43 chains to a stake marked No. 3. On running the lines, the end of the last one, instead of being at a stone marked No. 3, was 0.62 chain due E from it." Find the adjusted bearings and lengths of the old lines; also find the distance and direction from each station of the new survey to the corresponding corner of the old. DIVISION OF LAND. The method of latitudes and departures is especially useful in the division of land. The problem is usually as follows: Given the lengths and bearings of the sides of a field containing a certain area; it is required to divide the area into certain parts by a line running in a certain direction, in which case it is necessary to determine the starting point of the dividing line. Or it is required that the line shall begin at a certain point, in which case it is necessary to deter- mine the direction of the line. A certain field is described as follows: 1 N. 63 51' W. 6.91 Chs. 2 N. 63 44' W. 7.26 " 3 N. 69 35' W.. 3.34 " 92 PLANE SURVEYING 4 N. 77 50' W. 6.54 Chs. 5 N. 31 24' E. 14.38 " 6 N. 31 18' E. 16.81 " 7 S. 68 55' E. 13.64 " 8 S. 68 42' E. 11.54 " 9 S. 33 45' W. 31.55 " Beginning at a point upon the side 91, it is required to divide the area into two equal parts of 37 acres by a line having a bearing of N. 68 46' W. Some preliminary calculations will be necessary to determine the start- ing point of the dividing line such that the resulting area will be nearly that desired, and this dividing line will be practically parallel to a line connecting stations 7 and 9. Call the dividing line A B (see Fig. 67), and if this line begins at a point 15 chains from station 9, the 67 ' length of A B will be 24.528 chains; that of B 7 will be 14.90 chains. We shall now proceed to calculate the area by latitudes and departures, beginning with station 9. The calculations are shown in the following table, and the result, 36.5186 acres, is short of the required area by 0.4814 acre or 4.814 square chains. STATION BEARINGS DlS- IvATIl TANCE8 N. PUDE8 S. JJEPAI E. v In the tri- sin B < C angle A B 7 A we now have two sides and the included angle from which to calculate the length and bearing of A B. THE TRUE MERIDIAN. In order to ascertain the true meridian of a given place, several methods may be pursued. The general practice is to use the star Polaris at culmination or elongation. This star is on the meridian, nearly, when a plumb line covers it and the star Zeta, ^ DELTAV '-- I ' ^CASSIOPEIA / Fig. 68. the next to the end of the handle of ''The Dipper." See Fig. 68. When Polaris is on the meridian, as illustrated in this instance, it is said to be at "culmination." This star is often PLANE SURVEYING 95 referred to as the north or pole star. It is about 1^ from the pole, and revolves around the pole once every 23 hours and 56 minutes. Thus it is apparent that it comes on the true meridian twice each day. The arrows in the figure indicate the direction of the rotation. To Determine the True Meridian by the Compass. With Polaris at Eastern or Western Elongation. To determine the true meridian by means of the compass, take a plumb-line, and attach one end of the line to any suitable support situated as far above the ground as practicable, so as to have a clear field of view about 20 feet away. A board nailed on a telegraph pole, tree, or post at right angles, will suffice for this purpose. The plumb-bob maybe of any suitable material, of about 5 Ibs. in weight, as a brick, stone, iron ring, or coupling. It will serve the same purpose, with as accurate results, as the most highly polished or carefully manu- factured plumb-bob. The plumb-line should be about 25 feet in length, depending upon the latitude of the place, since the altitude of the pole above the horizon at any place is equal to the latitude of that place. Illuminate the plumb-line just below its support by means of a bull's-eye lantern, lamp, or candle, care being taken not to obliterate the line from the view of the observer. The best way is to screen the light, and throw the light on the plumb-line by means of a reflector. Next unfasten one of the uprights of the compass, and place it on a horizontal rest at some convenient point south of the plumb- line, say 30 feet in an east or west direction, and in such a position that when viewed through the peep-sight, Polaris will appear about two feet below the support of the plumb-line. It is customary to determine this position by trial the night before the observation. About 25 minutes before the time of elongation, as per table on page 130, bring the peep-sight into the same line of sight with the plumb-line and the star Polaris. Before reaching elongation, the star will move away from the plumb-line, to the east for eastern elongation, and to the west for western elongation. Hence, by moving the peep-sight in the proper direction that is, east or west the star can be kept on the plumb-line until it appears to PLAtfE SURVEYING remain stationary, thus indicating that it has reached its point of elongation. The peep-sight will now be secured in place by a clamp or weight, with its exact position marked on the rest. Now defer all further operations until the next day. The next morning place a slender flag or ranging pole at a distance of 200 or BOO feet from the peep-sight, and exactly in line with the plumb-line. Next carefully measure this distance, and take from the table (page 130) the azimuth of Polaris, corresponding to the latitude of the station of observation; find the natural tangent of this azimuth, and multiply it by the measured distance from the peep-sight to the rod. The product will express the distance to be laid off from the rod, exactly at right angles to the direction already determined that is, to the west for eastern elongation, and to the east for western elongation; and this point with the peep-sight, will define the direction of the meridian with sufficient accuracy for the needs of local surveyors. The position of the pole star may be found by means of the two stars /3 and a in the bowl of the "The Dipper' 1 (Fig. 68), which are called the "pointers" because of their pointing approximately to the pole star. THE TRANSIT. Construction. The transit is used for measuring horizontal and vertical angles directly, and for measuring bearings indirectly. It consists of a telescope mounted in standards attached to a divided horizontal plate, the telescope serving to define accurately the line of sight; while the horizontal plate, divided into degrees, minutes, and twenty or thirty seconds of arc, makes it possible to measure small horizontal angles. The instniment is provided with a three- or four-screw leveling base, by means of which it is attached to the tripod. The telescope is similar in construction to that of the Wye level, but is shorter and of less magnifying power, a power of from 24 to 26 diameters being about the average for the ordinary transit. The eye-piece may be either inverting or erecting, but the former is to be preferred. Since the principal function of the transit is to secure align- ment, the telescope must be capable of movement in a vertical 104 PLANE SURVEYING 97 plane, and to that end is supported in the standards by a transverse axis, permitting the telescope to be "transited," that is. turned through a complete vertical circle. For measuring horizontal angles the instrument is arranged with an upper and a lower motion, sometimes called the upper and the lower "limb." The lower limb is supported by the leveling base by means of a hollow conical axis; and into it is fitted, in turn, the conical axis of the upper limb. Each limb may be turned independently of the other, or they may be clamped together and to the leveling base. The lower limb carries the divided circle and the upper limb the vernier. For ordinary purposes the circle is divided to one-half degrees, and reads to single minutes by means of the vernier. It may also be divided so as to read to 20 or 30 seconds, and occasionally to 10 seconds. The divisions of the circle, however, should not be so crowded as to render the reading difficult, and the graduations should be properly adjusted to the magnifying power of the telescope. The verniers may be set at right angles to, or parallel with the line of sight, or at 30 thereto. With the verniers parallel with the line of sight that is to say, directly under the telescope or making an angle of 30 with the line of sight, the observer can read the angles without moving from his position, thereby avoiding the risk of disturbing the instrument by walking around it. See Fig. 69. For leveling the instrument, there are provided two level tubes set at right angles to each other. These are shown in the figure. One of them is attached to the upper plate, while the other may be attached either to the upper plate or to one of the standards. On account of lack of space these level tubes are quite short. The four-screw leveling base may consist of two parallel plates connected to each other by a one-half ball and socket joint, or the upper plate may be replaced by a ribbed casting. The four leveling screws rest in cups upon the lower plate and extend through the upper plate or casting. The leveling base is attached to the instrument proper, and the whole is attached to the tripod by screwing to a casting firmly attached to the legs. The vertical axis is furnished with a hook, to which may be attached a plumb-line for the accurate centering of the instru- 9S PLANE SURVEYING ment. A shifting center is also provided, by means of which, after the instrument has been approximately centered over a stake, it may be accurately adjusted by loosening the leveling screws and shifting the instrument upon the lower leveling base. See Fig. 69. Fig. 69. The three-screw leveling base is necessarily larger and differs in detail from the four-screw. The upper plate carrying the screws is permanently attached to the instrument; and the lower ends of 106 U. S. GEOLOGICAL SURVEY CAMP ON CHOPAKA MOUNTAIN. WASHINGTON PLANE SURVEYING 99 the screws rest upon the tripod casting, to which it is attached by a single center screw fitted with a strong spiral spring that engages upon a thread cut upon the vertical axis of the instrument. See Fig. 72. Fig. 70. The four-screw base commends itself from the fact that it can quickly be leveled approximately; and, no matter how much the threads are worn, the instrument can be brought to a -sol id bearing. 107 100 PLANE SURVEYING The three-screw base, however, is more easily manipulated, and all danger of binding the screws and springing the plates is obviated. Whichever type of instrument is preferred, the screws should work Pig. 71. smoothly and evenly, and the pitch should be adjusted to the sensitiveness of the bubbles. Most transits are fitted with a compass set in the upper plate 108 PLANE SURVEYING 101 Fig. 72. between the standards; but for city work, triangiilation, etc., the compass is dispensed with. 109 102 PLANE SURVEYING For measuring vertical angles, the transit is fitted with a vertical arc or circle divided usually to one-half degrees, and Pig. 73. reading to single minutes by the vernier. A level tube may also be attached to the under side of the telescope; and when this is 110 PLANE SURVEYING 103 provided the transit may be used as a leveling instrument. A striding level resting upon the standards may also be provided, by means of which the instrument can be more accurately leveled than by the short levels upon the upper limb. See Fig. 70. The telescope should always be provided with stadia wires, either fixed or adjustable, though the former are preferable. See article on "Stadia." The gradienter screw is a device attached to the clamp of the telescope, by means of which grades can be established, and horizontal distances, vertical angles, and differences of level can be measured with great rapidity. See article on " Gradienter" in Part III. Surveyor's Transit. This instrument is the plain transit, capable usually of measuring horizontal angles only, but occasion- ally fitted with a vertical circle or arc for measuring vertical angles. See Fig. 69. Engineer's Transit. When the instrument is provided with a vertical circle or arc, a level underneath the telescope, with or without gradienter screw, it is called the engineer's transit. See Fig. 70. Tachymeter. This term, meaning rapid measurer, has of recent years been applied to an instrument having a level attached to the telescope, a vertical arc or circle, and stadia wires. Such an instrument is adapted to the rapid location of points in a survey, since it is capable of measuring the three cp-ordinates of a point in space, i. e., the angular co-ordinates of altitude and azimuth, and the radius-vector or distance. The compass and gradienter are auxiliaries in the measurement of angles; arid an instrument having them in addition to the essential features mentioned above, is more perfectly adapted for tachymetric work. See Fig. 71. Theodolite. This term is applied to an instrument so con- structed that the telescope will not transit, but, in order to take backward sights, the telescope must be lifted out of its supports and turned end for end. See Fig. 73. Transit-Theodolite. This name is applied to an instrument in which the telescope not only can be transited, but also lifted out of its supports and turned end for end. See Fig. 72. Ill 104 PLANE SURVEYING Adjustment. When used merely as an angle-measurer, the following adjustments should be tested and, if necessary, corrected: 1st. To ascertain if the bubble tubes are perpendicular to the vertical axis of the instrument. To test, attach the instrument to the tripod and "set up" firmly on solid ground, preferably shaded from sun and wind. Revolve the transit upon its vertical axis so as to bring the bubble tubes parallel to a pair of diagonally opposite leveling screws. Bring the bubble of one of the tubes to the center by means of these screws. Do the same with the second bubble tube. Adjusting the second tube will throw the first one out, but repeat the alternate operations until each bubble stands in the center of its tube. Now revolve the instrument upon its vertical axis through 180, and note if the bubble of each tube still stands in the center. If so, the tubes are in adjustment. If the bubble of either tube runs to one end, bring it half-way back to the center by raising the opposite end of the tube by means of the capstan-headed screw. Relevel the instrument by the leveling screws, and again test the tubes. Repeat the operation, until the bubbles stand in the centers of the tubes in all positions of the instrument. It is advisable to carry out this adjustment as precisely as possible, as it will facilitate the remaining adjustments. If after several trials, it is found impossible to adjust the bubbles to the centers of the tubes, either the vertical axis is bent or the plates are sprung, and the instrument should be sent to the maker for correction. If one tube adjusts and the other does not. the fault is in the tube, and a new one should be ordered. 2d. To make the line of collimation revolve in a plane, or, in other words, to make the line of collimation perpendicular to the horizontal axis of the telescope* To test, having made the first adjustment, level the instrument carefully and clamp the upper limb. Drive a stake into the ground about 300 feet ahead of the instrument, and drive a tack in the head of the stake. By means of the lower motion revolve the instrument on its vertical axis until the intersection of the cross- hairs approximately covers the tack. Now clamp the lower motion, and carefully adjust the line of sight upon the tack by means of the lower tangent screw. Without disturbing either the upper 112 PLANE SURVEYING 105 or the lower limb, transit the telescope, that is, revolve it vertically, and sight to a tack in the head of a stake driven into the ground about 300 feet behind the instrument. Carefully adjust the tack to the intersection of the cross-hairs. Now unclamp the lower motion, and revolve the instrument upon its vertical axis until the intersection of the cross-hairs again covers the tack in the first stake. Clamp the lower motion, adjust the line of sight carefully by means of the tangent screw, again transit the telescope, and sight in the direction of the second stake. If the intersection of the cross-hairs falls upon the tack in the second stake, the line of collimation is in adjustment. If it does not, it will have to be adjusted. In Fig. 74, A is the position of the instrument, and B is the forward stake. If the instrument is in adjustment, the line of sight after transiting the telescope and revolving upon the vertical axis should strike the point B'. If the instrument is not in Fig. 74. adjustment, the line of sight after transiting the telescope will in the first instance strike some point as C. Drive a stake at this point and carefully center a tack. After revolving the instrument upon its vertical axis and again transiting the telescope, the line of sight will fall at a point C' as far on one side of B' as C was on the other. Drive a stake at C ' and carefully center it. Carefully measure the distance C C ' ; and center a stake at B ' , half-way between the two points. Now, by means of the capstan-headed screws attached to the diaphragm carrying the cross-hairs, move the cross-hairs until their intersection covers the point B", midway between B' and C'. Now repeat the operation of testing the adjustment and correcting the position of the line of collimation, until the points B and B ' are in the same straight line. It is necessary only that the line of collimation shall be accurately in adjustment; but for convenience in using the transit 113 106 PLANE SURVEYING as an angle-measurer it is desirable that the vertical cross-hair be at right angles to the horizontal axis of the telescope when the instrument is level. To test this, set up the instrument at some convenient point, 200 or 300 feet from a wall, tree, or other convenient object, upon which a point is clearly denned by a tack or otherwise. Carefully level the instrument, and cover this point accurately with the lower extremity of the vertical hair. Clamp the horizontal axis of the telescope; and by means of the tangent-screw slowly move the telescope in a vertical plane, and note if the hair continues to cover the point from one extremity to the other. If it does, the hair is in its proper position. If riot, loosen the diaphragm screws and turn the diaphragm vertically until the hair covers the point from end to end. This adjustment will disturb the last one and the two must be tested and corrected alternately until in perfect adjustment. If the transit is to be used for leveling, it is necessary that the horizontal cross-hair be in the optical center of the object glass. To test, set the instrument up firmly 200 or 300 feet from a wall, tree, or other convenient object, and, after leveling, carefully center the intersection of the cross-hairs upon a well-defined point. Clamp the axis of the telescope, turn the instrument upon its vertical axis through 180, and carefully center a point upon the intersection of the cross-hairs in this new position. Clamp the vertical axis, unclamp the telescope axis, transit the telescope, and carefully center the intersection of the cross-hairs upon the first point. Now clamp the telescope, loosen the vertical axis, and again revolve the instrument through 180. If the line of collimation again strikes the second point, the horizontal cross-hair is in adjustment. If not, carefully center this third point, bisect the distance between the second and third points, and move the cross- hair diaphragm until the intersection of the cross-hairs covers this fourth point. This adjustment will disturb the last two, and the three must be repeated in succession until accurately adjusted. 3d. To make the horizontal axis of the telescope perpendic- ular to the vertical axis of the instrument. To test, set up the instrument as explained for the last adjust- ment, and, after leveling, center carefully a point at each extremity of the horizontal cross-hair. Turn the instrument upon its vertical 114 PLANE SURVEYING 107 axis and transit the telescope, bringing one extremity of the horizontal cross-hair upon one of the points previously established. If the other extremity coincides with the second point, the axis of the telescope is in adjustment. If the axis is out of adjustment, the method of procedure is best illustrated by Fig. 75. A A ' is the line covered between the extremities of the hori- zontal wire or hair when the axis of the telescope is in adjustment. If it is not in adjustment, the wire will in the first position of the telescope cover the line A B, and in the second position the line A B ' . Therefore bisect the distance B B ' , and raise or lower the adjustable end of the telescope axis until the wire covers A A ' . Now repeat the test, and the correction if necessary. 4th. To make the axis of the telescope bubble tube parallel to the line of collimatioti of the telescope. This adjustment should be tested and corrected by the " peg " method as follows: Select a piece of comparatively level ground, drive a stake, "set up" the transit over it, and carefully level by ..*- 150' A* 150' Fig. 76. the plate levels. Next drive two stakes into the ground, one in front of the transit and the other at the same distance behind it. In Fig. 76, is the position of the transit, and A" and B are two stakes, each 150 feet from C. D is a fourth stake behind B and in line with it from C. The transit being leveled by the plate levels, bring the bubble of the telescope tube to the center of the tube, by means of the tangent-screw attached to the horizontal axis of the telescope. Hold a level rod upon A, adjust the target to the 115 108 PLANE SURVEYING horizontal cross-hair, and note the reading. Unclamp the lower motion, turn the transit upon its vertical axis, and note the reading of a rod held upon B. The difference of the readings of the rod held upon the two points will give the true difference of level, no matter how much the telescope level may be out of adjustment. Now take up the transit and remove it to the point D. Carefully level the transit by the plate levels, and again bring the bubble of the telescope tube to its center. Hold the rod upon the point B and note its reading. Do the same at the point A, and take the difference of the two readings. If the telescope level is in adjustment, this difference will be the same as found when the instrument was over the point C. Otherwise the tube is out of adjustment and will be corrected as follows: Let x represent the difference of level of A and B when the transit is at C. Let y represent the difference of level of A and B when the transit is at D. Let z represent the difference between a? and y. If y is greater than a?, subtract z from the rod reading upon A for the transit at D, and set the target at this new reading. Re- volve the telescope upon its horizontal axis, by means of the tangent-screw, until the horizontal wire accurately bisects the target. Now clamp the telescope axis, and bring the bubble to the center of the tube by means of the capstan-headed screw at one end of the tube. Again hold the rod upon B, and then upon A, and take the difference of their readings. If this difference now agrees with the true difference of elevation of the two points, the adjust- ment is complete. Repeat the operation as often as may be necessary. If y is less than x, add z to the rod reading upon A for the transit at D, and set the target at this new reading. Bisect the target by the horizontal cross-hair as before, clamp the horizontal axis, and bring the bubble to the center of the tube. Test and repeat as described before. Some transits are provided with an adjustable vernier to the vertical circle or arc, which should read when the telescope is horizontal. The former adjustment having been completed, the vernier can then be readily fixed in place. 116 PLANE SURVEYING 100 The cross-hair intersection should be in the center of the field ot vision of the eye-piece, and this adjustment may be made by means of the capstan-head screws attached to the eye-piece tube. To "set up" the transit. Lift the instrument out of the box by placing the hands underneath the plates. Avoid lifting it by the telescope or the standards. In attaching it to the tripod be careful that the threads engage properly, and screw it down firmly. Examine the tripod legs, and see that they are properly attached to the tripod head, neither too tight nor too loose. See that the tripod shoes are tight, and, before taking up the instrument, lightly clamp all the movable parts to prevent unnecessary wear and straining. Carry the instrument in the most convenient way, taking care not to hit it against trees, lamp-posts, doors, etc. To center the transit over a stake, rest one leg of the tripod upon the ground, and, grasping the other legs, pull the instrument in the proper direction to cover the stake. Now attach the plumb- line, and after bringing it to rest as close to the top of the stake as possible, note if the point is directly over the point in the stake. If it is not too far off the center, it may be brought closer by forcing the opposite leg into the ground or by a further spreading of the legs. After the instrument has been approxi- mately centered, it may be accurately adjusted by means of the shifting head. The operation of "setting up" is difficult of description, and facility can be attained only by practice. Avoid having the plates too much out of level, as this will result in unnecessary straining of the leveling screws and plates. Having centered the instrument over the stake, level it up by the levels upon the horizontal plate. To do this, turn the instrument upon its vertical axis until the bubble tubes are parallel to a pair of diagonally opposite plate-screws. Now, as you stand facing the instrument, grasp the screws between the thumb and forefinger, and turn the thumb of the left hand in the direction the bubble must move. Turn both thumbs in, or both thumbs out. Adjusting one tube will disturb the other, but adjust each alternately until the bubble of each remains in the center. To measure a horizontal angle by means of the transit. Set up the transit over the point C, Fig. 77. Set the verniers 117 110 PLANE SURVEYING to read 0, and clamp the upper limb. Now revolve the transit upon its vertical axis -by the lower motion, and sight to A. Clamp the lower motion, and accurately adjust the intersection of the cross-hairs to the point by means of the lower tangent-screw. Now unclamp the upper limb, and turn it upon its upper vertical axis to the point B. Clamp the upper limb and adjust the line of sight by the upper tangent-screw. The angle will now be found recorded upon the horizontal circle. If it becomes necessary to repeat the angle, loosen the lower motion, and, without disturbing the upper plate, turn the instrument back to the point A. Next clamp the lower motion, loosen the upper, arid turn the telescope to the point B. The sum of the two measured angles will now be found recorded upon the horizontal circle. Repeat as often as necessary, and divide the total horizontal angle by the number of repetitions for the probable value of the angle ABC. The operation of laying off a certain angle is essentially the same as the preceding, except that after the point A has been centered and the required angle laid off upon the horizontal circle, the tack in the stake B must be moved back and forth until it is accurately centered at the intersection of the cross-hairs. To survey a series of lines by means of the transit. "Set up" the transit over the point A, Fig. 78, and make the verniers read 0. Have the north end of the plate ahead, that is, in the direction of the survey. If a true meridian line has previously been laid out, the declination of the A needle may be determined from it. Without disturbing the upper plate, turn the transit upon its lower motion, and center upon B. Let the needle down upon its pivot, and as soon as it has come to rest take the bearing of the line Fi S- 78 ' AB. To measure the length of the line, hold one end of the tape or chain directly under the point of the plumb-bob, and send the head man in the direction of B. As he reaches the end of his tape, place him accurately in line by the vertical hair. For this purpose 118 PLANE SURVEYING 111 the telescope should be turned in a vertical plane to bring the intersection as closely as possible to the top of the stake. Now repeat this operation until the point B is reached. Move the transit to the point B, arid set it up as before with the north end of the plate ahead. Transit the telescope, and examine the verniers to see if they read 0. By means of the lower motion, center upon A, lower the needle, and take the bearing. Now clamp the lower motion, transit the telescope, and revolve the upper plate so the telescope points to C. Read the angle, which is that between A B produced and B C. Also read the bearing from the compass. Now measure B C as explained before, take up the instrument, and set it up at the point C. Continue thus until the series of lines have been surveyed. The angles measured are those indicated by the dotted lines (Fig. 78), and are called deflection angles. It is desirable in work of this character to calculate the bearing or azimuth of a line from the deflection angle, and to check the result by the compass. Referring to the figure, the records of the survey will be kept as follows, using the left-hand page of the transit or field book and starting from the bottom of the page: DEFLECTION. ; DKJ)UCKn LEFT. RIGHT. BEARINGS. 7 + 95 275 52 0' 852 45' E S5230'E 5 + 30 265 51 0' N7515' E N7530' E 2 + 90 240 31 30' 8 53 45' E 8 53 30' E 290 S85E It is not absolutely necessary A n the method described above, that the verniers should be set to read before aligning the instrument, provided that the verniers are read before turning off the angle. For instance, if the verniers read 40 15' after the instrument is set up over a stake, and after sighting along a certain line, read 60 15', the angle between the two lines is the difference of the reading of the verniers, or 20. As already explained under "the compass," the direction of a line may be given by its azimuth. If azimuths are computed from 110 112 PLANE SURVEYING the north to the right through 360, the azimuths for the preceding case will be as follows, as illustrated by the diagram : Since the bearing of the first line A B (Fig. 79), given by the compass, is S 85 E, its azimuth will evidently be the difference between 180 and 85, or 95. Since the second line B C deflects to the right by 31 30', its azimuth will evidently be the sum of the angles 95 and 31 30', or 126 30'. Since the third line C D deflects to the left 51 0', its azimuth will be the differ- ence of the angles 126 30' and 51 0', or 75 30'. The fourth line D E deflects to the right 52 0', arid there fore its azimuth will \.e the sum of the angles 75 30' and 52 0', = 127 30'. The same diagram may serve to illustrate the method of deducing the bearings of a series of lines from the deflection angles. Since the bearing of the first line as given by the compass is S 85 E, and the second line deflects to the right 31 30', it is evident that this second line decreases its easting by that amount, so that its bearing will be the difference between 85 and 31 30', = S 53 30' E. Since the third course deflects to the left by 51 0', its bearing to the east will be increased by this amount, but will pass into the northeast quadrant by 14 30', making its bearing 90 14 30', = N 75 30' E. The fourth course deflects to the right 52 0', returning to the southeast quadrant by 37 30', making its bearing 90 37 30', or S 52 30' E. 120 PLANE SURVEYING Traversing. This is a method of observing and recording the directions of a series of lines of a survey, so as to read off, upon the horizontal circle, the angles that the lines make with some other line of the survey, which may be either a true meridian or some line adopted as a meridian for that survey. Before starting out upon a traverse, it is best to lay out upon the ground a true meridian, either from observations on Polaris or by means of the "Solar transit," as will be explained later on. This line should be 300 or 400 feet in length, clearly denned by stakes carefully centered, one of the stakes preferably being the first station of the survey. The transit can now be set over this stake, and the line of sight carefully centered upon the second stake by means of the lower motion, the verniers first having been set to read 0. The subsequent operations can best be illustrated by a diagram. Fig. 80. First lay out a meridian through station 1 (Fig. 83), and define it by a stake driven 300 or 400 feet away towards N. Both of these stakes should be carefully " witnessed," that they may be recovered at any time. To begin the survey, carefully center the transit over station 1, with verniers set to zero; turn the instrument upon its lower motion until the line of sight approximately covers the stake at the north end of the meridian, and carefully center it by the lower tangent-screw; lower the compass needle (if there is one), and note and record the magnetic declination. Next, with the lower motion securely clamped, unclamp the upper motion, and revolve the upper plate in the direction of station 2. Clamp the upper motion, and carefully center the line of sight by the 121 114 PLANE SURVEYING upper tangent-screw. Note and record the angle upon the plate, which will be the azimuth of the line 1 2. Measure the distance from 1 to 2, and note and record the compass bearing as a check. Now, with the upper motion securely clamped, remove the transit to station 2; and carefully center it over the stake with the north end of the plate ahead, that is, in the direction from 1 to 2. Transit the telescope, unclamp the lower motion, and bring the line of sight to cover station 1. Carefully center it by the lower tangent-screw. Clamp the lower motion, transit the telescope, unclamp the upper motion, and revolve the upper plate until the line of sight falls upon station 3, carefully centering it by the upper tangent-screw. Read and record the plate angle, which will be the azimuth of the line 2 3. Measure the distance from 2 to 3, and read the bearing of the needle for a check. Now see that the upper plate is securely clamped, move the instrument to station 3, and proceed as before; and so on through- out the traverse. In the above example (Fig. 80), since the first line is in the north- east quadrant, its bearing N. 38 E. will be the same as its azimuth (38), and this angle is recorded upon the plate. The second line, however, passed into the southeast quadrant, and its azimuth will be recorded upon the plate as 38 + 81, or 119. Its bearing, how- ever, will be S. 61 E. The azimuth of the third line will be 38 + 81 - 23, or 96. Its bearing will be S. 84 E. In a similar manner, the bearing of the fourth line may be deduced from the azimuth as shown upon the plate. Fig. 81 illustrates a traverse beginning in the southeast quadrant. The bearing of the first line is S. 41 E., and therefore its azimuth will be 180 - 41 = 139. The bearing of the second line will be S. 73 E., and its azimuth will be 107; and so on with the remaining courses of the traverse. By a similar process of reasoning, the azimuths of courses in the southwest and northwest quadrants may be deduced from the bearings, and vice versa. We therefore establish the following rules: (a) Courses in northeast quadrant Azimuth = Bearing. (6) Courses in southeast quadrant Azimuth = Supplement of Bear- ing. 122 PLANE SURVEYING 115 (c) Courses in southwest quadrant Azimuth = 180 + Bearing. (d) Courses in northwest quadrant Azimuth = 360 - Bearing. Bearing north: Azimuth = or 360. " east: " = 90. " south: " == 180. " west: " = 270. For convenience in determining azimuths, the inside graduations of the horizontal circle may be numbered from to 360 to the Fig. 81. right, beginning at the north end; and the outside graduations from to 360 to the right, beginning at the south end. Then keeping the north end of the plate ahead that is, in the direction of the traverse for a traverse beginning in either the northeast or north- west quadrants, azimuths will be recorded directly upon the plate from the inside graduations; and for a traverse beginning in either the southeast or southwest quadrants, azimuths will be recorded directly upon the plate from the outside graduations. Traversing is particularly adapted to surveying roads, streets, railroads, shores of lakes, river banks, etc.; and in land surveying it possesses an advantage over the method by interior angles, on account of the readiness it affords in obtaining the bearings from the azimuths, and the greater rapidity with which the work may be platted, since the angle which each line makes with the assumed meridian or refer- ence line, may be taken at once from the field notes. In United States government surveys, when a traverse is run to mark the divisions between private estates and bodies of water retained as public property, it is called a meander line. 110 PLANE SURVEY ING Keeping Notes. All notes of measurements of angles and dis- tances should be recorded as soon as made, in a special notebook adapted to the purpose. Avoid the practice of making notes upon scraps of paper and in small pocket notebooks and of filling in details from memory. The notes will probably Jbe used by other persons unfamiliar with the locality, for platting and for general information, and these persons must depend entirely upon what is recorded, and how it is recorded, for their interpretation. To this end the notes should be clear and concise, yet full enough to give all necessary information. They should permit of only one interpretation, and that the correct one. The note keeper should bear in mind constantly the nature of the survey and the object to be attained, and this will enable him to determine what measurements are necessary. Do not crowd the notes. Use the left-hand page of the book for notes, and the right- hand page for such sketches and remarks as may be necessary. The record is usually made with a pencil, using a medium hard grade. If incorrect entries are made, erase them neatly; but avoid errors as much as possible, as too many erasures tend to discredit the work. After each notebook is filled, label it with the subject of the survey, the dates between which it was recorded, and all other information that may be of service in filing for future reference. Above all, do not lose a notebook, as it may contain information that cannot be recovered at any price. Referring to Fig. 80, the notes would be entered in the record as follows: STATION DISTANCES DEFLECTIONS AZIMUTH NEEDLE RIGHT LEFT 1 2 3 4 220 . 00' 225.00' 235 . 00' 190.00' 38 00' 81 00' 23 00' 40 00' 38 00' 11 9 00' 96 00' 56 00' N. 38 00' E. S. 61 00' E. S. 84 00' E. N. 56 00' E. Figs. 82 and 83 illustrate the method of keeping notes by means of sketches. Checking the Traverse. For an ordinary survey not involving unusual precision, a transit reading to single minutes will be sufficient. A single measurement will ordinarily give the angle with sufficient PLANE SURVEYING 117 accuracy; but should a check upon the measurement be deemed necessary the angle may be "repeated" as explained on page 109. As a further check against errors in angles, the magnetic bearing of each line should be read, showing the approximate directions of the lines, and by comparison with the azimuths or the deduced bear- ^, ings, will serve to point out gross errors, as for instance reading right for left when measuring deflec- tion angles. This check upon the Fig. 82. Fig. 88. angles should always be applied in the field, so that errors may be rectified before leaving the work. If the traverse involves a closed area, the accuracy of the transit work may be tested by adding together all of the measured angles. The sum of the interior angles should equal (n-2) X 180, n being the number of sides of the field. For the deflection angles, the sum of all the deflections to the right should differ from the sum of all the deflections to the left by 360; that is to say, the algebraic sum of the deflection angles should be 360. It is sometimes desirable to check the lengths of the courses before leaving the field. If there is any reason to question the accuracy PLANE SURVEYING of the measurement of any line, it should be measured, preferably in the opposite direction. In a closed traverse, it is well to run diagonal lines across the traverse as an additional check upon both the angles and the measured distances. For city work, the engineer should lay out a true meridian 300 or 400 feet long, and mark the extremities of the line by per- manent monuments set in the ground and carefully protected from disturbance. To do this, in some convenient place permit- ting an unobstructed line of eight, drive a large stake, and mark its center by a hollow -headed tack. Center the transit carefully over this point, and proceed to lay out a true meridian, preferably with the solar attachment. Mark the direction of this line by a second stake carefully centered by a tack as before. Now, about 25 feet from the first stake, and in line with it and the second stake, exca- vate a hole in the ground about three feet in diameter and five feet Fig. 84. deep, or deep enough to be below the frost line. Next build a foundation of concrete about two feet square and three feet deep. Before this has " set," insert in it a cut-stone post about nine inches square at its lower end and of such length that its top will come just below the surface of the ground, and having set into its top a copper bolt about inch by 4 inches. The post may be centered in the concrete by the transit, and should be set "plumb." Now locate and build a second monument in line with the second stake and a few feet from it. After the concrete has set firmly, again set the transit carefully over the center of the first stake, and accurately align it by the tack in the second stake. Now "plunge" (reverse) the telescope, and carefully center a point in the top of the copper bolt; mark this point with a steel punch. In the same way center a point in the top of the bolt of the second monument. The monuments may be protected by enclosing them in cast-iron valve-boxes with covers. Either one or both of these monuments may be used as "standard" bench-marks from which PLANE SURVEYING 111) all the levels and grades may be ascertained. For this purpose a city datum may be assumed, or better, the bench-mark may be connected by a line of levels with a bench-mark of the II. S. Coast and Geodetic Survey, or, if such is not available, with a bench- mark of the nearest railroad. THE STADIA. Attached to the diaphragm carrying the horizontal and vertical hairs, are two auxiliary horizontal hairs called "stadia'' hairs or wires. These hairs may be either fixed in position or adjustable, but the fixed hairs are the better for field use and cost much less. See Fig. 84. Any instrument-maker will equip a level or a transit with either fixed or adjustable stadia wires, and they should be included in every outfit. The stadia is used for measuring horizontal distances and differences of elevation, without the use of chain or tape or other apparatus except the leveling rod or a specially graduated stadia Fig. 86. rod. It is based upon the principle of the similarity of triangles. Thus, if the stadia hairs are spaced so as to intercept one foot upon a rod held at a distance of one hundred foot, the rod intercept for any other distance will be in direct proportion to the first. See Fig. 85. 120 PLANE SURVEYING Unfortunately the construction of the telescope of an engineer- ing instrument modifies the above simple statement, and a formula for the use of the stadia will now be deduced. Let O in Fig. 86 be the optical center of the object-glass of the telescope. This point may be assumed at the center of the lens, and the error involved in such assumption is inappreciable and may be neglected. Let SSi be a portion of the stadia rod covered by the stadia hairs CCi. From C and Ci draw the lines CSi and CiS through the optical center of the object-glass. Upon looking through the eye- piece of the telescope, O will be seen a^ at Si and Ci as at S. C 3 c, Fig. 87. Call i the distance between the stadia hairs, s the intercept upon the rod,/' the distance from O to the wires, and d the distance of the rod from O. The triangles COCi and SOSi are similar, and therefore we have the proportion i:s ::/' : a therefore d = (1) But/' varies with d. That is to say, if the rod were to be moved closer to the instrument, as at QJ), the lens would be moved farther from the wires, or the wires from the lens, and in either case the wire interval would intercept a shorter space upon the rod, as S 2 S 3 . The ratio -*C -, or its equal , will therefore vary I 1 S for each position of the rod. But however they vary, we have from a well-known principle of optics: PLANE SURVEYING 121 +- in which f is the principal focal distance of the lens, and f ' and d are any pair of conjugate focal distances. Substituting the value of - . 7 from (1) in (2), there results the equation d=ts+f. (3) Equation 3 gives the distance of the rod from the lens. We can establish some very important relations: In Fig. 87 lay off OF ' = OF = principal focal distance of lens =f. C and Ci being the stadia wires, draw C D and Ci E parallel to the axis of the lens, and through F' draw D Si and E S; then will S Si = the intercept upon the rod. The distance of the rod from the point F' is ] : cos 0. Hence S = S- C08( ' + * ) cos 6 cos cos d> - sin sin = b -r - - = b' (cos 9 - sin (f> tan 0): but tan ~ T/ITT ? an< ^ therefore S = S' (cos - sin = ,-^A 1UU J.UU ' Therefore II', the distance along the slope, = S' (100 cos (j> - sin ) ; and H, the horizontal distance, = II' cos = S' (100 cos 2 (f> - cos - $ sin 2 <) = 100 S' - S' (100 sin 2 < -I- 1 sin 2 0). It may be well to note that the lower reading of the rod need not nec- essarily be such as to make CD = AB, but only as a matter of convenience. EXAMPLE FOR PRACTICE. Upper rod reading = 7.49 Lower rod reading = 4.67 Vertical angle of lower rod reading = 15 35'. Required to find the distance parallel to the slope between B and D and the horizontal distance AG. Let it now be required to find the difference of elevation between B and D = CG. Evidently CG = II tan (j> = S' (100 cos 2 tan < - cos sin tan (). = S' (100 sin cos - sin 2 ] = S' (100 I sin 2 - sin 2 \ In the last example, determine the difference of level of B and D. It must not be forgotten of course, that the gradienter used in this way cannot give results eo accurately as the spirit level; PLANE SURVEYING 139 but nevertheless, for rapid work, the results will be sufficiently correct. If the student possesses a set of stadia reduction tables, the values of sin 2 and i sin 2 can be taken out at once and much labor saved. To Lay out a Meridian with the Transit. By means of the North, Star at Upper or Lower Culmination. Twice in 24 hours (more exactly, 23 hours 56 minutes) the north star "culminates"; that is to say, it attains to its maximum distance from the pole, above or below it. At the moment of culmination, the star is upon the meridian and if, therefore, a line be ranged out upon the ground in the same vertical plane, it will define a meridian. Set up the transit over a peg, in an open space, giving an unobstructed view of a line about 400 or 500 feet long. Level the instrument carefully (it should be in perfect ad justment), and, a few minutes before the time of culmination, as given in the table, focus the intersection of the cross-hairs upon the star ; clamp the plates, the vertical axis, and the horizontal axis of the telescope. Now by means of the tangent-screws attached to the vertical axis and to the vertical circle, move the telescope in azimuth and altitude, keeping the cross-hairs fixed upon the star. After a time it will be found that the position of the star no longer changes in altitude; it is then upon the meridian. Now clamp the vertical axis, plunge the telescope, and carefully center a stake 400 or 500 feet from the instrument; the line connecting the two stakes, will define the true meridian. , The whole operation may be repeated several nights in sue. cession, and the mean of all the results taken. By Means of the North Star at Eastern or Western Elon- gation. Twice in 24 hours, the north star attains to its maximum distance east or west of the pole, called its eastern or western " elon- gation." If a line be ranged out upon the ground in the direc- tion of the star at, say, the time of eastern elongation, and again at the time of western elongation and if the angle between these two lines be bisected by a third line, this last line will evi- dently be a true north and south line. Otherwise. Having laid out a line upon the ground in the direction of the north star say at western elongation take from 140 PLANE SURVEYING a table the azimuth (or bearing) of the star at such time, and upon the horizontal plate of the transit set off this angle to the east and range out a line which will therefore be a true north and south line. If the position of the-star is taken at eastern elongation, the azimuth must be turned off to the west. Set up the transit over a peg a few minutes before the star attains its maximum elongation, as given by the table. Level, and fix the line of collimation upon the star, following its movement as described under the previous method. After a time, it will be found that the movement of the star in azimuth ceases; the star has then attained its maximum elongation. Now clamp the ver- tical axis of the instrument, plunge the telescope, and center a stake in the proper direction. Now take from the table the proper azi- muth, revolve the upper plate through the given angle in the proper direction, and range out a line upon the ground for the true meridian. In order to determine the azimuth of the north star at eastern or western elongation, it is necessary to know the latitude of the place of observation. Definitions. The altitude of a star is the vertical angle at the instrument included between the plane of the horizon and the line from the instrument to the star, as given by the line of colli- mation. The latitude of a place is equal to the altitude of the pole. If, therefore, we have any method of determining the altitude of the pole, the latitude of the observer is known at once. The altitude of the pole may be determined by observing the altitude of the north star, first at its upper culmination and again at its lower culmination. The mean of these observations, cor- rected for refraction, will give the altitude of the pole and there- fore the latitude of the observer. See tables of refraction of Polaris. Set up the transit and level it, and proceed in the eame man- ner as described under the first method for laying out a true merid- ian. When the star has reached its maximum distance above or below the pole, as indicated by the line of collimation moving in a horizontal plane, clamp the horizontal axis of the telescope and read the angle upon the vertical circle. The result will be the altitude of the star, say at upper culmination. Repeat the operation PLANE SURVEYING 141 at lower culmination. Now, if A represents the altitude at upper culmination, and A l the altitude at lower culmination ; d the refrac- tion at upper culmination, and d l the refraction at lower culmina- tion, then A p , the altitude of the. pole (= latitude of the place), will be given by the following: . A p = i(A + Ai-d-4) It will be well to repeat these observations and take the mean of the results as the probable altitude of the pole. Fig. 99. THE SOLAR TRANSIT. The solar transit is an ordinary engineer's transit fitted with a solar attachment. Of the many forms of solars in use, that invented by G. N. Saegmuller, Washington, D. C., seems to be the favorite. In its latest form it is shown in Fig. 99, and consists of a telescope and level attached to the telescope of the transit (see Fig. 100) in such a manner as to be free to revolve in two direc- tions at right angles to each other. When the transit telescope is horizontal and the bubble of the solar in the center of its tube, the tmxiliary telescope with its bubble revolves in horizontal and ver- tical planes. 142 PLANE SURVEYING PLANE SURVEYING 143 If now the line of collimation of the transit be brought into the meridian, the telescope pointing to the south, then, if we lay off upon the vertical circle, upward, the co-latitude of the place, the polar axis of the solar will be parallel to the axis of the earth. If now the two lines of sight are parallel and the solar telescope is revolved upon its polar axis, it is evident that its line of sight will describe a plane parallel to the plane of the equator. If now the transit telescope be still maintained parallel to the equator, if we turn the Bolar telescope upon its horizontal axis until the angle between the two lines of collimation equals the declination of the sun, then when the solar telescope is revolved upon its polar axis, its line of collimation will follow the path of the sun for the grven day, provided there be no change in the sun's declination. If therefore the solar telescope is revolved until the image of the sun is brought between a pair of horizontal and vertical wires, pro- vided in the telescope for that purpose, at that instant the line of sight of the transit telescope is in the meridian. The horizontal axis of the solar telescope and the polar axis of the Bolar are provided with clamps and tangent-screws by means of which careful adjustments may be made.. Two pointers are at- tached to the solar telescope, so adjusted that when the shadow of the one is thrown upon the other, the sun will appear in the field of view. There are also provided colored glass shade's to the eye- piece to protect the eye when observing upon the snn. The objective and the cross-hairs are focused in the usual way. Adjustments of the Solar Transit. It is assumed in what follows that the transit is in perfect adjustment, particularly the plate levels, the horizontal axis of the telescope, and the zero of the vertical circle. 1. To adjust the Polar Axis. The polar axis should be vertical when- the line of collimation and the horizontal axis of the telescope are horizontal. To make this adjustment, level the tran- sit by means of the plate levels. If the telescope is not fitted with a level, make the vernier of the vertical circle read zero. Now bring the bubble of the solar to the center of its tube and clamp the horizontal axis. Loosen the clamp of the polar axis, and turn the solar upon its polar axis through 180. If the bubble remains in the center of the tube, the solar axis is in adjustment. If the 144 PLANE SURVEYING bubble runs toward one end of the tube, correct one-half of the error by revolving the solar telescope upon its horizontal axis and the other half by means of the capstan -headed screws at the base of the solar. If the telescope of the transit is fitted with a level, it will be better to test the vertically of the vertical axis by means of it, since it is longer and more sensitive than the bubbles upon the plate. To do this, revolve the telescope upon ita vertical axis until it is directly over a pair of diagonally opposite plate screws, and bring the bubble to the center by means of the tangent-screw attached to the horizontal axis of the telescope. Now revolve the telescope upon its vertical axis through 180, and note if the bub- ble runs to one end; if it does correct one-half the error by the parallel plate-screw and the other half by the tangent-screw of the horizontal axis, and repeat this test and correction until the bubble remains in the center in all positions. 2. To Adjust the Cross -Hairs of the Solar. The line of collimation of the solar telescope should be parallel to the line of collimation of the transit telescope. The first adjustment having been made, first bring the telescope into the same vertical plane by centering a stake by the transit telescope and clamping the verti- cal axis. Now turn the telescope of the solar upon the polar axis until the intersection of the cross-hairs covers the same point upon the stake, and clamp the polar axis. Now level both telescopes by bringing the bubbles to the center, and measure the distance between the axes of the two telescopes; draw at this distance two black parallel lines upon a piece of white paper. Tack up the paper against a wall, post, or other convenient object, adjusting it in position so that one black line is covered by the horizontal cross- hair of the transit telescope; notice if the other black line is cov- ered by the horizontal cross-hair of the solar; if so, the adjustment is completed; otherwise, move the diaphragm carrying the cross- hairs of the solar, until the second black line is covered. Adjust- ing the cross-hair diaphragm may displace the solar telescope ver- tically, so that the bubble should again be brought to the center of the tube, and the adjustment tested and repeated until the two lines of collimation are parallel, when the two bubbles are simul- taneously in the center of the tubes. PLANE SURVEYING 145 The Use of the Solar Transit. An observation with the solar transit involves four quantities as follows: 1. The time of day, that is to say, the hour-angle of the sun. 2. The declination of the sun. 3. The latitude of the place of observation. 4. The direction of the meridian. Any three of these quantities being known, the fourth may be determined by direct observation. The principal use of the solar transit is to determine a true meridian when the other three quan- titles are known. To Lay Out a True Meridian. Set up the transit over a stake; level the instrument carefully; and bring the lines of collimation of the telescopes, into the same vertical plane by the method pre- viously described. Take the declination of the sun as given in the Nautical Almanac for the given day, and correct it for refraction and hourly change. Revolve the transit telescope upon its hori- zontal axis so that the vertical circle will record this corrected dec- lination, turning it down if the declination ia north, and elevating it if the declination ia south. Now, without disturbing the posi- tion of the transit telescope, bring the solar telescope to a horizon- tal position by means of the attached level. ^ It is evident that the angles between the lines of collimation will equal the corrected declination of the Bun, and the inclination of the solar telescope to its polar axis will be equal to the polar distance of the sun. Next, without disturbing the relative positions of the two tele- scopes, set the vernier of the transit telescope to the co-latitude of the place, and clamp the horizontal axis. It is evident that the transit telescope is parallel to the equator, and that the solar tele- scope is in a position to describe the path of the sun when the line of collimation of the transit is in the true meridian; and unless the line of collimation is in the true meridian, the sun cannot be brought between the cross-hairs of the solar telescope. Therefore unclamp the vertical axis of the transit and the polar axis of the solar, and, maintaining the relative positions of the telescopes revolve the transit upon its vertical axis, and the solar upon its polar axis, until the sun is brought between the cross-hairs of the solar telescope. Now clamp the vertical axis of the transit and range out a line upon the ground for the true meridian. 146 PLANE SURVEYING The solar apparatus should not be used between 11 A. M. and 1 p. M. if the best results are desired. From 7 to 10 A. M. and from 2 to 5 P. M. in the summer will give the best results. The greater the hour-angle of the sun, the better the observation will be so far as instrumental errors are concerned. However, if the sun is too close to the horizon, the uncertainties in regard to refrac- tion will cause unknown errors of considerable magnitude. Observation for Time. If the two telescopes being in position, one in the meridian and the other pointing to the eun are now revolved upon their horizontal axes (the vertical remaining undisturbed) until each is level, the angle upon the horizontal plate between their directions, as found by sighting on a distant object, will give the time from apparent noon, reliable to within a few seconds. To Determine the Latitude. Level the transit carefully, and point the telescope toward the south, setting off the declination of the sun upon the vertical circle, elevating the object end if the dec- lination ia south, and depressing it if the declination is north. Bring the telescope of the solar into the same vertical plane with the transit telescope by the method previously described, level it carefully, and clamp it. The angle between the lines of collirnation will then equal the declination of the sun. With the solar tele- scope, observe the sun a few minutes before its culmination, by moving the transit telescope in altitude and azimuth until the image of the sun is brought between the cross-hairs of the solar, keeping it there by means of the tangent- screws until the sun ceases to rise. Then take the reading of the vertical circle, correct for refraction due to altitude by the table below, subtract the result from 90, and the remainder is the latitude sought. 154 TOPOGRAPHERS OF U. S. GEOLOGICAL SURVEY AT WORK ON MOUNTAIN SUMMIT IN ALASKA PLANE SURVEYING 147 Mean Refraction at Various Altitudes.* Barometer, 30 inches. Fahrenheit Thermometer, 50. Altitude. Refraction. Altitude. Refraction. 10 5' 19" 20' 2' 39" 11 4 51 25 2 04 12 4 27 30 1 41 13 4 07 35 1 23 14 3 49 40 1 09 15 3 34 45 58 16 3 20 50 49 17 3 08 60 34 18 2 57 70 21 19 2 48 80 10 Preparation of the Declination Settings for a Day's Work. The solar ephemeris gives the declination of the sun for the given day, for Greenwich mean noon. Since all points in America are west of Greenwich, by 4, 5, G, 7, or 8 hours, the declination found in the ephemeris is the declination at the given place at 8, 7, 6, 5, or 4 o'clock A. M. of the same date, according as the place lies in the "Colonial," " Eastern." Central,'' Mountain," or "Pa- cific" time belts respectively. The columns headed "Refraction Corrections" (see table) give the correction to be made to the declination, for refraction for any point whose latitude is 40. If the latitude is more or less than 40, these corrections are to be multiplied by the correspond- ing coefficient given in the table of " Latitude Coefficients " (page 148). Thus the refraction corrections in latitude 30 are G5 one- hundredths, and those of 50 142 one-hundredths of the correspond- ing ones in latitude 40. There is a slight error in the use of these latitude coefficients, but the maximum error will not amount to over 15 seconds, except when the sun is very near the horizon, and then any refraction becomes very uncertain. All refrac- tion tables are made out for the mean (or average) refraction whereas the actual refraction at any particular time and place may be not more than one-half or as much as twice the mean refraction, with small altitudes. The errors made in the use of these latitude coefficients are therefore very small compared with the errors re- * This table, as well as those following, is taken from the catalogue of George N. Saegmuller, Washington, D. C. 148 PLANE SURVEYING suiting from the use of the mean, rather than unknown actual, refraction which affects any given observation. Latitude Coefficients. LAT. COEFF. LAT. COEFF. LAT. COEFF. LAT. OOEFF. 15 .30 27 .5fi 39'-* .96 51 .47 16 .32 28 .59 40 1.00 52 .53 17 .34 29 .62 41 1.04 53 .58 18 .36 30 .65 42 1.08 54 .64 19 .38 31 .68 43 1.12 55 .70 20 .40 32 .71 44 1.16 50 .76 21 .42 33 .75 45 1.20 57 .82 22 .44 34 .78 46 1.24 58 .88 23 .46 a r > .82 47 1.29 59 1.94 24 .48 3C .85 48 1.33 CO 2.00 25 .50 37 .89 49 1.38 26 .53 38 .92 50 1.42 If the date of observation be between June 20 and September 20, the declination is positive and the hourly change negative; while if it be between December 20 and March 20, the declination is negative and the hourly change positive. The refraction cor- rection is always positive; that is, it always increases numerically the north declination, and diminishes numerically the south dec- lination. The hourly refraction corrections given in the ephem- eris are exact each for the middle day of the five-day period, cor- responding to that of hourly corrections. For the extreme days of any such period, an interpolation can be made between the adjacent hourly corrections, if desired. By using standard time instead of local time, a slight error is made, but the maximum value of this error is found at those points when the standard time differs from the local time by one- half hour, and in the spring and fall when the declination is chang- ing rapidly. The greatest error then, is less than 30 seconds, and this is smaller than can be set off on the vertical circle or declina- tion arc. Even this error can be avoided by using the true dif- ference of time from Greenwich in place of standard meridian time. EXAMPLES FOR PRACTICE. (1) Let it be required to prepare a *able of declination for June 10, 1904, for a point whose latitude is 40 20' , and which lies in the " Central Time" belt. PLA^E SURVEYING 149 Since the time is 6 hours earlier than that at Greenwich, the declination given in the ephemeris is the declination at the given place at 6 A. M. of the same date. This is found to be 23 0' 18". To this must be added the hourly chancre which is also plus and equal to 11.67". The latitude coefficient is 1.013. The following table may now be made out. I i i j j i HOUR DECLINATION IREF.COR. SF.TTINfl HOUR , DKCI.IN ATIOX REP. COR. 8KTTIXO 7 + 23 0' 30" + V 10" 23 1' 40" I 23 1/41" 18" 23 1' 59" 8 4- 23 0' 41" + 44" 23 1' 25" 2 23 1' 52" 22" 23 2' 14" 9 + 23 0' 53" + 29" 23 1' 22", 3 23 2' 04" 29" 23 2' 3.T' 10 -j-23r 5" + 22" 22 1' 27" 4 23 2' 16" 44" 23 3' (V 11 + 23 1' 17*i+ 18" 23 l f 35" 5 23 2' 28" 1' 10" 23 3' 38" PROBLEMS INVOLVING USE OF TRANSIT. Perpendiculars and Parallels. To erect ft perpendicular to a line at a given point of the line. Set up the transit over the given point, and with the verniers set to 0, direct the line of sight along the given line. Clamp the lower motion, unclamp the upper motion, and turn off an angle of 90 in the proper direction for the required line. To erect a perpendicular to an, inaccessible line at a given point of the line. Let AB, Fig. 101, be the given inaccessible line, and A the point of the line at which it is proposed to erect the perpendicular AD. Select some point II from which can be distinctly seen the points A and B of the inaccessible line. Setup the transit at the point II, and measure the angle AHB. Also from the point H run out and measure a line of any convenient length, and in such a direction that the points A and B can be seen from its extremity, as E. Now measure the angles AHE and BHE. Now set up the transit at E, and measure the angles BEIT, BEA, and AEH. In the triangle AHE, we know from measurement the length of the side HE, as also the angles AHE and AEII, from which may be calculated the length of the side AH, which is also one side of the triangle AIIB. From the triangle BEH, we have the length HE, known by measure Fig. 101. 150 PLANE SURVEYING Refraction Correction. Latitude, 40. January. February. March. April. May. Juno. 1 Ih. 1 58 1 1 Ih. l o:5 : 1 3h. 57 1 Ih. 28 1 5h. 1 11 2 2 16 2 1 10 ! 2 4 1 19 2 32 i '1 3 3 04 3 1 27 ; 3 5 2 18 2 3 39 g 1 19 8 4 623 1 1 54 3 Ih. 1 26 1 4 2 06 5 4 39 4 5 1 039 2 44 3 4 55 5 1 30 1 .7 2 23 3 30 5 1 2 1 37 K 1 59 6 3 054 i 1 26 8 4 043 2 2 11 5 3 2 04 6 2 1 06 7 4 1 14 :, 2 30 7 5 1 10 7 s 9 11 18 3 2 59 4 601 1 1 51 2 207 3 2 51 4 540 f> 7 8 9 JO 11 1J 4 321 1 1 21 2 1 31 3 1 56 4 3 04 7 g '.i 10 11 U 13 3 1 21 4 1 56 5 4 04 1 55 2 1 02 3 1 15 4 1 47 8 9 10 II U 13 M 5 2 08 1 036 2 41 3 51 4 1 10 5 1 58 1 34 7 s '. 11 13 3 37 4 53 5 1 26 1 25 2 29 3 36 4 51 5 1 22 s li 10 11 12 13 it 1 018 2 022 3 29 4 43 5 1 09 1 018 2 22 14 1 1 46 IS 1 1 16 1 1 o o o4 15 2 38 11 1 23 15 3 029 15 16 2 2 01 ll IS 3 1 48 IS 1C, 58 10 n 3 48 4 1 06 15 16 2 27 3 34 17 4 42 5 1 08 17 18 s 21 24 3 2 40 4 500 1 1 42 2 156 3 2 31 4 4 35 1 1 37 16 17 IS lit JO Jl JJ 23 4 2 47 5 8 39 1 1 12 1 1 20 3 1 40 4 2 31 5 6 49 1 1 07 11 19 JO 22 23 24 1 10 1 39 3 08 48 54 J 1 05 19 20 Jl JJ 23 5 1 49 1 032 2 036 3 45 4 1 02 5 142 1 030 2 34 17 11' JO Jl JJ 83 Jl 85 4 49 5 1 18 1 22 2 26 3 33 4 47 5 1 15 1 021 2 025 r.* JO Jl JJ 83 ji 25 1 018 2 022 3 028 4 42 5 1 08 1 018 2 022 3 029 25 28 J'.i 2 158 3 2 22 4 4 07 1 1 32 Jt j:. 26 27 JS 2 1 15 3 1 33 4 2 18 5h. 5 28 28 29 45 50 : 1 01 ! 2 34 26 28 ., so 3 42 4 58 5 1 36 1 028 2h. 32 JO 27 88 89 3 032 4 46 5 1 13 1 020 2 Off 88 L'7 28 j'." 4 42 5 1 08 1 018 2 022 3 29 2 1 44 30 1 42 80 4 31 80 4h. 43 .')" 3 2 13 31 2h. 47 4 44 Ml 4h. 3 41 31 5h. 1 11 July. August. September. October. November. December. j 5h. 1 09 1 1 Ih. 39 1 Ih. 59 1 2h. 3 21 1 Ih. 1 54 2 2 44 2 2 1 06 2 3 13 57 2 2 2 11 3 4 5 1 19 2 23 3 30 3 4 Ih. 26 2 30 3 37 3 4 3 54 4 1 14 5 208 a 4 3 1 21 4 1 56 5 4 04 3 4 4 1 1 32 3 4 3 2 59 4 601 5 6 4 43 5 4 53 6 1 42 fl 1 1 03 ,, 2 1 44 5 1 1 58 7 5 1 10 6 5 1 26 2 47 | 2 10 7 3 2 13 6 2 2 16 8 3 57 8 3 27 8 4 3 41 ^ 3 3 04 8 9 10 1 20 2 24 3 31 8 9 1 28 2 32 3 39 9 10 4 1 19 5 2 18 '. 4 06 5 39 9 10 5 1 1 37 8 g 4 623 11 4 44 10 4 55 ll 1 45 ll 1 07 11 2 1 50 10 1 200 12 5 1 11 M 5 1 30 2 50 U 2 15 U ll 13 It 15 1 021 2 25 3 32 18 it 1 30 2 34 3 42 13 ll 15 3 1 01 4 1 K 5 2 34 13 i: r, 3 33 4 18 5 39 18 ll 4 4 07 5 1 1 42 18 it 3 309 4 638 16 4 46 4 58 16 1 048 16 1 12 16 2 1 56 15 1 201 17 5 1 13 16 5 1 36 17 2 54 17 2 20 17 3 2 31 10 2 220 18 19 20 1 022 2 26 3 0X3 17 is in 1 032 2 036 3 45 is 19 JO 3 1 05 4 1 32 5 2 51 18 19 JO 3 40 4 31 5 29 18 19 JO 4 4 35 5 1 1 46 17 11' 3 3 11 4 647 5 21 4 47 JO 4 1 02 Jl 1 52 Jl 1 16 "1 2 201 JO 1 201 JJ 5 1 15 Jl 5 1 42 JJ 2 58 2 25 JJ 3 2 40 Jl 2 220 23 Jl 25 1 023 2 027 3 34 ..., 23 Jt 1 34 2 038 3 48 88 Jl 25 3 1 10 4 1 39 5 3 08 j. 1 ; jt 89 3 48 4 47 5 39 Jl 25 4 4 59 5 1 1 50 88 Jl 3 3 11 4 6 49 26 4 49 4 1 06 26 1 055 Jo 1 21 88 2 2 06 25 1 200 27 5 1 18 y, 5 1 49 2 1 02 87 2 31 3 2 49 88 2 2 19 88 29 SO 81 1 25 2 029 3 036 4 051 5h. 1 22 21 JS j'.i .",0 31 1 036 2 41 3 51 4 1 10 5h. 1 58 89 80 3 1 15 4 1 47 5h. 3 34 JS J'.I 80 81 3 56 4 3 04 5 11 01 Ih. 1 26 1 37 2 04 28 J'.i n 4 5 33 5h. 27 JS J'.l :;i 3 309 4 6 43 5h. PLANE SURVEYING 151 ment, as well as the angles BHE and BEH, from which we can calculate the length of the Bide BH, which is also one side of the triangle AIIB. Therefore in the triangle AHB, we have the lengths of the two sides All and BII by calculation; and the angle AIIB by measurement. We can therefore calculate the angle IIAB, which equals the angle AHD. Set up the transit at II, sight to A, and turn off the angle AHD ( IIAB), measuring off II D of a length equal to AH cos AHD. Then AD will be the perpendicular required, and its length will equal AH sin AIID. The calculation is as follows : In the triangle AHE, the angle HAE = 180 - (AHE + AEII), and therefore All : HE : : sin AEII : sin HAE; or, AH = HE si " f, sin HAE In the triangle 1IEB, IIBE = 180 - (BHE + BEH), and therefore HB : HE : : sin HEB : sin IIBE; sin HBE In the triangle AHB, the sum of the angles HAB and HBA = 180 AHB. Let x represent the difference of the angles IIAB and HBA. Then, from trigonometry, AH + HB : All - HB : : tan -J (IIAB + HBA) : tan A (IIAB - HBA); or, All + HB : All HB :: tan (ISO 1 - AHB) : tan i .r; or, AH + HB : AH- IIB :: cot -: tan J .- From this last proportion we find #, the difference of the two angles HAB and IIB A. We then have the simultaneous equations: IIAB + IIB A = c (say) HAB - HBA = (I (say) Therefore IIAB = -^t~; and HBA = ~~. 2 & EXAMPLE FOR PRACTICE Given HE (Fig. 101) = 125 feet; AHE = 122; AHB = 94; BHE =28; BEH = 121; BE A = 80; AEII =41. It is required to find the angle AHD, the length of HI), and the length of AD Aus. AHD'= 50" 54'; II D = 170.93 feet; AD = 217.92 feet 152 PLANE SURVEYING To let full a perpendicular to a line from a given point. Let AB, Fig, 102, be the given line, and C the point. Set up the transit at some point A of the given line, and measure the angle BAG. Take the instrument to C, sight to A and turn off an angle AOB = 90 5 - BAG. The instrument will then sight in the direc- tion of the required perpendicular CB. To let fall a perpendicular to a line from, an, inaccessible point. LetBC, Fig. 103, be the given line and A the inaccessible point from which it is desired to let fall the perpendicular upon BC. Set up the instrument, as at B; and, after measuring the AT- D Fig. 102. Fig. 103. length of BC, measure the angle ABC. Take the instrument to G, and measure the angle AGB. Then in the triangle ABC, AB : BG : : sin AGB : sin (AGB + ABC); and, BD = ABcos ABC; tan AGB BD = BC tan ACB + tan ABC' EXAHPLE FOR PRACTICE. Given BC (Fig. 103) = 250 feet; ABC = 6315'; ACB = 55^40'. Calculate the length of BD, and the length of AD. j BD = 126.2 feet. S ' \ AD = 250.7 feet. To let fall a perpendicular to an inaccessible line from a given point outside of the line. Let AB, Fig. 104, be the inacces- sible line, and G the point from which it is desired to let fall the perpendicular to AB. Through G run out and measure a line of any convenient length, as CD, and measure the angles ACB PLANE SURVEYING 153 DCB, and DCA. Set up the instrument at D, and measure the angles ADC and BDC. In the triangle BDC, we have given two angles and the included side, from which can be calculated the length of the side CB, In the triangle ADC, we have given two angles and the included side, from which can be calculated the length of the side AC. Then, in the triangle ACB, we have the lengths of the sides AC and CB, and the included angle ACB, from which can be calculated the angle CAB. If, then, the instrument be set up at C, and an angle ACE be turned off equal to 90 - BAC, the line of sight will point in the direction of the A required perpendicular, and the length of the perpendicular will be given by AC cos ACE. This same method will serve to trace a, line through a given point parallel to an inaccessi- Fig. 104. l>le line. For if, with the instru- ment at C, an angle ACA' be turned off equal to CAB, the line A' B' will be parallel to AB. Obstacles to Alignment. By Perpendiculars: "When a tree, house, or other obstacle obstructing the line of sight (see Fig. 105) is encountered, set up the transit at the point B, turn off a right angle, and measure the length of the line BC. Erect a second perpendicular CD at C, and measure its length. At D erect a third perpendicular DE, making DE = BC. Then the fourth perpendicular EF will be in the direction of the required line. The distance from B to E will be given by CD. If perpendic- ulars cannot be conveniently set off, let BC and DE make any equal angle with the line AB, so that CD will be parallel to it. By an Equilateral Triangle. At B turn off from the direc- tion of AB produced, an angle of 60 in the direction of BC (see Fig. 106), and make BC any convenient length sufficient to clear the obstacle. Set up the instrument at C and turn off an angle of 60 from BC to CD and make CD of a length equal to BC Finally at D turn off a third angle of 00 from CD to DB, and the line DE will be in the direction of AB produced.- The dis- tance BD will equal BC or CD. 154 PLANE SUKVEYING JJy Tricmgulation. Let AB, Fig. 107, be the line to be prolonged beyond tlie obstacle. Choose some point as C from which can be seen the line AB as well as some point D beyond the obstacle. With the transit at A, measure the length of AB and measure the angle BAG. Set up the transit at C, and meas- Fig. 105. Fig. 106. ure the angles BOA and ACD. Then, in the triangle ACB, we have one side and two angles known, from which can be cal- culated the lengths of AC and BC. In the triangle ACD, we know the length AC and the angles BAC and ACD, from which can be calculated the length CE, the angle BEC, and the distance AE. Therefore from C measure the distance CE, set up the transit at E, and turn off the angle CEF equal to 180 minus the angle BEC, for the direction of the required line. The length of BE will evidently equal AE - AB. Jjy a Random L'ine. When a wood, hill, or other obstacle prevents one end of a line (as B, Fig. 108) being seen from the other end A, run out and measure a random line, as AC, as Fig. 108. nearly in the required direction as may be guessed, until a point C is reached from which B can be seen. Xow, if convenient, measure the perpendicular offset from AC to the point B, from BC which can be calculated the angle CAB. Tan. CAB = r-p . If a right angle cannot be turned off at C, turn off any convenient angle and measure the distance CB; then, in the triangle ACB, there are given two sides and the included angle, from which can be calculated the anjj;le CAB. Kovv taking the transit back to A, the angle CAB can be turned off in the proper direction from PLANE SURVEYING 155 AC, and the correct line AB can be run out and measured in the proper direction. By Latitudes and Departure*. When- a single line such as AC cannot be run so as to come opposite the given point B (Fig. 109), a series of zigzag lines (as AC, CD, DE, EF, and FB) can be run in any convenient direction, so as at last to arrive at the desired point B. Any one of these lines (as, for instance, AC) may be taken as a meridian to which all of the others may be referred,and their bearings there- from deduced. Calculate the total latitudes and departures of Fig. 109. these lines, as AX and BX; then the bearing of the required line BA with respect to AC will be T>V given by Tan. BAG = -j^. By Triangulation. When obstacles prevent the use of either of the preceding methods, if a point C can be found from which A and B are accessible (see Fig. 110), measure the dis- tances CA and CB, and the angle ACB, from which can be calcu- lated the length of the side AC and the angle CAB. Now measure the angle ACD to some point D beyond the obstacle; then, in the triangle ACD, we have two angles and the included side, from which may be calcu- lated the length of the side CD. Fig, 110. Measure the distance CD in the proper direction, set up the tran- sit at D, and turn off an angle CDB equal to the supplement of ADC, for the direction of the required line. The distance from A to D may also be calculated from the triangle ACD, the stake at D given its proper number, and the line continued. If the distances CA and CB cannot be meas- ured, it will be necessary to measure a base-line through C, from the extremities of which the angles to A and B can be measured and the required distances calculated as before. 156 PLANE SURVEYING The following problem, as illustrated in Fig. Ill, is of fre- quent occurrence in line surveys. The line AB of the survey having been brought up to one side of a stream, it is desired to continue the line of the survey across the stream to the point C, the latter point being visible from B and accessible. It is required to find the length of the line BC, that the stake at may be given its proper number, and the survey continued from that point. With the transit at B, turn off the required angle to locate the point 0, and drive a stake at that point. If possible, deflect from BC a right angle to some point E, and measure the length of BE. Take the transit to E, and measure the angle BEC. The dis- tance BC is therefore: BC; = BE tan BEC. If it is not possible to turn off a right angle at B, then through B run a line (as BE') in any convenient direction, and measure its length; measure also the angles E'BC and BE'C. In the triangle CBE', there are then given two angles and the included side, from which the side BC can be calculated. Should it be nec- essary to take soundings at cer- tain intervals (as, say, 50 or 100 feet across the stream), then in the triangle BE X there are given the distance BX, the distance E'X, and the angle XBE', from which can be calculated the angle BE'X. With the transit at E', turn off from BE' the angle BE'X. Now. starting a boat from the shore, direct it in line from B to C until it comes upon the line of sight of the transit from E' to X. At that point take soundings, and similarly for the point X', etc. If the point C is not visible from B, find some point, as E (see Fig. 112), from which B and C are visible, and measure the angle BEC and the distance EB. Find a second point, as F, from which E and C are visible, and measure the angles CEF and EFC and the distance EF. Then, in the triangle ECF, there are given two angles and the included side, from which can be calculated the distance EC. In the tri- angle BCE, then, there are given two sides and the included PLANE SURVEYING 157 angle, and from these the third side BO and the angle EBC can be found. The stake C can now be numbered, and the bearing of BC deduced. EXAMPLES FOR PRACTICE. 1. In Fig. Ill, given BE' = 210 feet; angle CBE' = 110 15'; angle BE'C = 3420'; stake B numbered 8 + 54. It is required to -find the number of the stake C. 2. (a) In Fig. 112, given EF = 250 feet; BE = 128 feet; angle EFC = 46 40'; angle CEF - 103 30'; angle BEG = 39 10'. If the stake at B is numbered 12 + 20, it is required to find the number of the stake at C. (5) If the bearing of the line AB is S 75E, and the deflection angle of BE from AB is 104 to the right, find the bearing of BC. To Supply Omissions. Any two omissions in a closed sur- vey whether of the direction or of the length, or of both, of one or more lines of the survey can always be supplied by the application of the principle of lati- tude and departures, although this method should be resorted to only in cases of absolute necessity, since any omission renders the checking of the field work im- possible. In the following para- graphs, the methods outlined will apply equally whether the survey has been made with the transit or with the compass. CASE 1. When the length and bearing of any one side are wanting. In Fig. 113, let the Fig. 113. dotted line FG represent the course whose length and bearing are wanting. Calculate the latitudes and departures of the remain- 158 PLANE SURVEYING ing courses; and since in a closed survey the algebraic sum of the latitudes and departures should equal zero, therefore the difference of the latitudes will be the latitude of the missing line, and the difference of the longitudes will be the required longitude. The latitude and longitude of the line, form the sides of a right triangle, from which we have: Longitude Tangent of Bearing = 114. The required length will be given Latitude by L = 7^ =R : . J Cos Bearing CASE 2. When the length of one side and the bearing of an- other are wanting. (a) WHEN THE DEFICIENT SIDES ADJOIN EACH OTHER. Ill Fig. 114, let the bearing of DE, and the length of FE, be lacking. Draw DF. From the preceding proposition we can calculate the bearing and length of DF, as though DE and EF did not exist. Then, in the triangle DEF, we have given the lengths DF and DE and the angle DEF, from which can be calculated the angle FDE and the length EF. (1} WHEN THE DEFICIENT SIDES AliE SEPARATED FROM EACH OTHER. Iii Fig. 115 let ABCDE FGA represent a seven-sided field, in which the length of CD, and the bearing of FG, are want- - ing. Draw DB', B'A', A'G', of the same lengths, and parallel respectively to CB, BA, and AG. 115. Connect G' with GE and.F. Then, in. the figure DB'A'G'E, there are given the lengths and PLANE SURVEYING 159 bearings of all of the courses but G'E. The length and bearing of the last course can be calculated by the principles of Case 1. Then, in the triangle EFG , there are given the lengths and bearings of EF and EG', from which can be calculated the length and bearing of FG'. Therefore, in the triangle GFG', since GG' is equal in length and parallel to CD, there are given the lengths of GF and FG', and the bearings of GG' and FG', from which can be cal- culated the length of GG' and the bearing of GF. CASE 3. When thelengths of two sides arc (ranting. (a) WHEN THE DEFICIENT SIDES ADJOIN EACH OTHER. In the seven-sided Fig. 116, let the lengths of DE and EF be wanting. Calculate the length and bearing of DF by the principles of Case 1. Then, in the triangle . EDF, there are given the angles at D and F, and the length of DF, from which can be calculated the lengths of DE and EF. (i) WHEN THE DEFICIENT SIDES ARE SEPARATED FROM EACH OTHER. In Fig. 115, let the lengths of CD and GF be want- ing. As before, having calculated the length and bearing of FG', in the triangle FGG', the angle at G can be calculated from the bear- ings of FG and GG' ; the angle at G' from the bearings of GG'and FG'; and the angle at F from the bearings of FG and FG'. There are given then the three angles of the triangle, and the length of one side, from which can be calculated the lengths of the other sides. CASE 4. When the hearings of two sides are wanting. () WHEN THE DEFICIENT SIDES ADJOIN EACH OTHER. In Fig. 116, find the length and bearing of DF as before. Then, in the triangle DEF, there are given the lengths of the three sides, from which can be calculated the required angles. (J) WHEN THE DEFICIENT SIDES ARE SEPARATED FROM EACH OTHER. In Fig. 115, let the bearings of CD and GF be wanting. Calculate the length and bearing of FG' as before. Then, in the \ Fig. 116. PLANE SURVEYING triangle FGG', there are three sides known, from which can be cal- culated the three angles, and therefore the bearings can be deduced. UNITED STATES PUBLIC LAND SURVEYS. The first surveys of the public lands of the United States were carried out in Ohio, under an act of Congress approved May 20th, 1785. This act provided for townships 6 miles square, containing 36 sections of 1 mile square. The townships 6 miles square, were laid out in ranges, extending northward from the Ohio River, the townships being numbered from south to north, and the ranges from east to west. The territory embraced in these early surveys forms a part of the present state of Ohio and is known as " The Seven Ranges." The sections were numbered from 1 to 36 com- W 36 30 24 16 12 6 35 29 23 17 II 5 34 28 22 16 10 4 33 27 21 15 9 3 32 26 20 14 8 2 31 25 19 13 7 1 W 6 5 4 3 2 1 7 6 9 to II 12 id 17 16 15 14 13 19 20 21 22 23 24 30 29 28 27 26 25 31 32 33 34 35 36 Fig. 117. Fig. 118. mencing with No. 1 in the southeast corner of the township, and running from south to north in each tier, to No. 36 in the north- west corner of the townships as shown in Fig. 117. A subsequent act of Congress, approved May 18th, 1796, pro- vided for the appointment of a surveyor general, and directed the survey of the lands northwest of the Ohio River, and above the mouth of the Kentucky River. This act provided that ** the sec- tions shall be numbered respectively, beginning with the number one in the northeast section, and proceeding west and east alter- nately, through the township, with progressive numbers till the thirty-sixth be completed." This method is shown in Fig. 118 and is still in use. An act of Congress, approved Feb. llth 1805, directs the sub- division of the public lands into quarter sections, and provides that PLANE SURVEYING 161 all the corners marked in the public surveys shall be established as the proper corners of sections, or subdivisions of sections, which they were intended to designate, and that corners of half and quarter sections not marked shall be placed, as nearly as possible, " equidistant from those two corners which stand on the same line." This act further provides that ' the boundary lines actually run and marked * * * shall be established as the proper bound- ary lines of the sections or subdivisions for which they were intended; and the length of such lines as returned by * * * the surveyors * * * shall be held and considered as the true length thereof " An act of Congress, approved April 24th, 1820, provides for the sale of public lands in half-quarter sections, and requires that " in every case of the division of a quarter section the line for the division thereof shall run north and south. An act of Congress, approved April 5th, 1832, directed the subdivision of the public lands into quarter -quarter-sections and that in every case of the division of a half-quarter section, the dividing line should run east and west; and that fractional sections should be subdivided under rules and regulations prescribed by the Secretary of the Treasury. By an act of Congress, approved March 3rd, 1849, the Depart- ment of the Interior was created, and the act provided "That the Secretary of the Interior shall perform all the duties in relation to the General Land Office, of supervision and appeal now discharged by the Secretary of the Treasury. * * *" By this act the General Land Office was transferred to the Department of the Interior where it still remains. The division of the public lands is effected by means of merid- ian lines and parallels of latitudes established six miles apart. The squares thus formed are called Towns/tips, and contain 36 square miles, or 23,040 acres " as nearly as may be. " All of the townships situated north or south of each other, form a Range and are named by their number east or west of the principal meridian. Thus, the first range west of the meridian would be designated as Range 1 West ( E. 1. W.). Each tier of townships is named by its number north or south of the base line, as Township 2 North (T. 2. N.). 162 PLANE SURVEYING Existing laws further require that each township shall be divided into thirty-six sections, by two sets of parallel lines, one governed by true meridians and the other by parallels of latitude, the latter intersecting the former at right angles, at intervals of one mile ; and each of these sections must contain, as nearly as possible, six hundred and forty acres. These requirements are evidently inconsistent because of the convergency of the meridians, and the discrepancies will be greater as the latitude increases. In view of these facts, it was provided in section 3 of the act of Congress approved May 10th, 1800, that " in all cases where the exterior lines of the townships, thus to be subdivided into sections and half-sections, shall exceed, or shall not extend six miles, the excess or deficiency shall be specially noted, and added to or deducted from the western or northern ranges of sections or half- sections in such township, according as the error may be in running lines from east to west, or from south to north ; the sections and half-sections bounded on the northern and western lines of such townships shall be sold as containing only the quantity expressed in the returns and plots, respectively, and all others as containing the complete legal quantity." To harmonize these various requirements as fully as possible, the following methods have been adopted by the general land office. Initial points are first establis hed astronomically under special instructions, and from this initial point a "principal meridian " is laid out north and south. Through this initial point a "base line" is laid out as a parallel of latitude running east and west. On the principal meridian and base lines, the half-mile, mile and six-mile corners are permanently located, and in addition, the meander corners at the intersection of the line with all streams, lakes or bayous prescribed to be meandered. These lines may be run with solar instruments, but their correctness should be checked by observations with the transit upon Polaris at elongation. Standard parallels, also called correction lines, are run east and west from the principal meridian at intervals of twenty-four miles north and south of the base line, and the law provides that " where standard parallels have been placed at intervals of thirty or thirty-six miles, regardless of existing instructions, and where gross irregularities require additional standard lines, from which to TOPOGRAPHER OF U. S. GEOLOGICAL SURVEY AT WORK ON THE PLAINS OF COLORADO PLANE SURVEYING 163 initiate new, or upon which to close old surveys, an intermediate correction line should be established to which a local name may be given: and the same will be run, in all respects, like the regular standard parallels." Guide meridians are extended north from the base line, or standard parallels, at intervals of twenty-four miles east and west of the principal meridian. When conditions are such as to require the guide meridians to run south from a standard parallel or a correction line, they are initiated at properly established closing corners of the given paral- lel. That is to say, they are begun from the point on the parallel at which they would have met it if they had been run north from the next southern parallel. This point is obtained from computa- tion, and is less than twenty-four miles from the next eastern or western meridian by the convergence of the meridians in twenty- four miles. In case guide meridians have been improperly located too far apart, auxiliary meridians may be run from standard corners, and these may be designated by a local name. The angular convergence of two meridians is given by the equation (f> = m sin L (1) where m is the angular difference in longitude of the meridians, and L is the mean latitude of the north and south length under consideration. The linear convergence in a given length I is c = I sin 4> (2) The radius of a parallel at any latitude L is given by the equation r = cos L (3) where R is the mean radius of curvature of the earth. The distance between meridians is usually given in miles and this must be reduced to degrees. To do this it is first necessary to find the linear value of one degree of longitude at the mean latitude from the proportion. 1 : 360 : : a> : 2nr (4) the value of / being found from (3) 164 PLANE SURVEYING Equation (4) will give results sufficiently accurate, although in strict accuracy R should be the radius of curvature at the mean latitude. For full details of public-land surveying, see " Manual of Surveying Instructions for the Survey of the Public Lands of the United States," issued by the Commissioner of the General Land Office. These " Instructions " are prepared for the direction of those engaged on the public land surveys, and new editions are issued from time to time. Much of the foregoing in very condensed form is taken from the edition of 1894. The following table gives the convergency both in angular units and linear units for township 6 miles square, between lati- tudes 30 and 70 north: Let it be required to find from the table the linear converg- ence for a township situated in latitude 38 29' north. Looking in the table opposite 39 we find the linear con- vergence. For 39 = 58.8 links For 38 = 56.8 links Difference for 1 = 2.0 links Difference for 1' = 2.0 -^ 60 = .0333 links Difference for 29' = .0333 X 29 = .97 links Therefore total convergence for latitude 38 29' = 56.8 4 0.97 links = 57.77 links. 172 PLANE SURVEYING 165 BASE HEASUREHENT. It is not intended in what follows to go into the details of the measurement of a base for an extended system of triangulation, as that properly belongs to Geodetic Surveying. Some description of base measuring apparatus will be given, with illustrations of Lat. Convergency. Lat. Convergency On the Parallel. Angle. On the Parallel. An gle. Degrees. Links. Minutes. Seconds. Degrees. Links. Minutes. Seconds. 30 41.9 3 50 86.4 6 12 31 43.G 3 7 51 896 6 25 32 45.4 3 15 52 92.8 6 39 33 47.2 ! 3 23 53 96.2 (5 54 34 49.1 3 30 54 99.8 9 35 50.9 3 38 55 103.5 7 25 36 52.7 3 46 56 107.5 7 42 37 54.7 3 55 57 111.0 8 38 56.8 4 04 58 116.0 8 19 39 58.8 4 13 59 120.6 8 38 40 60.9 4 22 60 125.5 8 59 41 63.1 4 31 61 130.8 !) 22 42 65.4 4 41 62 136.3 <) 46 43 67.7 4 51 63 142.2 10 11 44 70.1 5 1 64 148.6 10 38 45 72.6 5 12 65 155.0 11 8 46 75.2 5 23 66 162.8 11 39 47 77.8 5 23 67 170.7 12 13 48 80.6 5 46 68 179.3 12 51 49 83.5 5 59 69 188.7 13 31 70 199.1 14 15 various devices, and special attention will be given to the use of the tape in the accurate measurement of lines such as occur in usual field operations of Plane Surveying. Much of what follows is from the excellent treatise on Topo- graphic Surveying by Herbert M. Wilson. A trigonometric survey is usually carried over a country where the direct measurement of distances is impracticable, and since the calculations of these distances proceeds from the direct measure- ment of the base-line, this base line should be so located as to permit of its length being determined with any degree of accuracy consistent with the nature of the work involved. To attain the desired results, the site should be reasonably level and afford room for a base of proper length so that its ends may be intervisible. and permit of the development of a scheme of 166 PLANE SURVEYING primary triangulation giving the best-conditioned figures possible. Other things being equal, that site is best that includes solid ground; both for permanency of monuments and facility and accuracy of measurement. Base Apparatus. In early days, base-lines were measured by means of wooden rods, varnished and tipped with metal. The rods were supported in trestles, the contacts between the ends being made with great care. Later, compensated rods were employed, as for instance the Contact-Slide Apparatus of the U. S. Coast Survey and the Repsold primary base bars of the U. S. Lake Survey, see Fig. 119, resulting in greater accuracy in the measurement of base lines. The use of the iced bar (see Fig. 120) by the U. S. Coast Survey, represents the highest development of base-measuring apparatus. Fig. 119. Within recent years the steel tape has become popular as the accuracy attainable with its use has become more fully appreciated. Errors in Base Heasurement. The following are the chief sources of error in base measurement: 1. Changes of temperature ; 2. Difficulties of making contact ; 3. Variations of the bars or tape from the standards. The refinements of measurement consist especially in a. Standardizing the measuring apparatus, or its comparison with a standard of lengtn. b. Determination of temperature, or its neutralization by the use of compensating bars. c. Means adopted for reducing the number of contacts to the fewest possible, and of making these with the greatest degree of precision. PLANE SURVEYING 167 The inherent difficulties of measurement with lar* of any kind are : 1. Necessity of measuring short bases because of the number of times which the bar must be moved. 2. Expense, as a considerable number of men are required. 3. Slowness, the measurement often occupying from a month to six weeks. The advantages of measurement made with a steel tape are : 1. Great reduction in the number of contacts, as the tapes are about three hundred feet long as compared with bars of about twelve feet. 2. Comparatively small cost because of the few persons required. 3. Shortness of the time employed, an hour to a mile being an ordinary record in actual measurement 4. Errors in trigonometric expansion may be reduced by increasing the length of the base from 5 miles, the average length of a bar-measured base, to 8 miles, not an uncommon length for tape-measured bases. Fig. 120. Steel tapes offer a means of measuring base lines which is superior to that obtained by measuring bars, because they combine the advantages of great length and simplicity of manipulation, with the precision of the shorter laboratory standards, providing only that means be perfected for eliminating the errors of tem- perature and of sag in the tape. Base lines can be so conveniently^ and rapidly measured with long steel tapes as to permit of their being made of greater length than has been the practice with lines measured by bars, and as a result, still greater errors may be introduced in tape-measured bases and yet not affect the ultimate 168 PLANE SURVEYING expansion any more than will the errors in the latter, because of the greater length of the base. The tapes used for this work are of steel, either 300 feet or 100 meters in length. The tapes used by the Coast Survey are 101.01 meters in length, 6.34 millimeters by 0.47 millimeters in cross -section, and weigh 22.3 grams per meter of length. They are subdivided into 20 meter spaces by graduations ruled on the sur- face of the tape, and their ends terminate in loops obtained either by turning back and an- nealing the tape on itself, or by fastening them into brass hand- les. When not in use, the tapes are rolled on reels for easy trans- portation. The steel tapes used by the Geological Survey are similar to those used by the Coast Survey, excepting in their length, whicb is a little over 300 feet. They are graduated for 300 feet and are subdivided every 10 feet, the last 5 feet of which at either end is subdivided to feet and tenths. The various instrument-makers now carry such tapes in stock, wound on hand -reels. All tapes must be standardized before and after use, by comparison with laboratory standards, and, if possible, thereafter frequently in the field by means of an iced-bar apparatus. In measuring with steel tapes, a uniform tension must be applied. In order to get a uniform tension of 20 to 25 pounds, some form of stretcher should be used. That used by the U.S. Coast Survey consists of a base of brass or wood, 2 or 3 feet in length by a foot in width, upon which is an upright metallic standard, and to this is attached by a universal joint, an ordinary spring-balance, to which the handle of the tape is fastened. See Fig. 121. The upright standard is hinged at its junction with the base, so that when the tape is being stretched, the tapeman Fig. 121. PLANE SURVEYING (T 1 ?! ^ ~ t - -o x a x s M ^ be ^ c ts. -r 17 o PLANE SURVEYING can put the proper tension on it by taking hold of the upper end of the upright standard and using it as a lever, and by pulling it back toward himself he is enabled to use a delicate leverage on the balance and attain the proper pull. The thermometers used are ordinary glass thermometers, around the bubbles of which should be coiled thin annealed steel wire, so that by passing them in the air adjacent to the tape, a temperature corresponding to that of the tape can be obtained. Experience with such thermometers shows that they closely fol- low the temperature of the steel tape. For the best results, two thermometers should be used, each at about one-fourth of the distance from the extremities of the tape. The stretching device used by the U. S. Geological Survey is much simpler and more quickly manipulated than that of the 1 Coast Survey. The chief object to be at- I tained in tension is steadiness and uni- Fig. 123. formity of tension; the simplest device which will attain this end is naturally the best. Two general forma of such devices are employed by the U. S. Geological Survey, one for the measurement of base lines along railways, where the surface of the ties or the roadbed furnishes support for the tape, and the device must therefore be of such kind as to permit of the ends being brought close to the surface; the other is employed in measurements made over rough ground, where the tape may frequently be raised to considerable heights above the surface and be supported upon pegs. The stretcher used by the Geological Survey for measuring on railways is illustrated in Fig. 122, and was devised by Mr. He L. Baldwin. It consists of an ordinary spring-balance attached to. the forward end of the tape, where a tension of twenty pounds is applied, the rear end of the tape being caught over a hook which is held steadily by a long screw with a wing-nut, by which the zero of the tape may be exactly adjusted over the mark scratched on the zinc plate. The spring-balance is held by a wire running over a wheel, which latter is worked by a lever and held by ratchets in any desired position, so that by turning the wheel, a PLANE SURVEYING 171 uniform strain is placed on the spring-balance, which is held at the desired tension by the ratchets. The tape- stretcher used by the U. S. Geological Survey off railways consists of a board about 5 feet long, to the forward end of which is attached by a strong hinge, a \vooden lever about 5 feet in length, through the larger portion of the length of which is a slot. See Fig. 123. Through the slot is a bolt with wing- nut, which can be raised or lowered to an elevation corresponding with the top of the hub over which measurement is being made; hung from the bolt is the spring- balance, to which the forward tapeman gives the proper tension by a direct pull on the lever, the weight of the lever and the friction in the hinge being such as to make it possible to bring about a uniform tension without dif- ficulty. The zero on the rear end of the tape is adjusted over the contact mark on the zinc by means of a similar lever with hook- bolt and wing-nut, but without the use of spring-balance. LAYING OUT THE BASE. The most laborious operation in base measurement is its preliminary preparation, which consists of: 1. Aligning with transit or theodolite; 2. Careful preliminary measurement for the placing of stakes on rough ground; 3. Placing of zinc marking-strips on the stakes. Base lines measured with steel tapes across country are aligned with transit or theodolite, and are laid out by driving large hubs of 3 X 6 scantling into the ground, the tops of the same project- ing to such a height as will permit a tape-length to swing free of obstructions. These large hubs are placed by careful preliminary measurement at exact tape-lengths apart, and between them as sup- ports, long stakes are driven at least every 50 feet. Into the sides of these near their tops are driven horizontally, long nails, which are placed at the same level by eye, by sighting from one terminal hub to the next. The tape rests on these nails and on the surface of the terminal hubs are tacked strips of zinc on which to make the contact marks. A careful line of spirit-levels must be run over the base-lines, and the elevation of the hub or contact-mark of each tape-length must be determined in order to furnish data for reduction to the horizontal. In measuring over rough ground, six men are necessary: two tape -stretchers, two markers, two observers of thermometers, one 172 PLANE SURVEYING of whom will record. The co-operation of these men is obtained by a code of signals, the first of which calls for the application of the tension; then the two tape- stretchers by signal announce when the proper tension has been applied; then the rear observer ad justs the rear graduation over the determining mark on the zinc plate and gives a signal, upon hearing which, the thermometer recorder near the middle of the tape lifts it a little and lets it fall on its supports, thus straightening the tape. Immediately thereafter the front observer marks the position of the tape graduation on the zinc plate, and at the same time the thermometers are read and recorded. After the measurement of the base line has been completed in the field, the results of the measurement have to be reduced for various corrections, among which are: Comparison with standard measure: Corrections for inclination and sag of tape if such is used; Correction for temperature. The first correction to be applied is that of reducing the tape- line to the standard, " standardizing " the tape as it is called. By sending the tape to the National Bureau of Standards at Washing- ton, D. C., a statement may be had of the length of the tape com- pared with the standard. For this service a small fee is charged. For an additional fee a statement may be had of the temperature and pull at the ends for which the tape is a standard. As the length of a steel tape varies with the temperature, one of the most uncertain elements in the measurement of a base with the steel tape, is the change in the length of the standard due to changes of temperature. Corrections, therefore, must be made for every tape-length as derived from readings of one or more ther- mometers applied to the tape in the course of measurement. Steel expands .0000063596 of its length for each degree Fahrenheit. This decimal multiplied by the average number of degrees of temperature above or below 62 degrees at the time of the measurement, gives the proportion by which the base is to be diminished or extended on account of temperature changes. This correction is applied usually by obtaining with great care, the mean of all thermometer readings taken at uniform intervals of distance during the measurement. PLANE SURVEYING 173 The data for the correction for inclination of base are obtained by a careful line of spirit-levels over the base-line. In the course of this leveling, elevations are obtained for every plug upon which the tape rests. The result of this leveling is to give a profile showing rise or fall in feet or fractions thereof between the points of change in inclination of the tape-line. From this and measured distances between these points, the angle of inclination is com- puted by the formula In which D is the length of the tape or measured base : and // is the difference in height of the ends of tape or measured base, expressed in feet. is the angle of slope expressed in minutes. The correction in feet to the distance is that computed by the equation Correction = D 8in ' *' ffi An approximate formula for reducing distances measured upon sloping ground to the horizontal is expressed by the rule : Divide the square of the difference of level by twice the measured distance, subtract the quotient thus found from the measured distance, and the remainder equals the distance required ; thus in which d equals the horizontal or reduced distance. When the base measurement is made with steel tape across country, and accordingly is not supported in every part of its length, there will occur some change in its length, due to sag. As previously explained, the tape should be rested upon supports not less than 50 feet apart. With supports placed even this short dis- tance apart, however, a change of length will occur between them, while even greater changes will occur should one or more supports be omitted as in crossing a road, ravine, etc. Since tapes are standardized by laying them upon a flat standard, it is necessary to determine the amount of shortening from the above causes. 174 PLANE SURVEYING The following reduction formulae apply : Let w = weight per unit of length of tape : t = tension applied w a = n = number of sections into which tape is divided by supports. I length of any section L normal length of tape or right-line distance be- tween n marks when under tension : = nl ap- proximately. If a tape be divided by equidistant supports, the difference in distance between the end graduations, due to sag, or the correction for sag = dL becomes If one or more supports are omitted, then the omission of m consecutive supports shortens the tape by J__m(m + 1) (m + 2) a 2 / 3 : when I is the length of the section when no supports are omitted. Example. Let n = 6 ; I == 50 feet ; w .0145 = weight in pounds per foot found by dividing whole weight of tape by whole length ; t = 20 pounds. ,/ = = 0.0162 feet, 24 V t ' which is the amount of shortening of each tape-length. This cor- rection is always negative. If there had been 86 full tape-lengths in measured base-line, the total corrections for sag would be 86 X .0162 = 1.393 feet. THE PLANE-TABLE. Construction. The plane-table consists essentially of a draw- ing-board mounted upon a tripod. This board is usually twenty- four by thirty inches, constructed in sections to prevent warping; it is attached to the tripod by a three-screw leveling base arranged PLANE SURVEYING 176 to permit the board to be turned in azimuth and to be clamped in any position. The instrument is designed to at once sketch in the field, to scale, the lengths and relative directions of all lines and the posi- tions of objects to be included in the survey. For drawing straight lines, a steel ruler is provided upon which is mounted at each end, a pair of open sights like those of the compass, or, a tele- scope is mounted at the center of the ruler, fitted with stadia wires, a vertical arc and a longitudinal striding level. The eye. piece should be inverting, and whether the open sights or the tele- scope is used, the line of sight should always be parallel to the edge of the ruler. The straight edge with the attached telescope or open sights is called the alidade. For leveling the instrument, two cylindrical levels, at right angles to each other, are mounted upon the alidade and either an attached or detached compass is provided for determining the bear- ing of lines. For attaching the paper to the board, various devices are used. One consists of a roller at each end of the table upon one of which the paper is wound up as it is unrolled from the other, the edges of the paper being held close to the board by spring clips. This arrangement permits the paper to be used in a con- tinuous roll and to be tightly stretched over the board. The use of the continuous roll of paper is undesirable, however, and separate sheets of proper size should be used, attached to the board and held firmly in place by the spring clips provided with the instrument. The use of thumb-tacks should be avoided. Under the most favorable conditions, the plane-table is a very awkward instrument and difficult to handle, but it is admirably adapted to filling in the details of a topographical survey. For this purpose it is the standard instrument of the United States Geo- detic Survey and is also largely used by the United States Geolog- ical Survey. It cannot be used on damp or very windy days and is not therefore, of as general utility as the transit and stadia. Fig. 124 shows one form of construction of the plane table with leveling screws and Fig. 124 shows a plane table with a much simpler form of leveling head. This latter was designed by Mr. W. D. Johnson and has received the approval of the topographers 176 PLANE SUKVEYING of the United States Geological Survey. The whole arrangement is very light, but does not permit of as close leveling as does the usual form with leveling screws. Adjustments. 1st. To determine whether the edge of the ruler is straight. Fig. 124. Place the ruler upon a smooth surface, and draw a line along its edge, and also lines at its ends. Reverse the ruler on these lines, and draw another line along its edge. If these two lines coincide, the ruler is straight. PLANE SURVEYING 177 2nd. To make the plane of the table horizontal when the bubbles are in the center of the tubes. Assuming the table to be plane, set the alidade in the middle of the table, level by means of the leveling screws, draw lines along the edge and ends of the ruler, and reverse the alidade on these lines. If the bubbles remain in the center of the tubes, they are in adjustment. If they Fig. 124a. do not, correct one-half of the error by means of the leveling screws and the remainder by means of the capstan -headed screws of the level tubes. Repeat the operation until the bubbles remain in the center of the tubes in both positions of the alidade. 3rd. To make the line of collimation perpendicular to the horizontal axis of the telescope. 178 PLANE SURVEYING Level the table and point the telescope towards some small and well-defined object. Remove the screws which confine the axis of the telescope in its bearings, reverse the telescope in its bearings, that is, change the axis end for end. being careful not to disturb the position of the alidade upon the table, and again sight upon the same object. If the intersection of the cross hairs bisects the object, the adjustment is complete. If not, correct one-half of the error by means of the horizontal screws attached to the reticle. Sight on the object again and repeat the operation until the line of collimation will bisect the object in both positions of the telescope. 4th. To make the line of collimation parallel to the axis of the bubble tube. Attach the longitudinal striding level to the telescope and carry out the adjustment by the " peg " method as described for the transit. 5th. To make the horizontal axis of the telescope parallel to the plane of the table. Level the table and point the telescope to a well-defined mark at the top of some tall object, as near as possible consistent with distinct vision. Turn the telescope on its horizontal axis, and point to a small mark at the base of the same object. Draw lines on the table at the edge and ends of the ruler. Reverse on these lines, point the telescope to the lower object and turn the telescope upon its horizontal axis. If the line of collimation again covers the higher point, the adjustment is complete. If it does not, correct one-half of the error by means of the screws at one end of the horizontal axis. 6th To make the vertical arc or circle read zero when the line of collimation is horizontal. Level the table and measure the angle of elevation or depres- sion of some object. Remove the table to the object, level as before, and measure the angle of depression or elevation of the first point Half the difference, if any, of the readings is the error of the adjustment. Correct this by means of the screws attached to the vernier plate, and repeat the operation until the angles as read from the two stations are equal. 186 PLANE SURVEYING 173 Fig. 125. Use. The plane-table is used for the immediate mapping of a survey made with it, no angles being measured, but the direction and length of lines being plotted at once, upon the paper. The simplest case is the location of a number of points from one central point, called the method of radi- ation. The table is "set up" so that some convenient point upon the paper is over a selected spot upon the ground and is then clamped in azimuth. Mark the point upon the table by sticking a needle into the board. Now bring the edge of the alidade in contact with the needle and swing it around until the line of sight, which is parallel to the edge of the ruler, is directed to the point to be located. Having determined the scale of the plat, aline is drawn along the edge of the ruler to scale, equal to the distance to the desired point, such distance having been measured either with the tape or stadia. In the same way locate all of the other points, which may include houses, trees, river banks, etc. If the plane- table is set up in the interior of a field at a point from which all of the corners are visible, the corners can be thus located and after being connected, there results a plot of the area. Instead of occu- pyinga point in the interior of the field, one corner may be selected from which all of the others are visible, or a point outside of the field may be chosen from which to measure the lines to the several corners. Evidently from such a survey, data is lacking from which to calculate the area, and either the map must be scaled for addi- Fig. 126. tional data or the area measured with the planimeter. The Fig. 125 illustrates the method of surveying a closed area by the method of radiation. The plane-table is at the point 180 PLANE SURVEYING o and drawn to an exaggerated scale. The area abode representing to scale, the area ABODE. It may be desirable to set up the table at some other point, as for instance one of the corners of the field, and run out some of the lines to the other corners as a check upon the work. Traversing, or the Method of Progression. This method is practically the same as the method of surveying a series of lines with the transit, but requires that all of the points be accessible. It is the best method of working as it provides a complete check upon the survey. Let ABODE, Fig. 126, be the series of lines to be surveyed by traversing. Set up the table at B, the second angle of the line, so that the point b upon the paper will be directly over the point B upon the ground. (The point b should be so chosen as to leave room upon the paper for as much of the traverse as possible.) Stick a needlo at the point b and place the edge of the alidade against it. Swing the alidade around until the line of sight covers the point A. Measure BA and lay it off to the proper scale as ba. Now turn the alidade around the point b and sight to and measure the distance BC and plot it to scale as be. Remove the instrument to c with the point c upon the paper directly over C upon the ground, and c b in the direction of CB. This is diffi- cult to accomplish with the plane-table, but if the plot is drawn to a large scale, it must be done. If the plot is drawn to a small scale, it will be sufficiently accurate to set the table over the point C as nearly as possible in the proper direction and then turn the board in azimuth until b is in the direction of B. Stick a needle at c and check the length of cb. Swing the alidade around c until the line of sight covers D, measure CD and plot cd. Remove to D and proceed as before and so on through the traverse. If the survey is of a closed field, the accuracy of the work will be checked by the closure of the survey. Fig. 127. 188 PLANE SURVEYING 181 The method of progression is especially adapted to the survey of a road, the banks of a river, etc., and often many of the details may be sketched in with the eye. When the paper is tilled, put on a new sheet, and on it, fix two points, such as D and E, which were on the former sheet and from them proceed as before. The sheets can afterward be united so that all points on both shall be in their true relative positions. flethod of Intersection. This is the most rapid method of using the plane-table. Set up the instrument at any convenient point, as A in Fig. 127 and sight to all the desired points as D, E, F, etc., which are visible, and draw indefinite lines in their direc- tions. Measure any line as AB, B being one of the points sighted to, and plot the length of this line upon the paper to any convenient scale. Move the instrument to B so that b upon the paper will be directly over B upon the ground, and so that la upon the paper will be in the direction of BA upon the ground as explained under the method of progression. Stick a needle at the point 1) and swing the alidade around it, sighting to all the former points in succession, and draw lines in their direction. The intersection of these two sets of lines to the several points will determine the position of the points. Connect the points as d, e,f, g, in the figure. In surveying a field, one side may be taken as the base line. In choosing the base line, care must be exercised to avoid very acute or obtuse angles: 30 and 150 being the extreme limits. The impossibility of always doing this, sometimes renders this method deficient in precision. TOPOGRAPHICAL SURVEYING. A topographical map is one showing the configuration of the surface of the ground of the area to be mapped and includes lakes, rivers, and all other natural features, and sometimes artificial features as well. A topographical survey is one conducted for the purpose of acquiring information necessary for the production of a topograph- ical map of the area surveyed. Nearly all engineering enterprises involve a topographical survey more or less extended, depending upon the nature and 182 PLANE SURVEYING importance of the contemplated work. The construction of an important building may involve a survey of the foundation site to determine the amount of cut and fill ; the construction of a bridge will involve a hydrographic survey of a body of water to acquire information in regard to direction and velocity of current, depth of water, nature of bottom, and proper site for piers and abutments. A proposed railroad will not only involve a survey of the line itself, but a topographical survey extending from 200 to 400 feet upon each side. The design of a sewer system or a waterworks system, dams, reservoirs, canals, irrigation channels, tunnels, etc , all involve topographical surveys. In what follows it is intended to outline the methods of con- ducting field operations, based partly upon the nature and impor- tance of the problem involved, and partly upon the instruments used. The different methods of representing topography and the involved drafting-room work will be fully treated in Topographical Drawing. The field operations, in so far as the methods and instruments are concerned, may be classified as follows : 1. Sketching by the eye, without or with the tape for measuring dis- tances. 2. Sketching with the aid of the Locke hand-level or clinometer, hori- zontal distances being measured either by pacing or with the tape. 3. Determining the elevation of points with the wye-level, horizontal distances being determined either with the stadia or tape. 4. Determining points with the transit and stadia. 5. Topographical sketching with the plane-table and stadia. 6. Photography. 7. Triangulation. It is evident that the first method is entirely lacking in accu- racy, and such work should be done only when speed is the most important consideration, only the roughest approximation to the topographical features being attempted ; contour lines cannot be located. Work of this nature is of value principally for purposes of promoting an enterprise ; artistic, showy plates being desired. Little can be said descriptive of the manner of carrying out the field work, since this will require considerable artistic ability as well as the ability to " see " things and estimate distances. Com- paratively few men possess the ability to carry out topography of this nature. It necessarily follows that the work must be done PLANE SURVEYING 183 entirely by sketching in the field, and for this purpose the following equipment is needed : 2 or 3 medium pencils, kept well sharpened. Rubber eraser. Thumb-tacks. Several sheets of drawing paper, 14" x 14". One light drawing board, 15" x 15". A pocket compass will be useful in determining the bearing to prominent objects to tie in the stations of the survey. A Locke hand-level or Abney clinometer will also be useful for finding approximate heights, and either of these instruments can be readily carried in the pocket. It will be more convenient to have the paper cross-ruled into one-fourth inch squares, the center line being ruled in red, but if drawing paper is used, it will be neces- sary to add an engineer's scale to the equipment. The back of the drawing board should be fitted with a leather pocket, with flap and button, in which the blank sheets and the finished topographic sheets should be kept. A strap attached to the board and to go over the shoulder, will prove a great convenience. A waterproof cover should be provided to protect the board and sheets in case of rain. A compass or transit survey forms the backbone of the topog- raphy, and the sketching should include an area upon each side of the line so surveyed, and running parallel with it. A separate sheet should be used for each course (by course is intended the straight line from one turning point to the next), no matter how short it may be. Begin at the bottom of the sheet and sketch the topography up the sheet, that is, in the direction of the progress of the survey, and number the sheets in order. Begin each new sheet with the same station that ended the preced- ing sheet. After the field work is completed, the sheets can be laid down in order, the angles between their center lines corre- sponding to the deflection angles as given by the transit notes of the survey. The topography can now be traced upon tracing cloth in a continuous sheet. The method above outlined will result in a saving of time, especially in w r orking up the topographic plat. The second method commends itself in connection with a preliminary survey of a highway, steam or electric road, irriga- 184 PLANE SURVEYING tion channels, canals, -etc. The equipment should be as follows : 1 or 2 straight edges, about 12 feet in length. 1 or 2, 100-foot steel tapes. 1 or 2 plumb-bobs. 1 pocket compass. 2 or 3 medium pencils, kept well sharpened. Rubber eraser. Thumb-tacks. Several sheets of drawing paper or cross-section paper, 14" X 14". One light drawing board, 15" X 15" with waterproof cover. The topographic party should be made up of the topographer arid one or two assistants, depending somewhat upon the nature of the survey and the country traversed. If the country permits of rapid progress of the transit and level party, two assistants will be necessary to keep the topography abreast of the survey. Rapid work may, however, be done with one assistant, provided the topography does not extend more than 200 feet each side of the transit line. The Abney clinometer is well adapted for this class of work, on account of its portability, which is an important item in a rough country with steep side slopes. It can be used in the same way as the Locke hand-level, if necessary, but is a more generally useful instrument, as is described in Part 1. The straight edge should be of well-seasoned, straight-grained material, as light as possible, but so constructed as to prevent warping. It should be divided into spaces of one foot each, painted alternately red and white. The tapes should be of band steel, as they are subjected to rough usage, and they should be divided to feet and tenths at least. A plumb-bob is necessary for plumbing down the end of the tape on steep slopes. The pocket compass is a necessary adjunct in work of this character. The drawing paper should preferably be cross-section paper ruled into one-fourth inch squares with a heavy center line in red, but if ordinary drawing paper is used, it will be necessary to include in the outfit an engineer's scale, by means of which distances may be platted upon the sheet. Enough of these sheets should be carried to cover a day's work, but no more. The drawing board should be fitted up as described tinder the previous method. Method of Procedure. The transit line furnishes, of course, the backbone of the survey, and the topography will be taken for 192 PLANE SURVEYING 185 the proper distance upon each side of this line, by locating points both as to distance and elevation, upon perpendiculars from the transit stations. In rough country, it may be necessary to locate these points intermediate between the transit stations. Before starting out upon a day's work it is necessary to procure from the level party, the elevation of the transit stations, or if the topog- raphy keeps pace \dth the transit survey, the elevation may be ob- tained from the leveler at each station. For points intermediate between transit stations, the elevations may be gotten closely enough with the clinometer or hand-level. The number of each station as well as its elevation, should be noted upon the topo- graphic sheet, and the topography will include the location of contour lines, at proper vertical intervals, as well as all streams, lakes, property lines, etc. An example showing the method of keeping the field notes, will at the same time best serve to explain the methods of conducting the survey. Beginning with station at the bottom of the sheet, the number and elevation of the station are noted. See Fig. 128. Sending the assistant out upon one side of the transit line and at right angles thereto, he holds the rod at points to be designated by the topographer, the distances to be determined by pacing, or with the tape, and the elevations determined either by sighting upon the rod with the clinometer, or by laying the straight edge upon the ground at right angles to the line and applying the clinometer to it to determine the slope, from which elevations can at once be determined. Contour points are then readily interpolated and the distance out platted to scale upon the sheet and a note made of the elevation of the contour lines. If a lake or stream intervenes within the limits of the topographic survey, determine the distance to and elevation of the shore line and plat upon the sheet. Deter- mine points upon the other side of the transit line in the same way. If one or more contour lines cross the transit line between stations, determine the points of crossing and plat the points upon the sheet, to scale, as shown between stations and 1. It will be noticed in this case that the elevation of station 0, is 138 feet and of station 1, is 141 feet. If contours are to be taken at vertical intervals of five feet, it is apparent that the 140-foot contour line must cross the transit line between these stations. If the slope of 186 PLANE SURVEYING A B C Fig. 128. the ground is uniform, the point of crossing may be taken at two- thirds of the distance from to 1. Otherwise, locate the point with the clinometer. PLANE SURVEYING 187 \ \ \ 145 \ 140 \ Fig. 128. Now go to station 1 and locate contour points and other topographic features as before described, and connect points in the 188 PLANE SURVEYING same contour line, sketching in the curve of the line with the eye. Use a separate sheet for each portion of the transit line from turn- ing point to turning point; this will require that the turning points appear upon two consecutive sheets. Likewise, if the length of the line between turning points is too long to be platted upon a single sheet, begin the second sheet with the same station that completed the first sheet and so continue throughout the survey. As each sheet is completed, number it and return to the pocket on the back of the drawing board. The pocket compass should be used for determining the bearing of property lines, roads, streams, etc., crossed by the survey, and to take the bearings to prominent objects. The topographic sheets should be filed away in such a manner as to make them easily accessible at any time, as the engineer in charge of the transit survey may wish to consult them from time to time. The office work of preparing the topographic plat can be very expeditiously carried out as before described. The use of the wye-level as a topographic instrument is limited, but for certain kinds of work the instrument is the most satisfactory, as for instance, the survey of a dam-site; the survey of a reservoir-site; the survey of a town preparatory to planning sewer and waterworks systems and the planning of street pave- ments. The instrument should be fitted with stadia wires for measur- ing horizontal distances, and this will usually prove a great conven- ience, resulting in saving of both time and expense. A steel tape should, however, be included in the equipment for field work, for the purpose of checking measurements with the stadia. In addition to the above there should be provided, the following equipment: Self-reading level rod, capable of being read to hundredths of a foot. Hatchet. Marking crayon. 2 or 3 medium pencils, kept well sharpened. Plumb-bob. Rubber eraser. Portable turning point. The method of using the level rod in connection with the stadia for measuring distances has been fully discussed in Part II. PLANE SURVEYING 189 The portable turning point will prove of great convenience and may be made from a triangular piece of thin steel, with the corners turned down to project about one inch. If the level is to be used with the tape, the party will be made up of the levelman, two tapemen and a rodman, unless the nature of the work will permit of the rodman carrying the rear end of the tape. If the stadia is used for measuring distances, only the rodman will be required in addition to the levelman. The levelman carries the note book and enters into it all rod readings both for elevation and distances. These notes should be entered upon the left-hand page, the right-hand page being re- served for notes and sketches, which should be as full as possible. The levelman should cultivate the practice of calculating the ele- vations of the stations as the work progresses, at least of the turning points and bench-marks, in order that the results may be checked and errors discovered at once and corrected. If this work is left to be afterward carried out in the office, errors may be dis- covered that may require considerable time to locate and correct. If the area to be surveyed is, for instance, a reservoir site, it will be found most convenient to cover the area with a system of rectangles as shown in the figure, the parallel lines being spaced from 200 to 400 feet apart as may be most desirable. These lines should be run in with the transit, stakes being set at the inter- sections of the cross lines, or if the area is not very extended and is comparatively level,, by means of the level itself, the perpen- dicular distances between the parallel lines being measured with the tape. These lines having been laid down, the next step is to estab- lish a system of bench-marks over the area. Begin by establishing a "standard" bench-mark at some central point upon a permanent object, easily identified, and from thence radiate in all directions, returning finally to the original bench-mark for purposes of checking. Having located and satisfactorily checked the bench- marks, begin by running the level over all the lines running in one direction, as from A to B, back from C to D and so on, taking rod readings at every fifty or one hundred feet, in addition to the readings at the stakes at intersections of cross lines. It is to be understood that stakes are not to be driven at the intermediate 190 PLANE SURVEYING points. Next run the level over the lines at right angles to the former ones and in the same way, checking the levels at inter- sections. Advantage should be taken of every opportunity to check upon bench-marks previously located, and to establish others. In keeping the field records, the notes of the two sets of lines should be kept in separate books ; that is to say, if, for instance, one set of lines run north and south, and, therefore, the other east and west, the notes of the north and south lines should be entered in one set of books and the notes of the east and west lines in an- other set, and a note should be made of the direction in which a line is run, as from north to south or from east to west. In conducting a survey for the preparation of a topographical map necessary to the design of a sewer or waterworks system, much the same method is to be followed, but now the streets and alleys take the place of the rectangular system referred to above. As before, all the streets and alleys running in parallel directions are to be gone over in a systematic way, readings being taken fifty or one hundred feet apart in addition to street and alley intersec- tions. (By street and alley intersections is intended the intersec- tions of the center lines, the lines of levels being run along these center lines.) If a fairly accurate map of a town is available, the distances measured with the tape along the center line of the streets and alleys will serve as a check upon the map. If, however, dis- crepancies occur or there is no map available, it will be necessary to use the transit for staking out street lines and for determin- ing the relative directions of these lines. It follows that the topography of the ground between streets and alleys can only be approximated, but sufiicient points accurately determined will have been established to permit the platting of a contour map, from which the system can be laid down. The office work involved in the survey of an area, as above described^ consists in preparing profiles of the level lines and pre- paring a plat of the lines surveyed. From the profiles the contour points can be laid down in their proper position upon the plat, and as each point is laid down, its elevation should be noted in pencil, and after all the points have been platted, the points in the same contour line can be connected preferably free-hand producing the con- PLANE SURVEYING 191 tour map. The scale to be adopted will depend upon the nature of the work, but should be as large as possible, consistent with the convenient handling of the map. Transit and Stadia. The method by transit and stadia is of more general application than the preceding method, points being located by " polar co-ordinates," that is to say, by direction and distance from a known point, the elevation being determined at the same time. Method of conducting field operations. If the area to be sur- veyed is small, the preceding method, based upon a system of rectangles, will prove satisfactory, and the elevations of the corners and salient points can be determined at the same time that the lines forming the rectangles are being laid down. Especial care should be taken to check the elevations of the corners. In making a survey for a sewer or a waterworks system, the transit and stadia method will be found efficient, especially in cases where no survey has previously been made, the map, if it exists at all, having been compiled from the records in the County Record- er's office. The bench-marks necessary in a survey of this kind, however, should be established with the wye-level, and it may be desirable to determine the elevation of street intersections in the same way. If the area to be surveyed is too large, or of uneven topography, proceed as follows: Choose a point, as the intersection of two streets, the corner of a farm, or an arbitrary point conveniently located and drive a stake firmly at this point, "witnessing" it from other easily recognized points or stakes. The transit should be set over this point with the vernier reading zero, and the instru- ment pointed by the lower motion in the direction of the meridian. This may be the true meridian previously determined, the mag- netic meridian as shown by the needle, or an arbitrary meridian assumed for the purpose of the survey. It will generally be more satisfactory to run out a true meridian by means of the solar attachment, but in any event the direction of the line taken as a meridian should be defined by stakes, firmly driven into the ground, and "witnessed" by stakes or other objects easily recognized. The elevation of .the starting point, if not known, is assumed and recorded in the notebook. A traverse line should now be run, its 192 PLANE SURVEYING position and direction chosen with a view to obtaining from each station the largest possible number of pointings to salient features of the area under survey, and these pointings are taken while the instrument is set at any station, and before the traverse is com- pleted. The length of each course is measured with the stadia, and together with the azimuth and the vertical angle, it should be recorded in the notebook. The length, azimuth, and vertical angle of each course should be read' from both ends to serve as a check. The additional pointings taken from each course of a traverse are usually called " side shots", and for each there are required the distance, azimuth, and vertical angle. These will locate the point and determine its elevation. The method of using the stadia has already been quite fully discussed in Part II., and need not be repeated. The points selected for side shots should be such as will enable the contours to be platted intelligently aLd accurately upon the map of the area under survey. They should be taken along ridges and hollows and at all changes of slope. They should be taken at frequent intervals along a stream to indicate its course, or along the shore of a lake. It is usually required that the location of artificial structures, such as houses, fences, roads, etc., be determined that they may be mapped in their proper position. Pointings, therefore, should be taken to all fence corners and angles, and to enough corners and angles of buildings, to permit of their being platted. Sufficient points should be taken along roads to determine their direction. Wooded lands, swamps, etc., may be indicated by pointings taken around their edges. In addition to the notes above described, the recorder should amplify the notes with sketches, to aid the memory in mapping. The traverse, of course, forms the backbone of such a survey, and the accuracy of the resulting topographical map will depend upon the degree of care bestowed upon running the courses. Over uneven ground, it is often desirable to run a secondary trav- erse from the first, for the more rapid and accurate location of points. The organization of a party will depend upon the nature of the country traversed and of the results required. Changes in PLANE SURVEYING 193 the make-up of parties, as given below, will suggest themselves for any special work. For economy and speed, the party for taking topography with transit and stadia will consist of a transitman or observer, a recorder in charge of the notebook, who should be capable of making such sketches as are necessary, and two to four men with stadia rods. The greater the distances to be traversed by the stadia men between points taken, the greater number the observer can work to advantage. One or two axemen may be employed if clearing is to be done. The party may be reduced to two men one to handle the instrument, record notes and make sketches, the other to carry the rod. The Plane Table and Stadia. The plane table is an instru- ment intended for topographic purposes only and is used for the immediate mapping of a survey made with it, no notes of angles being taken, but the lines being platted at once upon the paper. The use of the plane table has been fully described. In topo- graphical work over an extended area, it may be used for filling in details, based upon a previous traverse made with a transit, or based upon a system of triangulation as will be described. Over small areas, the traverse itself may be run with the plane table and the details filled in at the same time. It is the standard in- strument of the United States Geological Survey and is largely used upon the United States Geodetic Survey. The points in favor of the plane table are : Economy, since the map is made at once without the expense of notes and sketches; and as the mapping is all done upon the ground to be represented, all of its peculiarities and characteristics can be correctly repre- sented. On the other hand, the plane table is an instrument useful only for taking topography ; the rodmen are idle while the map- ping is being done ; the instrument is more unwieldy than the transit, particularly upon difficult ground ; the record of the work for a long period is constantly exposed to accident ; the distortion of the paper with the varying dampness of air, introduces errors in the map ; while the area exposed makes it too unstable to use in high winds. 194 PLANE SUKVEYING The organization of a party for the taking of topography, using the plane table, is much the same as with the transit and stadia ; however, on account of the weight of the instrument, means of transportation must be provided. A less number of rodmen can be employed than with the stadia, owing to the time required for mapping. An observer, a man to reduce stadia notes and sketch topog- raphy around points determined by intersection or stadia from the plane table station, and one rodman, will make the minimum working party, in addition to which, axemen and a team for trans- portation will be required. Photography. The following is taken from Gillespies Sur- veying (Staley). " Photography has long been successfully employed by European engineers, notably those of Italy, for the purpose of taking topography. The Canadian Government has also employed it successfully in the survey of Alaska. The recommendation of this method is the great saving of time in the field, while giving topographic features with all the accuracy required for maps to be platted on a scale of 1 to 25,000. JV1. Javary states that the maximum error both for horizontal distances and elevations, using a camera with a focal length of twenty inches and a microscope in examining the points, was only 1 in 5,000 as deduced from a number of cases. M. Laussedat, in his work, found that this method did not require more than one-third the time necessary by the usual methods. This makes it especially suitable in all mountainous regions, where so much time is lost in getting to and from stations, that little is available for observations and sketching. A single occupation of a station with photographic apparatus would suffice to complete work that with the ordinary methods would require several days." Instruments. The ordinary camera may be used, if it is pro- vided with a level. A tripod head for leveling the instrument, and a roughly graduated horizontal circle for reading the direction of the line of sight, when photographing different parts of the horizon, are convenient attachments. 0.00 Hi III ai in lla ill is II PLANE SURVEYING 195 A camera is sometimes used upon a plane table, the record of the work being* made upon the paper in connection with a set of radial lines drawn from the point representing the station occupied. Many special forms of instrument combining the camera and theodolite have been devised, some one of which should be used if work of this kind is to be undertaken on a large scale. For a description of these instruments, and a complete treatise on this subject, comprising a discussion of the requirements of the appa- ratus, the fundamental principles of photography, methods of field work, forms of notes, reduction of notes and making of the map, together with the bibliography of the subject, see United States Coast and Geodetic Survey Report, 1893, Part II., Appendix 3. The camera tripod as ordinarily constructed is too unstable for purposes of topographic surveying, and it is desirable to have a tripod constructed especially for this class of work. Glass plates are heavy and awkward to carry aside from their fragile nature. Cut films can be procured in any of the standard sizes, and as they are light and stand rough handling and give ordinarily as good results as the glass plates, they are to be preferred. Their cost is about double that of the glass. TRIANQULATION. This method of surveying is sometimes called " Trigonometric Surveying" and sometimes "Geodetic Surveying", though this latter is properly applied only when the area to be surveyed is so extensive that allowance must be made for the curvature of the earth. Since this instruction paper is devoted to Plane Surveying only, the curvature of the earth will be neglected. Triangulation, or Triangular Surveying, is founded upon the method of determining the position of a point at the apex of a- triangle of which the base and two angles are measured. Thus in Fig. 129 the length of the base line AB is measured and the angles PAB and PB A are measured, from which can be calculated the lengths of the sides PA and PB. This calculated length of PA will then be taken as the side of a second triangle, and the angles PAC and PCA measured, from which the other sides of the triangle can be calculated. By an extension of this principle, 196 PLANE SURVEYING a field, farm, or a country can be surveyed by measuring a base line only, and calculating all of the other desired distances, which are made the sides of a connected series of imaginary triangles whose angles are carefully measured. Measuring the base line. For a base line, a fairly level stretch of ground is selected, as nearly as possible in the middle of the area to be surveyed, and a line from one thousand feet to one-half mile, or longer, is very carefully meas- ured. The ends of this line are marked with stone monuments or solid stakes. If the survey 'is of sufficient importance, the ends of the base line and the apexes of the Fig. 129. triangles should be permanently preserved by means of stones not less than six inches square in cross-section and two feet long, these stones being set deep enough to be beyond the disturbing action of frost. Into the top of this stone should be leaded a copper bolt about one-half inch in diam- eter, the head of the bolt being marked with a cross to designate the exact point. The point may be brought to the surface by a plumb-line for use in the survey. The location of each monument should be fully described with reference to surrounding objects of a permanent character, so as to be easily recovered for future use. The measurement of the base line for the areas of limited extent should be made with a precision of from one in five thousand to one in fifty thousand, depending upon the scale of the map, the extent of the area under survey, and the nature and importance of the work. The two ends of the base line having been determined and marked, the transit is set over one end and a line of stakes ranged out between the two ends, especial care being taken to make the alignment as perfect as possible. These stakes should be not less than two inches square, driven firmly into the ground, preferably at even tape lengths apart, or at least at one-half or one-quarter tape lengths, center to center; the centers should be marked by 204 PLANE SURVEYING 19V fine scratches upon strips of tin or zinc tacked to the top of the stakes, For ordinary work the base line may be measured with a tape, notes being made of the temperature, pull, grade, and distances between supports, the tape having been previously standardized. For a degree of precision, such as is attempted upon the work of the United States Coast and Geodetic Survey, more refined methods are used, but as this properly belongs to geodetic surveying, it is unnecessary to consider it here. Measuring the angles. After establishing and measuring the base line, prominent points are chosen for triangulation points or apexes of triangles, and from the extremities of the base line angles are observed to these points, care being taken to BO choose the points that the angles shall in no case be less than 30, nor more than 120. The distances to these and between these points are then calculated by trigonometric methods, the instrument being then placed at each of these new stations and angles observed from them to still more distant stations, the calculated lines being used as new base lines. This process is repeated and extended until the entire district included in the survey is covered with a network of "primary triangles " of as large sides as possible. One side of the last triangle should be so located that its length can be determined by direct measurement as well as by calculation ; the accuracy of the work can thus be checked. Within these primary triangles secondary or smaller triangles' are formed to serve as the starting points for ordinary surveys with the transit and tape, transit and stadia, plane table, etc., to fix the location of minor details. Tertiary triangles may also be formed. When the survey is not very extensive, and extreme accuracy is not required, the ordinary methods of measuring angles may be employed. Otherwise there are two methods of measuring angles, called, respectively, the method of repetition and the method by continuous reading. When an engineer's transit is used for measuring angles, the method by repetition is the simplest and best and is carried out as follows : The vernier is preferably set at zero degrees and then by the lower motion turned upon the left- hand station; the lower motion is then clamped and the instrument turned by the upper motion upon the right-hand station: the 205 198 PLANE SURVEYING upper motion is then clamped and the instrument turned by the lower motion upon the left-hand station; lower motion clamped and instrument again turned by upper motion upon right-hand station. This process is repeated as often as may be necessary to practically cover the entire circle of 360 and the circle is then read. This reading divided by the number of repetitions will give the value of the angle. Now reverse the telescope and repeat the observations described above, but from right to left; the readings being taken in both directions to eliminate errors due to clamping and unclamp- ing and personal errors due to mistakes in setting upon a station. The readings should be taken with the telescope both direct and reverse to eliminate errors of adjustments. Both verniers should be read in order to eliminate errors due to eccentricity of verniers, and the entire circle is included in the operation in order to elim- inate errors due to graduation. The second method, by continuous reading, consists in point- ing the telescope at each of the stations consecutively, and reading the vernier at each pointing; the difference between the consecu- tive readings being the angle be- tween the corresponding points. Thus in Fig. 130 with the instru- ment at zero, the telescope is first directed to A and the vernier is read; then to B, C, D, E, etc., in succession, the vernier being read at each pointing. The reading of the vernier on A, subtracted from that on B, will give the angle AOB and so on. It is necessary in this method, to read both to the right and to the left, and with the telescope both direct and inverted. Since each angle is measured on only one part of the limb, it is necessary after completing the readings once around and back, to shift the vernier to another part of the limb and repeat the readings in both directions, and with the telescope direct and inverted. This is done as many times as there are sets of readings. Each complete set of readings to right and left, with 206 PLANE SURVEYING 199 the telescope direct and inverted, gives one value for each angle. The lengths of the sides of the triangles should be calculated with extreme accuracy in two ways if possible, and by at least two persons. Plane trigonometry may be used for even extensive surveys; for though these sides are really arcs and not straight lines the error under ordinary circumstances will be inappreciable. Radiating Triangulation. This method as is illustrated in Fig. 131 consists in choosing a conspicuous point O, nearly in the center of the area to be surveyed. Other points as A, B, C, D, etc., are so chosen that the signal at () can be seen from all of them, and that the triangles ABO, BCO, etc., shall be as nearly equilateral as possible. Measure one side, as AB for example, and at A measure the angles OAB and OAG; at B measure the angles OB A and OBC; and so on around the polygon. The correctness of these measure- ments may be tested by the sum of the angles. It will seldom be the case, however, that the sum of the angles will come out just even, and the angles must then be adjusted, as will be explained later. The calculations of the lengths of the unknown sides are readily made by the usual trigonometric methods; thus in the tri- angle AOB, there are given one side and all of the angles of the triangle from which to calculate AO and BO. Similarly all of the triangles of the polygon may be solved, and finally the length of OA may be measured and compared with the calculated length, &p found from the first triangle. A farm or field may be surveyed by the previously described method, but the following plan will often be more convenient : Choose a base line as AB within the field and measure its length. Consider first the triangles which have AB for a base, and the corners of the field for vertices. In the triangle ACB for example (see Fig. 132), we measure the angles CAB and CBA and the length of the base line AB, We can therefore calculate the length 207 200 PLANE SURVEYING of AC and BC. Next consider the field as made up of triangles with a common vertex A. In each of them, two sides and the included angle are given, to find the third side. If now the point B at the other end of the base line be taken for a common vertex, a check w'ill be obtained upon the work. A field or a farm or any inaccessible area such as a swamp, a lake, etc., may be surveyed without entering it. For a farm or any area permitting unobstructed vision, it will only be necessary to choose a base line AB, from which all of the corners of the farm, or all of the salient points of the area, can be seen. Take their bearings, or the angles be- tween the base line and their directions. The distances from A and B to each of them can be calculated as described, and the figure will then sho.w in what manner the content of the field is the difference between the contents of the triangles having A or B for a vertex, which lie outside of it, and those which lie partly within the field and partly outside of it. Their contents can be calculated, and their difference will be the desired content. See Fig. 133. Evidently the entire area included between the cor- ners of the field and the base line is the sum of the triangles A2B, 2B3 and 3B4. Subtracting from this sum the areas of the tri- angles 2A1, 1AB, 1B6, 56B and 5B4, there will remain the required area of the field, 123456. In all of the operations which have been explained, the posi- tion of a point has been determined by taking the angles, or bear- ings, of two lines passing from the two ends of a base line to the unknown point, but the same determination may be effected inversely by taking from the point the bearings by compass of the two ends of the base line or any two known points. The unknown point will then be fixed by plotting from the two known points, the opposite bearings, for it will be at the intersection of the lines thus determined. 208 PLANE SURVEYING 201 The determination of a point by the method founded on the intersection of lines, has the serious defect that the point sighted to will be very indefinitely determined if the lines which fix it meet at a very acute or a very obtuse angle, which the relative position of the points observed from and to often render unavoid- able. Intersections at right angles should therefore be sought for,^so far as other con- siderations will permit. Adjusting the Triangle. All of the angles of a given tri- G O angle are measured. If but two have been measured, and the third computed, the entire error of measurement of the two angles will be thrown into the third angle. It will be found, upon adding together the measured angles of a triangle, that the sam of the three angles is almost invariably more or less than 180. With *-> J the engineer's transit the error should be less than one minute. If there is no reason to suppose that one angle is measured more carefully than another, this error should be divided equally among the three angles of the triangle, and the corrected angles are used in computing the azimuths and lengths of the sides. This distri- bution of the error is called "adjusting" the triangle. With the large systems of extensive geodetic surveys much more elaborate methods are employed, since a large number of triangles must be adjusted simultaneously so that they will all be geometrically con- sistent, not only each by itself, but one with another. 209 MEDIUM PRICED DRAWING OUTFIT SHOWING THE INSTRUMENTS NECL&. SARY FOR PURSUING A COURSE IN MECHANICAL DRAWING. MECHANICAL DRAWING PART I The subject of mechanical drawing is of great interest and importance to all mechanics and engineers. Drawing is the method used to show graphically the small details of machinery; it is the language by which the designer speaks to the workman; it is the most graphical way to place ideas and calculations on record. Working drawings take the place of lengthy explana- tions, either written or verbal. A brief inspection of an accurate, well-executed drawing gives a better idea of a machine than a large amount of verbal description. The better and more clearly a drawing is made, the more intelligently the workman can com- prehend the ideas of the designer. A thorough training in this important- subject is necessary to the success of everyone engaged in mechanical work. The success of a draftsman depends to some extent upon the quality of his instruments and materials. Begin- ners frequently purchase a cheap grade of instruments. After they have become expert and have learned to take care of their instruments they discard them for those of better construction and finish. This plan has its advantages, but to do the best work, strong, well-made and finely finished instruments are necessary. INSTRUMENTS AND MATERIALS. Drawing Paper. In selecting drawing paper, the first thing to be considered is the kind of paper most suitable for the pro- posed work. For shop drawings, a manilla paper is frequently used, on account of its toughness and strength, because the draw- ing is likely to be subjected to considerable hard usage. If a finished drawing is to be made, the best white drawing paper should be obtained, so that the drawing will not fade or become discolored with age. A good drawing paper should be strong, have uniform thickness and surface, should stretch evenly, and should neither repel nor absorb liquids. It should also allow con- siderable erasing without spoiling the surface, and it should lie smooth when stretched or when ink or colors are used. It is, of Copyright, 1'JOU, by American School of Correspondence. 211 MECHANICAL DRAWING course, impossible to find all of these qualities in any one paper, as for instance great strength cannot be combined with fine surface. In selecting a drawing paper the kind should be chosen which combines the greatest number of these qualities for the given work. Of the better class Whatman's are considered by far the best. This paper is made in three grades; the hoi pressed has a smooth surface and is especially adapted for pencil and very fine line drawing, the cold pressed is rougher than the hot pressed, has a finely grained surface and is more suit- able for water color drawing; the rough is used for tinting. The cold pressed does not take ink as well as the hot pressed, but erasures do not show as much on it, and it is better for general work. There is but little difference in the two sides of Whatman's paper, and either can be used. This paper comes in sheets of standard sizes as follows: Cap, 13X17 inches. Elephant, 23 X 28 inches Demy, 15 X 20 Medium, 17 X 22 Royal, 19 X 24 Super-Royal, 19 X 27 Imperial, 22 X 30 lleph 'ohm Columbia, 23 X 34 Atlas, 26 X 34 Double Elephant, 27 X 40 Antiquarian, 31 X 53 Emperor, 48 X 68 The usual method of fastening paper to a drawing board is by means of thumb tacks or small one-ounce copper or iron tacks. In fastening the paper by this method first fasten the upper left hand corner and then the lower right pulling the paper taut. The other two corners are then fastened, and sufficient number of tacks are placed along the edges to make the paper lie smoothly. For very fine work the paper is usually stretched and glued to the board. To do this the edges of the paper are first turned up all the way round, the margin being at least one inch. The whole surface of the paper included between these turned up edges is then moistened by means of a sponge or soft cloth and paste or glue is spread on the turned up edges. After removing all the surplus water on the paper, the edges are pressed down on the board, commencing at one corner. During this process of laying down the edges, the paper should be stretched slightly by pulling the edges towards the edges of the drawing board. The drawing board is then placed horizontally and left to dry. After the paper has become dry it will be found to be as smooth and tight as a 212 MECHANICAL DRAWING drum head. If, in stretching, the paper is stretched too much it is likely to split in drying. A sliglit stretch is sufficient. Drawing Board. The size of the drawing board depends upon the size of paper. Many draftsmen, however, have several boards of various sizes, as they are very convenient. The draw- ing board is usually made of soft pine, which should be well sea- soned and straight grained. The grain should run lengthwise of the board, and at the two ends there should be pieces about If or 2 inches wide fastened to the board by nails or screws. These end pieces should be perfectly straight for accuracy in using the T-square. Frequently the end pieces are fastened by a glued DRAWING BOARD matched joint, nails and screws being also used. Two cleats on the bottom extending the whole width of the board, will reduce the tendency to warp, and make the board easier to move as they raise it from the table. Thumb Tacks. Thumb tacks are used for fastening the paper to the drawing board. They are usually made of steel either pressed into shape, as in the cheaper grades, or made with a head of German silver with the point screwed and riveted to it. They are made in various sizes and are very convenient as they can be easily removed from the board. For most work however, 213 MECHANICAL DRAWING draftsmen use small one-ounce copper or iron tacks, as they can be forced flush with the drawing paper, thus offering no obstruction to the T-square. They also possess the advantage of cheapness. Pencils. In pencilling a drawing the lines should be very fine and light. To obtain these light lines a hard lead pencil must be used. Lead pencils are graded according to their hardness, and are numbered by using the letter H. In general a lead pencil of 5H (or HHHHH) or 6H should be used. A softer pencil, 4H, is better for making letters, figures and points. A hard lead pencil should be sharpened as shown in Fig. 1. The -wood is cut away so that about 1 or ^ inch of lead projects. The lead can then be sharpened to a chisel edge by rubbing it against a bit of sand paper or a fine file. It should be ground to a chisel edge and the corners slightly rounded. In making the straight lines the chisel edge should be used by placing it against the T-square or triangle, and because of the chisel edge the lead will remain sharp much longer than if sharpened to a point. This chisel edge enables the draftsman to draw a fine line exactly through a given point. If the drawing is not to be inked, but is made for tracing or for rough usage in the shop, a softer pencil, 3H or 4H, may be used, as the lines will then be somewhat thicker and heavier. The lead for compasses may also be sharpened to a point although some draftsmen prefer to use a chisel edge in the compasses as well as for the pencil. In using a very hard lead pencil, the chisel edge will make a deep depression in the paper if much pressure is put on the pencil. As this depression cannot be erased it is much better to press lightly on the pencil. Erasers. In making drawings, but little erasing should be necessary. However, in case this is necessary, a soft rubber should be used. In erasing a line or letter, great care must be exercised or the surrounding work will also become erased. To prevent this, some draftsmen cut a slit about 3 inches long and 4 to inch wide in a card as shown in Fig. 2. The card is then 214 MECHANICAL DRAWING placed over the work and the line erased without erasing the rest of the drawing. An erasing shield of a form similar to that shown in Fig. 3 is very convenient, especially in erasing letters. It is made of thin sheet metal and is clean and durable. For cleaning drawings, a sponge rubber may be used. Bread crumbs are also used for this purpose. To clean the drawing o o o big. 2. Fig. scatter dry bread crumbs over it and rub them on the surface with the hand. T-Square. The T-square consists of a thin straight edge called the blade, fastened to a head at right angles to it. It gets Fig. 4. its name from the general shape. T-squares are made of various materials, wood being the most commonly used. Fig. 4 shows an ordinary form of r i '-square which is adapted to most work. In Fig. 5 is shown a T-square with edges made of ebony or mahogany, as these woods are much harder than pear wood or maple, which is generally used. The head is formed so as to fit against the left- hand edge of the drawing board, while the blade extends over the surface. It is desirable to have the blade of the T-square form a right angle with the head, so that the lines drawn with the T-- square will be at right angles to the left-hand edge of the board. This, however, is not absolutely necessary, because the lines drawn with the T-square are always with reference to one edge of the 215 MECHANICAL DRAWING. board only, and if this edge of the board is straight, the lines drawn with the T-square will be parallel to each other. The T- square should never be used except with the left-hand edge of the board, as it is almost impossible to find a drawing broad with the edges parallel or at right angles to each other. The T-square with an adjustable head is frequently veiy con- venient, as it is sometimes necessary to draw lines parallel to each Fig. 5. other which are not at right angles to the left-hand edge of the board. This form of T-square is similar to the ordinary T-square already described, but the head is swiveled so that it may be clamped at any desired angle. The ordinary T-square as shown in Figs. 4 and 5 is, how ever, adapted to almost any class of drawing. * Fig. 6 shows the method of drawing parallel horizontal lines witli the T-square. With the head of the T-square in contact with the left-hand edge of the board, the lines may be drawn by moving the T-square to the desired position. In using the T-square the upper edge should always be used for drawing as the two edges may not be exactly parallel and straight, and also it is more convenient to use this edge with the triangles. If it is neces- sary to use a straight edge for trimming drawings or cutting the paper from the board, the lower edge of the T-square should be used so that the upper edge may not be marred. For accurate work it is absolutely necessary that the working edge of the T-square should be exactly straight. To test the 216 MECHANICAL DRAWING. straightness of the edge of the T-square, two T-squares may be placed together as shown in Fig. 7. This figure shows plainly that the edge of one of the T-squares is crooked. This fact, how- ever, does not prove that either one is straight, and for this deter- mination a third blade must be used and tried with the two given T-squares successively. Triangles. Triangles are made of various substances such as wood, rubber, celluloid and steel. Wooden triangles are cheap but are likely to warp and get out of shape. The rubber tri- angles are frequently used, and are in general satisfactory. The transparent celluloid triangle is, however, extensively used on ac- count' of its transparency, which enables the draftsmen to see the work already done even when covered with the triangle. In using a rubber or celluloid triangle take care that it lies perfectly flat or Fig. 7. TRIANGLES. is hung up when not in use ; when allowed to lie on the drawing board with a pencil or an eraser under one corner it will become warped in a short time, especially if the room is hot or the sun happens to strike the triangle. Triangles are made in various sizes, and many draftsmen have several constantly on hand. A triangle from 6 to 8 inches on a side will be found convenient for most work, although there are many cases where a small triangle measuring about 4 inches 217 10 MECHANICAL DRAWIKO. on a side will be found useful. Two triangles are necessary for every draftsman, one having two angles of 45 degrees each and one a right angle ; and the other having one angle of 60 degrees, one of 30 degrees and one of 90 degrees. The value of the triangle depends upon the accuracy of the angles and the straightness of the edges. To test the accuracy of the right angle of a tri- angle, place the triangle with the lower edge rest- ing on the edge of the T-square, as shown in Fig. 8. Now draw the line C D, which should be perpendicular to the edge of the T-square. The same triangle should -then be placed in the position shown at B. If the right angle of the triangle is exactly 90 degrees the left-hand edge of the triangle should exactly coincide with the line C D. To test the accuracy of the 45-degree triangles, first test the right angle then place the triangle with the lower edge resting on the work- ing edge of the T-square, and draw the line E F as shown in Fig. 9. Now without moving the T-- square place the triangle Fig. 8. Fig. 9. so that the other 45-degree angle is in the position occupied by the first. If the two 45-degree angles coincide they are accurate. Triangles are very convenient in drawing lines at right angles to the T-square. The method of doing this is shown in Fig. 10. Triangles are also used in drawing lines at an angle with the horizontal, by placing them on the board as shown in Fig. 11. Suppose the line E F (Fig. 12) is drawn at any anjle, and we wish to draw a line through the point P parallel to \i 218 MECHANICAL DRAWING. li First place one of the triangles as shown at A, having one edge coincidkg with the given line. Now take the other triangle and place one of its edges in contact with the bottom edge of triangle A. Holding the triangle B firmly with the left hand the triangle A may be slipped along to the right or to the left until the edge of the triangle reaches the point P. The line M N may then be drawn along the edge of the triangle passing through the point P. In place of the tri- angle B any straight edge such as a T-square may be used. A line can be drawn perpendicular to another by means of the triangles as follows. Let E F (Fig. 13) be the given line, and suppose we wish tc draw a line perpendicular to E F through the point D. Place the longest side of one of the triangles so that it coincides with the lina E F, as the triangle is snown in. posi- tion at A. Place the other triangle (or any straight edge) in the position ot the triangle as shown at B, one edge resting against the edge of the triangle A. Fig, 10. Fig. 11. Then holding B with the left hand, place the tn angle A in the position shown at C, so that the longest side passes through the point D. A line can then be drawn through the point D perpendicular to E F. In previous figures we have seen how lines may be drawn making angles of 30, 45, 60 and degrees with the horizontal. If it is desired to draw lines forming angles of 15 and 75 degrees the triangles may be placed as shown in Fig. 14. In using the triangles and T-square almost any line may b drawn. Suppose we wish to draw a rectangle having one side 219 i'j MECHANICAL DRAWING. horizontal. First place the T-square as shown in Fig. 15. By moving the T-square up or down, the sides A B and D C may be drawn, because they are horizontal and parallel. Now place one of the triangles resting on the T-square as shown at E, and hav- ing the left-hand edge passing through the poirt D. The vertical Fig. 12. Fig. 13. line D A may be drawn, and by sliding the triangle along the edge of tlie T-square to the position F the line B C may be drawn by using the same edge. These positions are shown dotted in Fig. 15. If the rectangle is to be placed in some other position on the drawing board, as shown in Fig. 16, place the 45-degree triangle F so that one edge is parallel to or coincides with the side D C. Now holding the triangle F in position place the triangle H so that its upper edge coincides with the lower edge of the triangle F. By holding H in position and sliding the triangle F along its upper edge, the sides A B and D C may be drawn. To draw the sides A D and B C the triangle should be used as shown at E. Compasses. Compasses are used for drawing circles and arcs of circles. They are made of various materials and in various sizes. The cheaper class of instruments are made of brass, but they are unsatisfactory on account of the odor and the tendency to tarnish The best material is German silver. It does not soil L Fig. 14. 220 MECHANICAL DRAWING. readily, it has no odor, and is easy to keep clean. Aluminum in- struments possess the advantage of lightness, but on account of the soft metal they do not wear well. The compasses are made in the form shown in Figs. 17 and 18. Pencil and pen points are provided, as shown in Fig. 17. Either pen or pencil may be inserted in one leg by menns of a shank and socket. The other leg is fitted with a needle point which is placed at the center of the circle. In most instru- ments the needle point is separate, and is made of a piece of round steel wire having a square shoulder at one or both ends. Be- r A* o K hK I ." o o Fig. 15. low this shoulder the needle point projects. The needle is made in this form so that the hole in the paper may be very minute. In some instruments lock nuts are used to hold the joint firmly in position. These lock nuts are thin discs of steel, with notches for using a wrench or forked key. Fig. 19 shows the detail of the joint of high grade instruments. Both legs are alike at the joint, and two pivoted screws are inserted in the yoke. This permits ample movement of the legs, and at the same time gives the proper stiff- ness. The flat surface of one of the legs is faced with steel, the other being of German silver, in order that the rubbing parts may be of different metals. Small set screws are used to prevent the pivoted screws from turning in the yoke. The contact surfaces of this joint are made cir- cular to exclude dust and dirt and to prevent rusting of the steel face. Figs. 20, 21 and 22 show the detail of the socket; in some Fig. 16. MECHANICAL DRAWING. instruments the shank and socket are pentagonal, as shown in Fig. 20. The shank enters the socket loosely, and is held in place by means of the screw. Unless used very carefully this arrange- ment is not durable because the sharp corners soon wear, and the pressure on the set screw is not sufficient to hold the shank firmly in place. In Fig. 21 is shown another form of shank. This is round, having a flat top. A set screw is also used to hold this in posi- tion. A still better form of socket is shown in Fig. 22 ; the hole Fig. 17. Fig. 18. is made tapered and is circular. The shank fits accurately, and is held in perfect alignment by a small steel key. The clamping screw is placed upon the side, and keeps the two portions of the split socket together. Figs. 17 and 18 show that both legs of the compasses are jointed in order that the lower part of the legs may be perpen- dicular to the paper while drawing circles. In this way the ueedle point makes but a small hole in the paper, and both nibs of MECHANICAL DRAWING. the pen will press equally on the paper. In pencilling circles it is not as necessary that the pencil should be kept vertical ; it is a good plan, however, to learn to use them in this way both in pen- cilling and inking. The com- passes should be held loosely be- tween the thumb and forefinger. If the needle point is sharp, as it should be, only a slight pres- sure will be required to keep it in place. While drawing the circle, incline the compasses slightly in the direction of revolution and press lightly on the pencil or pen. In removing the pencil or pen, it should be pulled out Fi - 19 - straight. If bent from side to side the socket will become en- larged and the shank worn; this will render the instrument inac- curate. For drawing large circles the lengthening bar shown in Fig. 17 should be used. When using the lengthening bar the Fig. 20. Q Fig. 21. one hand and the circle needle point should be steadied with described with the other. Dividers. Dividers, shown in Fig. 23, are made similar to the compasses. They are used for laying off distances on the draw- ing, either from scales or from other parts of the drawing. They _ may also be used for dividing a line ( 1 1 . V-^_ i into equal parts. When dividing a line into equal parts the dividers should be turned in the opposite direc- tion each time, so that the moving point passes alternately to the right and to the left. The instrument can then be operated readily with one hand. The points of the dividers should be very sharp so that the holes made in the paper will be small If large holes are made in the paper, and the distances betweer Fig. 22. 223 16 MECHANICAL DRAWING. the points are not exact, accurate spacing cannot be done Sometimes the compasses are furnished with steel divider points in addition to the pen and pencil points. The compasses may then be used either as dividers or as compasses. Many drafts- men use a needle point in place of dividers for making measure- ments from a scale. The eye end of a needle is first broken off and the needle then forced into a small handle made of a round piece' of soft pine. This instrument is very convenient for indicating the intersection of lines and marking off distances. Bow Pen and Bow Pencil. Ordinary large compasses are too heavy to use in making small circles, fillets, eta The leverage of the long leg is so great that it is very difficult to draw small circles accurately. For this reason the bow compasses shown in Figs. 24 and 25 should be used on all arcs and circles having a radius of less than three-quarters inch. The bow compasses are also con- venient for duplicating small circles such as those which represent boiler tubes, bolt holes, etc., ince there is no tendency to slip. The needle point must be adjusted to the same length as the pen or pencil point if very small circles are to be drawn. The adjustment for altering the radius of the circle can be made by turning the nut. If the change in radius is considerable the points should be pressed to- gether to remove the pressure from the nut which can Pig. 23 tnen ke turned in either direction with but little wear on the threads. Fig. 26 shows another bow instrument which is frequently used in small work in place of the dividers. It has the advantage of retaining the adjustment. Drawing Pen. For drawing straight lines and curves that are not arcs of circles, the line pen (sometimes called the ruling pen) is used. It consists of two blades of steel fastened to a handle as shown in Fig. 27. The distance between the pen points can be adjusted by the thumb screw, thus regulating the width of line to be drawn. The blades are given a slight curvature so that there will be a cavity for ink when the points are close together. MECHANICAL DRAWING. 17 The pen may be filled by means of a common steel pen or with the quill which is provided with some liquid inks. The pen should not be dipped in the ink because it will then be necessary to wipe the outside of the blades before use. The ink should fill the pen to a height of about or | inch ; if too much ink is placed in the pen it is likely to drop out and spoil the drawing. Upon finishing the work the pen should be carefully wiped with Fig. 24. Fig. 25. Fig. 26. jhamois or a soft cloth, because most liquid inl.s corrode the steel. In using the pen, care should be taken that both blades bear equally on the paper. If the points do not bear equally the line will be ragged. If both points touch, and the pen is in good condition the line will be smooth. The pen is usually inclined slightly in the direction in which the line is drawn. The pen Fig. 27. should tounh the triangle or T-square which serve as guides, but it should not be pressed against them because the lines will then be uneven. The points of the pen should be close to the edge of the triangle or T-square, but should not touch it. To Sharpen the Drawing Pen. After the pen has been used for some time the points become worn, and it is impossible 225 18 MECHANICAL DRAWING. to make smooth lines. This is especially true if rough paper is used. The pen can be put in proper condition by sharpening it. To do this take a small, flat, close-grained oil-stone. The blades should first be screwed together, and the points of the pen can be given the proper shape by drawing the pen back and forth over the stone changing the inclination so that the shape of the ends will be parabolic. This process dulls the points but gives them the proper shape, and makes them of the same length. To sharpen the pen, separate the points slightly and rub one of them on the oil-stone. While doing this keep the pen at an angle of from 10 to 15 degrees with the face of the stone, and give it a slight twisting movement. This part of the operation requires great care as the shape of the ends must not be altered. After the pen point has become fairly sharp the other point should be ground in the same manner. All the grinding should be done on the outside of the blades. The burr should be removed from the inside of the blades by using a piece of leather or a piece of pine wood. Ink should now be placed between the blades and the pen tried. The pen should make a smooth line whether fine or heavy, but if it does not the grinding must be continued and the pen tried frequently. Ink. India ink is always used for drawing as it makes a permanent black line. It may be purchased in solid stick form or as a liquid. The liquid form is very convenient as much time is saved, and all the lines will be of the same color; the acid in the ink, however, corrodes steel and makes it necessary to keep the pen perfectly clean. Some draftsmen prefer to use the India ink which comes in stick form. To prepare it for use, a little water should be placed in a saucer and one end of the stick placed in it. The ink is ground by giving it a twisting movement. When the water has become black and slightly thickened, it should be tried. A heavy line should be made on a sheet of paper and allowed to dry. If the line has a grayish appearance, more grinding is necessary. After the ink is thick enough to make a good black line, the grinding should cease, because very thick ink will not flow freely from the pen. If, however, the ink has become too 226 MECHANICAL DRAWING. 16 thick, it may be diluted with water. After using, the stick should be wiped dry to prevent crumbling. It is well to grind the ink in small quantities as it does not dissolve readily if it has once become dry. If the ink is kept covered it will keep for two or three days. Scales. Scales are used for obtaining the various measure- ments on drawings. They are made in several forms, the most convenient being the flat with beveled edges and the triangular. The scale is usually a little over 12 inches long and is graduated for a distance of 12 inches. The triangular scale shown in Fig. 28 has six surfaces for graduations, thus allowing many gradua- tions on the same scale. The graduations on the scales are arranged so that the drawings may be made in any proportion to the actual size. For mechanical work, the common divisions are multiples of two. Thus we make drawings full size, half size, J, , Jg, gL, g^, etc. If a drawing is ^ size, 3 inches equals 1 foot, hence 3 inches is divided into 12 equal parts and each division represents one inch. If the smallest division on a scale represents Jg inch, the scale is said to read to Jg inch. Scales are often divided into y 1 ^, $, ^, 3^, etc., for archi- tects, civil engineers, and for measuring on indicator cards. The scale should never be used for drawing lines in place of triangles or T-square. Protractor. The protractor is an instrument used for laying off and measuring angles. It is made of steel, brass, horn and paper. If made of metal the central portion is cut out as shown in Fig. 29, so that the draftsman can see the drawing. The outer edge is divided into degrees and tenths of degrees. Some- times the graduations are very fine. In using a protractor a very sharp hard pencil should be used so that the lines will be fine and accurate. The protractor should be placed so that the given line ( pro 227 MECHANICAL DRAWING. duced if necessary ) coincides with the two O marks. The center of the circle being placed at the point through which the desired line is to be drawn. The division can then be marked with the pencil point or needle point. Irregular Curve, One of the conveniences of a draftsman's Fig. 29. outfit is the French or irregular curve. It is made of wood, hard rubber or celluloid, the last named material being the best. It is made in various shapes, two of the most common being Fig. 30. shown in Fig. 30. This instrument is used for drawing curves other than arcs of circles, and both pencil and line pen can be used. To draw the curve, a series of points is first located and then the curve drawn passing through them by using the part of the irregular curve that passes through several of them The MECHANICAL DRAWING. 2] curve is shifted for this work from one position to another. It frequently facilitates the work and improves its appearance to draw a free hand pencil curve through the, points and then use the irregular curve, talcing care that it always fits at least three points. In inking the curve, the blades of the pen must be kept Fig. 31. tangent to the curve, thus necessitating a continual change of direction. Beam Compasses. The ordinary compasses are not large enough to draw circles having a diameter greater than about 8 or 10 inches. A convenient instrument for larger circles is found in the beam compasses shown in Fig. 31. The two parts called channels carrying the pen or pencil and the needle point are clamped to a wooden beam ; the distance between them being equal to the radius of the circle. Accurate adjustment is obtained by means of a thumb nut underneath one of the channel pieces. LETTERING. No mechanical drawing is finished unless all headings, titles and dimensions are lettered in plain, neat type. Many drawings are accurate, well-planned and finely executed but do not present a good appearance because the draftsman did not think it worth while to letter well. Lettering requires time and patience; and if one wishes to letter rapidly and well he must practice. Usually a beginner cannot letter well, and in order to pro duce a satisfactory result, considerable practice is necessary. Many MECHANICAL DRAWING. think it a good plan to practice lettering before commencing a drawing. A good writer does not always letter well ; a poor writer need not be discouraged and think he can never learn to make a neatly lettered drawing. In making large letters for titles and headings it is often necessary to use drawing instruments and mechanical aids. The small letters, such as those used for dimensions, names of materials, dates, etc., should be made free hand. There are many styles of letters used by draftsmen. For titles, large Roman capitals are frequently used, although Gothic and block letters also look well and are much easier to make. ABCDEFGHIJ KLMNOPQR STUVWXYZ 1234567890 Almost any neat letter free from ornamentation is acceptable in the regular practice of drafting. Fig. 32 shows the alphabet oi vertical Gothic capitals. These letters are neat, plain and easily made. The inclined or italicized Gothic type is shown in Fig. 33. This style is also easy to construct, and possesses the advantage that a slight difference in inclination is not apparent. If the ver- tical lines of the vertical letters incline slightly the inaccuracy is very noticeable. The curves of the inclined Gothic letters such as those in the B, C, Gr, e7, etc., are somewhat difficult to make free hand, especially if the letters are about one-half inch high. In the alphabet shown in Fig. 34, the letters are made almost wholly of MECHANICAL DRAWING. straight lines, the corners only being curved. These letters are very easy to make and are clear cut. The first few plates of this work will require no titles; the only lettering being the student's name, together with the date and plate number. Later, the student will take up the subject of ABCDETGH/J KLMNOPQFt STUVWXYZ Fig. 3.3. lettering again in order to letter titles and headings for drawings showing the details of machines. For the present, however, in- clined Gothic capitals will be used. To make the inclined Gothic letters, first draw two parallel lines having the distance between them equal to the desired height of the letters. If two sizes of letters are to be used, the smaller should be about two-thirds as high as the larger. For the letters A BCDETGH/JKLM NOPQFISTUVWXYZ /23456789O Fig. 34. to be used on the first plates, draw two parallel lines ^ inch apart. This is the height for the letters of the date, name, also the plat^ number, and should be used on all plates throughout this ^-ork, unless other directions are given. In constructing the letters, they should extend fully to these lines, both at the top and bottom. They should not fall short of 231 24 MECHANICAL DRAWING. the guide lines nor extend beyond them. As these letters are inclined they will look better if the inclination is the same for all. As an aid to the beginner, he can draw light pencil lines, about \ inch apart, forming the proper angle with the parallel lines already drawn. The inclination is often made about 70 degrees ; but as a 60-degree triangle is at hand, it may be used. To draw these lines place the 60-degree triangle on the T-square as shown in Fig. 36. In making these letters the 60-degree lines will be found a great aid as a large proportion of the back or side lines have this inclination. Capital letters such as E, jP, P, T, Z, etc., should have the top lines coincide with the upper horizontal guide line. The bottom lines of such letters as Z>, E, L, Z, etc., should coincide with the lower horizontal guide line. If these lines do not coin- cide with the guide lines the words will look uneven. Letters, of which (7, 6r, 0, and Q, are types, can be formed of curved lines or of straight lines. If made of curved lines, they should have a little greater height than the guide lines to prevent their appear- ing smaller than the other letters. In this work they can be made of straight lines with rounded corners as they are easily constructed and the student can make all letters of the same height. To construct the letter A, draw a line at an angle of 60 degrees to the horizontal and use it as a center line. Then from the intersection of this line and the upper horizontal line drop a vertical line to the lower guide line. Draw another line from the vertex meeting the lower guide line at the same distance from the center line. The cross line of the A should be a little below the center. The F"is an inverted A without the cross line. For the letter M, the side lines should be parallel and about the same distance apart as are the horizontal lines. The side lines of the IF are not parallel but are farther apart at the top. The 7is not quite as wide as such letters as H, E^ N, R, etc. To make a Y. draw the center line 60 degrees to the horizontal ; the diverg- ing lines are similar to those of the V but are shorter and form a larger angle. The diverging lines should meet the center line a little below the middle. Tlie lower-case letters are shown in Fig. 35. In such letters 232 MECHANICAL DRAWING. 26 as 7/1, n, r, etc., make the corners sharp and not rounding. The letters a, b, c, e, g, o, p, q, should be full and rounding. The figures (see Fig. 32) are made as in writing except the 4, tf, 8 and 9. The Roman numerals are made of straight lines as they are largely made up of I, F"and X. At first the copy should be followed closely and the letters flrawn in pencil. For a time, the inclined guide lines may be used. obcctefgh/jk/mn Fig. 35. but after the proper inclination becomes firmly fixed in mind they should be abandoned. The horizontal lines are used at all times by most draftsmen. After the student has had consider- able practice, he can construct the letters in ink without first using the pencil. Later in the work, when the student has become pro- ficient in the simple inclined Gothic capitals, the vertical, block and Roman alphabets should be studied. PLATES, To lay out a sheet of paper for the plates of this work, the sheet, A B G F, (Fig. 36) is placed on the drawing board 2 or 3 inches from the left-hand edge which is called the working edge. If placed near the left-hand edge, the T-square and triangles can be used with greater firmness and the horizontal lines drawn with the T-square will be more accurate. In placing the paper on the board, always true it up according to the long edge of the sheet. First fasten the paper to the board with thumb tacks, using at least 4 one at each corner. If the paper has a tendency to curl it is better to use 6 or 8 tacks, placing them as shown in Fig. 36. Thumb tacks are commonly used ; but many draftsmen prefer one-ounce tacks as they offer less obstruction to the T-square and triangles. After the paper is fastened in position, find the center of the 233 MECHANICAL DRAWING. Fig. 234 MECHANICAL DRAWING. 27 sheet by placing the T-square so that its upper edge coincides with the diagonal corners A and G and then with the corners F and B, drawing short pencil lines intersecting at C. Now place the T-square so that its upper edge coincides with the point C and draw the dot and dash line D E. With the T-square and one of the triangles (shown dotted) in the position shown in Fig. 36, draw the dot and dash line H C K. In case the drawing board is large enough, the line C H can be drawn by moving the T- square until it is entirely off the drawing. If the board is small, produce (extend) the line K C to H by means of the T-square or edge of a triangle. In this work always move the pencil from the left to the right or from the bottom upward ; never (except for some particular purpose) in the opposite direction. After the center lines are drawn measure off 5 inches above and below the point C on the line JI G K. These points L and M may be indicated by a light pencil mark or by a slight puncture of one of the points of the dividers. Now place the T-- square against the left-hand edge of the board and draw horizontal pencil lines through L and M. Measure off 7 inches to the left and right of C on the center line DOE and draw pencil lines through these points (N and P) perpendicular to D E. We now have a rectangle 10 inches by 14 inches, in which all the exercises and figures are to be drawn. The lettering of the student's name and address, date, and plate number are to be placed outside of this rectangle in the ^-inch margin. In all cases lay out the plates in this manner and keep the center lines D E and K H as a basis for the various figures. The border line is to be inked with a heavy line when the drawing is inked. Pencilling. Inlaying out plates, all work, is first done in pen cil and afterward inked or traced on tracing cloth. The first few plates of this course are to be done in pencil and then inked ; later the subject of tracing and the process of making blue prints will be taken up. Every beginner should practice with his instruments until he can use them with accuracy and skill, and until he under- stands thoroughly what instrument should be used for making a given line. To aid the beginner in this work, the first three plates of this course are designed to give the student practice : they do MECHANICAL DRAWING. not involve any problems and none of the work is difficult. The student is strongly advised to draw these plates two or three times before making the one to be sent to us for correction. Dili- gent practice is necessary at first; especially on PLATE I as it involves an exercise in lettering. PLATE I. Pencilling. To draw PLATE 7, take a sheet of drawing paper at least 11 inches by 15 inches and fasten it to the drawing board as already explained. Find the center of the sheet and draw fine pencil lines to represent the lines D E and H K of Fig. 36. Also draw the border lines L, M, N and P. Now measure | inch above and below the horizontal center line and, with the T-square, draw lines through these points. These lines will form the lower lines D C of Figs. 1 and 2 and the top lines A B of Figs. 3 and 4- Measure | inch to the right and left of the vertical center line ; and through these points, draw lines parallel to the center line. These lines should be drawn by placing the triangle on the T-square as shown in Fig. 36. The lines thus drawn, form the sides B C of Figs. 1 and 3 and the sides A D of Figs. 2 and 4- Next draw the line A B A B with the T-square, 4 1 inches above the horizontal center line. This line forms the top lines of Figs. 1 and 2. Now draw the line D C D C 4| inches below the horizontal center line. The rectangles of the four figures are completed by drawing vertical lines 6| inches from the vertical center line. We now have four rectangles each 6 \ inches long and 4| inches wide. Fig. 1 is an exercise with the line pen and T-square. Divide the line AD into divisions each J inch long, making a fine pencil point or slight puncture at each division such as E, F, G, H, I, etc. Now place the T-square with the head at the left-hand edge of the drawing board and through these points draw light pencil lines extending to the line B C. In drawing these lines be careful to have the pencil point pass exactly through the division marks so that the lines will be the same distance apart. Start each line iu the line A D and do not fall short of the line B C or run over it. Accuracy and neatness in pencilling insure an accurate drawing. Some beginners think that they can correct inaccuracies while _<. _ Q I < n MECHANICAL DRAWING. inking; but experience soon teaches them that they cannot do so. Fig. 2 is an exercise with the line pen, T-square and triangle. First divide the lower line D C of the rectangle into divisions each | inch long and mark the points E, F, G, H, I, J, K, etc., as in Fig. 1. Place the T-square with the head at the left-hand edge of the drawing board and the upper edge in about the position shown in Fig. 36. Place either triangle with one edge on the upper edge of the T-square and the 90-degree angle at the left. Now draw fine pencil lines from the line D C to the line A B passing through the points E, F, G, H, I, J, K, etc. To do this keep the T-square 1 f r " ///// 5!Z52Za! / /-ft. SLJBiJ ^ (-?'{-? /- A / A N/CA Z_ f- jcf- / ? t y /7/t c 7/ i " Sf-* ABODE / ^ ^4- / // /// /i/ z X B rigid and slide the triangle toward the right, being careful to have the edge coincide with the division marks in succession. Fig. 3 is an exercise with the line pen, T-square and 45-degree triangle. First lay off the distances A E, E F, F G, G H, H I, IJ, J K, etc., each \ inch long. Then lay off the distances B L, L M, M N, N O, O P, P Q, Q R, etc., also \ inch long. Place the T- square so that the upper edge will be below the line D C of Fig. 3. With the 45-degree triangle draw lines from A D and D C to the points E, F, G, H, I, J, K, etc., as far as the point B. Now draw lines from D C to the points L, M, N, O, P, Q, R, etc., as 239 MECHANICAL DRAWING. far as the point C. In drawing these lines move the pencil away from the body, that is, from A D to A B and from D C to B C. Fig. 4 i s an exercise in free-hand lettering. The finished exercise, with all guide lines erased, should have the appearance shown in Fig. 4 of PL A TE I. The guide lines are drawn as shown in Fig, 87. First draw the center line E F and light pencil lines Y Z and T X, f inch from the border lines. Now, with the T- square, draw the line G, \ inch from the top line and the line H, ^ inch below G. The word "LETTERING-" is to be placed between these two lines. Draw the line I, fa inch below H. The lines I, J, etc., to K are all fa inch apart. We now practice the lower-case letters. Draw the line L, -^ inch below K and a light line | inch above L to limit the height of the small letters. The space between L and M is fa inch. The lines M and N are drawn in the same manner as K and L. The space between N and O should be ^ inch. The line P is drawn fa inch below O. Q is also fa inch below P. The lines Q and R are drawn J g inch apart as are M and N. The remainder of the lines S, U, V and W are drawn fa inch apart. The center line is a great aid in centering the word LETTERING" the alphabets, numerals, etc. The words "THE" and "Proficiency" should be indented about f inch as they are the first words of paragraphs. To draw the guide lines, mark off distances of inch on any line such as J and with the 60-degree triangle draw light pencil lines cutting the parallel lines. The letters should be sketched in pencil, the ordin- ary letters such as E, F, H, N, R, etc. being made of a width equal to about f the height. Letters like A, M and W are wider. The space between the letters depends upon the draftsman's taste but the beginner should remember that letters next to an A or an L should be placed near them and that greater space should be left on each side of an I or between letters whose sides are parallel; for instance there should be more space between an N and E than between an E and H. On account of the space above the lower line of the L, a letter following an L should be close to it If a T follows a T or the letter L follows an L they should lie placed near together. In all lettering the letters should be placed BO that the general effect is pleasing. After the four figures are 240 MECHANICAL DRAWING. 31 completed, the lettering for name, address and date should be pencilled. With the T-square draw a pencil line ^ inch above the top border line at the right-hand end. This line should be about 3 inches long. At a distance of fa inch above this line draw another line of about the same length. These are the guide lines for the word PLATE I. The letters should be pencilled free hand and the student may use the 60-degree guide lines if he desires. The guide lines of the date, name and address are similarly drawn in the lower margin. The date of completing the drawing should be placed under Fig. 3 and the name and address at the right under Fig. 4- The street address is unnecessary. It is a good plan to draw lines fa inch apart on a separate sheet of paper and pencil the letters in order to know just how much space each word will require. The insertion of the words ' Fig. jf," " Fig. 2" etc., is optional with the student. He may leave them out if he desires ; but we would advise him to do this extra lettering for the practice and for convenience in reference. First draw with the T-square two parallel line fa inch apart under each exercise ; the lower line being ^ inch above the horizontal center line or above the lower border line. Inking. After all of the pencilling of PLATE I has been completed the exercises should be inked. The pen should first be examined to make sure that the nibs are clean, of the same length and come together evenly. To fill the pen with ink use an ordi- nary steel pen or the quill in the bottle, if Higgin's Ink is used. Dip the quill or pen into the bottle and then inside between the nibs of the line pen. The ink will readily flow from the quill into the space between the nibs as soon as it is brought in contact. Do not fill the pen too full, if the ink fills about \ the distance to the adjusting screw it usually will be sufficient. If the filling has been carefully done it will not be necessary to wipe the outsides of the blades. However, any ink on the outside should be wiped off with a soft cloth or a piece of chamois. The pen should now be tried on a separate piece of paper in order that the width of the line may be adjusted. In the first work where no shading is done, a firm distinct line should be used. The beginner should avoid the extremes ; a very light line makes 241 MECHANICAL DRAWING. the drawing have a weak, indistinct appearance, and very heavy lines detract from the artistic appearance and make the drawing appear heavy. In case the ink does not flow freely, wet the finger and touch it to the end of the pen. If it then fails to flow, draw a slip of thin paper between the nibs (thus removing the dried ink) or clean thoroughly and fill. Never lay the pen aside without cleaning. In ruling with the line pen it should be held firmly in the right hand almost perpendicular to the paper. If grasped too firmly the width of the line may be varied and the draftsman soon becomes fatigued. The pen is usually held so that the adjusting screw is away from the T-square, triangles, etc. Many draftsmen incline the pen slightly in the direction in which it is moving. To ink Fig. 1, place the T-square with the head at the work- ing edge as in pencilling. First ink all of the horizontal lines moving the T-square from A to D. In drawing these lines con- siderable care is necessary ; both nibs should touch the paper and the pressure should be uniform. Have sufficient ink in the pen to finish the line as it is difficult for a beginner to stop in the middle of the line and after refilling the pen make a smooth con- tinuous line. While inking the lines A, E, F, G, H, I, etc., greater care should be taken in starting and stopping than while pencil- ling. Each line should start exactly in the pencil line A D and stop in the line B C. The lines A D and B C are inked, using the triangle and T-square. Fig. 2 is inked in the same manner as it was pencilled; the lines being drawn, sliding the triangle along the T-square in the successive positions. In inking Fig. 3, the same care is necessary as with the pre- ceding, and after the oblique lines are inked the border lines are finished. In Fig. 4 the border lines should be inked in first and then the border lines of the plate. The border lines should be quite heavy as they give the plate a better appearance. The intersections should be accurate, as any running over necessitates erasing. The line pen may now be cleaned and laid aside. It can be 242 r* MECHANICAL DRAWING. cleaned by drawing a strip of blotting paper between the nibs or by means of a piece of cloth or chamois. The lettering should be done free-hand using a steel pen. If the pen is very fine, accu- rate work may be done but the pen is likely to catch in the paper, especially if the paper is rough. A coarser pen will make broader lines but is on the whole preferable. Gillott's 404 is as fine a pen as should be used. After inking Fig. 4, the plate number, date and name should be inked, also free-hand. After ink- ing the words " Fig. 1" " Fig. 2," etc., all pencil lines should ba erased. In the finished drawing there should be no center lines, construction lines or letters other than those in the name, date, etc. The sheet should be cut to a size of u inches by 15 inches, the dash line outside the border line of PLATE I. indicating the edge. PLATE II. Pencilling. The drawing paper used for PLATE //should be laid out as described with PLATE I, that is, the border lines, center line and rectangles for Figs. 1 and 2. To lay out Figs. 3, 4 and 5 proceed as follows : Draw a line with the T-square parallel to the horizontal center line and | inch below it. Also draw another similar line 4f below the center line. The two lines o will form the top and bottom of Figs. 3, and 5. Now measure off 2^ inches on either side of the center on the horizontal center line and call the points Y and Z. On either side of Y and Z and at a distance of J inch draw vertical parallel lines. Now draw a vertical line A D, 4^ inches from the line Y and a vertical line B C 4| inches from the line Z. We now have three rectangles each 4 inches broad and 4| inches high. Figs. 1 and 2 are pen- cilled in exactly the same way as was Fig. 1 of PLATE /, that is, horizontal lines are drawn | inch apart. Fig. 3 is an exercise to show the use of a 60-degree triangle with a T-square. Lay off the distances A E, E F, F G, G H, etc. to B each J inch. With the 60 degree triangle resting on the upper edge of the T-square, draw lines through these points, E, F, G, H, I, J, etc., forming an angle of 30 degrees with the hori- zontal. The last line drawn will be A L. In drawing these lines move the poncil from A B to B C. Now find the distance 245 .34 MECHANICAL DRAWING. between the lines on the vertical B L and mark off these distances on the line B C commencing at L. Continue the lines from A L to N C. Commencing at N mark off distances on A D equal to those on B C and finish drawing the oblique lines. Fig. 4 is an exercise for intersection. Lay off distances of | inch on A B and A D. With the T-square draw fine pencil lines through the points E, F, G, H, I, etc., and with the T-square and triangle draw vertical lines through the points L, M, N, O, P, etc. In drawing this figure draw every line exactly through the points indicated as the symmetrical appearance of the small squares can be attained only by accurate pencilling. The oblique lines in Fig. 5 form an angle of 60 degrees with the horizontal. As in Figs. 3 and 4 mark off the line A B in divisions of ^ inch and draw with the T-square and 60-degree triangle the oblique lines through these points of division moving the pencil from A B to B C. The last line thus drawn will be A L. Now mark off distances of ^ inch on C D beginning at L. The lines may now be finished. Inking. Fig. 1 is designed to give the beginner practice in drawing lines of varying widths. The line E is first drawn. This line should be rather fine but should be clear and distinct. The line F should be a little wider than E ; the greater width being obtained by turning the adjusting screw from one-quarter to one- half a turn. The lines G, H, I, etc., are drawn ; each successive Jine having greater width. M and N should be the same and quite heavy. From N to D the lines should decrease in width. To complete the inking of Fig. 1, draw the border lines. These lines should have about the same width as those in PLATE I. In Fig. 2 the first four lines should be dotted. The dots should be uniform in length (about -^ inch) and the spaces also uniform (about -gig- inch). The next four lines are dash lines similar to those used for dimensions. These lines should be drawn with dashes about | inch long and the lines should be fine, yet distinct. The following four lines arc called dot and dash lines. The dashes are about | inch long and a dot between as shown. In the regular practice of drafting the length of the dashes depends upon the size of the drawing ^ inch to 1 inch being common. The last tour lines are similar, two dots being used between the 248 MECHANICAL DRAWING. 35 dashes. After completing the dot and dash lines, draw the border lines of the rectangle as before. In inking Fig. 3, the pencil lines are followed. Great care should be exercised in starting and stopping. The lines should begin in the border lines and the end should not run over. The lines of Fig. 4- must be drawn carefully, as there are so many intersections. The lines in this figure should be lighter than the border lines. If every line does not coincide with the points of division L, M, N, O, P, etc., some will appear farther apart than others. Fig. 5 is similar to Fig. 3, the only difference being in the angle which the oblique lines make with the horizontal. After completing the five figures draw the border lines of the plate and then letter the plate number, date and name, and the figure numbers, as in PLATE I. The plate should then be cut to the required size, 11 inches by 15 inches. PLATE III. Pencilling. The horizontal and vertical center lines and the border lines for PLATE III are laid out in the same manner as were those of PLATE II. To draw the squares for the six figures, proceed as follows : Measure off two inches on either side of the vertical center line and draw light pencil lines through these points parallel to the vertical center line. These lines will form the sides A D and B C of Figs. 2 and 5. Parallel to these lines and at a distance of I inch draw similar lines to form the sides B C of Figs. 1 and If and A D of Figs. 3 and 6. The vertical sides A D of Figs. 1 and 4 and B C of Figs. 3 and 6 are formed by drawing lines perpen- dicular to the horizontal center line at a distance of 6^ inches from the center. The horizontal sides D C of Figs. 1, 2 and 3 are drawn with the T-square A inch above the horizontal center line. To draw the top lines of these figures, draw (with the T-square) a line 4J inches above the horizontal center line. The top lines of Figs. 4, 5 and 6 are drawn inch below the horizontal center line. The squares are completed by drawing the lower lines D C, 4| inches below the horizontal center line. The figures of PLATES I and U 247 MECHANICAL DRAWING. were constructed in rectangles; the exercises of PLATE JZ/are, however, drawn in squares, having the sides 4 inches long. In drawing Fig. 1, first divide A D and A B (or D C ) into 4 equal parts. As these lines are four inches long, each length will be 1 inch. Now draw horizontal lines through E, F and G and vertical lines through L, M and N. These lines are shown dotted in Fig. 1. Connect A and B with the intersection of lines E and M, and A and D with the intersection of lines F and L. Similarly drav D J, J C, I B and I C. Also connect the points P, O, I and J forming a square. The four diamond shaped areas are formed by drawing lines from the middle points of A D, A B, B C and D C to the middle points of lines A P, A O, O B, I B etc., as shown in Fi, by Amtrjcan School of Co 855 40 MECHANICAL DRAWING. ANGLES. An angle is formed when two straight lines meet. An angle is often defined as being the difference in direction of two straight lines. The lines are called the sides and the point of meeting is called the vertex. The size of an angle depends upon the amount of divergence of the sides and is independent of the length of these lines. BIGHT ANGLE. ACUTE ANGLE. OBTUSE ANGLE. If one straight line meet another and the angles thus formed are equal they are right angles. When two lines are perpendic- ular to each other the angles formed are right angles. An acute angle is less than a right angle. An obtuse angle is greater than a right angle. SURFACES. A surface is produced by the motion of a line; it has two dimensions, length and breadth. A plane figure is a plane bounded on all sides by lines; the space included within these lines (if they are straight lines) is called & polygon or a rectilinear figure. TRIANGLES. A triangle is a figure enclosed by three straight lines. It is a polygon of three sides. The bounding lines are the sides, and the points of intersection of the sides are the vertices. The angles of a triangle are the angles formed by the sides. A right-angled triangle, often called a right triangle, is one that has a right angle. An acute-angled triangle is one that has all of its angles acute, An obtuse-angled triangle is one that has an obtuse angle. In an equilateral triangle all of the sides are equal. 350 MECHANICAL DRAWING. 11 If all of the angles of a triangle are equal, the figure is called an equiangular triangle. A triangle is called scalene, when no two of its sides are equal. In an isosceles triangle two of the sides are equal. BIGHT ANGLED TRIANGLE. ACUTE ANGLED TRIANGLE. OBTUSE ANGLED TRIANGLE. The base of a triangle is the lowest side ; however, any side may be taken as the base. In an isosceles triangle the side which is not one of the equal sides is usually considered the base. The altitude of a triangle is the perpendicular drawn from the vertex to the base. EQUILATERAL TRIANGLE. ISOSCELES TRIANGLE. SCALENE TRIANGLE. QUADRILATERALS. A quadrilateral is a plane figure bounded by four straight lines. The diagonal of a quadrilateral is a straight line joining two opposite vertices. QUADRILATERAL. PARALLELOGRAM. A trapezium is a quadrilateral, no two of whose sides are parallel. A trapezoid is a quadrilateral having two sides parallel 257 42 MECHANICAL DRAWING. The bases of a trapezoid are its parallel sides. The altitude is the perpendicular distance between the bases. A parallelogram is a quadrilateral whose opposite sides are parallel. The altitude of a parallelogram is the perpendicular distance between the bases which are the parallel sides. There are four kinds of parallelograms: RECTANGLE. A rectangle is a parallelogram, all of whose angles are right angles. The opposite sides are equal. A square is a rectangle, all of whose sides are equal. A rhombus is a parallelogram which has four equal sides; but the angles are not right angles. A rhomboid is a parallelogram whose adjacent sides are anequal ; the angles are not right angles. POLYGONS. A polygon is a plane figure bounded by straight lines. The boundary lines are called the sides and the sum of the sides is called the perimeter. Polygons are classified according to the number of sides. A triangle is a polygon of three sides. A quadrilateral is a polygon of four sides. A pentagon is a polygon of five sides. A hexagon is a polygon of six sides. A heptagon is a polygon of seven sides. An octagon is a polygon of eight sides. A decagon is a polygon of ten sides. A dodecagon is a polygon of twelve sides. An equilateral polygon is one all of whose sides are equal. An equiangular polygon is one all of whose angles are equal. A regular polygon is one all of whose angles are equal and all sf whose sides are equal. 258 MECHANICAL DRAWING. 43 CIRCLES. A circle is a plane figure bounded by a curved line, every point &f which is equally distant from a point within called the center. The curve which bounds the circle is called the circumference Any portion of the circumference is called an arc. The diameter of a circle is a straight line drawn through the center and terminating in the circumference. A radius is a straight line joining the center with the circumference. It has a length equal to one half the diameter. All radii (plural of radius) are equal and all diameters are equal since a diameter equals two radii. An arc equal to one-half the circumference is called a semi- circumference^ and an arc equal to one-quarter of the circumfer- ence is called a quadrant. A quadrant may mean the sector, arc or angle. A chord is a straight line joining the extremities of an arc. It is a line drawn across a circle that does not pass through the center. A secanjt is a straight line which intersects the circumference in two points. CIRCLE. A tangent is a straight line which touches the circumference at only one point. It does not intersect the circumference. The point at which the tangent touches the circumference is called the point of tangency or point of contact. 359 MECHANICAL DRAWING. A sector of a circle is the portion or area included between an arc and two radii drawn to the extremities of the arc. A segment of a circle is the area included between an arc and its chord. Circles are tangent when the circumferences touch at only one point and are concentric when they have the same center. CONCENTRIC CIRCLES. INSCRIBED POLYGON An inscribed angle is an angle whose .vertex lies in the cir- cumference and whose sides are chords. It is measured by one- half the intercepted arc. A central angle is an angle whose vertex is at the center of the circle and whose sides are radii. An inscribed polygon is one whose vertices lie in the circum- ference and whose sides are chords. MEASUREflENT OF ANGLES. To measure an angle describe an arc "with the center at the vertex of the angle and having any convenient radius. The por- tion of the arc included between the sides of the angle is the measure of the angle. If the arc has a constant radius the greater the divergence of the sides, the longer will be the arc. If there are several arcs drawn with the same center, the intercepted arcs will have different lengths bub they will all be the same fraction of the entire circumference. In order that the size of an angle or arc may be stated with- MECHANICAL DRAWING. 45 7J out saying that it is a certain fraction of a circumference, the cir- cumference is divided into 360 equal parts called degrees. Thus we can say that an angle contains 45 degrees, which means that it is jfg 5 o" | of a circumference. In order to obtain accurate measure- ments each degree is divided into 60 equal parts called minutes and each minute is divided into 60 equal parts called seconds. Angles and arcs are usually measured by means of an instrument called protractor which has already been explained. /eo SOLIDS. A polyedron is a solid bounded by planes. The bounding planes are called the faces and their intersections edges. The intersections of the edges are called vertices. A polygon having four faces is called a tetraedron ; one having six faces a hexaedron ; of eight faces an octaedron ; of twelve faces a dodecaedron, etc. RIGHT PRISM. TRUNCATED PRISM. A prism is a polyedron, of which two opposite faces, called bases, are equal and parallel ; the other faces, called lateral faces are parallelograms. The area of the lateral faces is called the lateral area. The altitude of a prism is the perpendicular distance between the bases. Prisms are triangular, quadrangular, etc., according to the shape of the base. A right prism js one whose lateral edges are perpendicular to the bases. 261 46 MECHANICAL DRAWING. A regular prism is a right prism having regular polygons for A parallelepiped is a prism whose bases are parallelograms. If the edges are all perpendicular to the bases it is called a right parallelepiped. A rectangular parallelepiped is a right parallelepiped whose bases are rectangles ; all the f ices are rectangles. PAKALLELOP1PKD. OCTAEDRON. A cube is a rectangular parallelepiped all of whose faces are squares. A. truncated prism is the portion of a prism included between the base and a plane not parallel to the base. PYRAMIDS. A pyramid is a polyedron one face of which is a polygon (called the base) and the other faces are triangles having a com- mon vertex. REGULAR PYRAMID. FRUSTUM OF PYRAMID. The vertices of the triangles form the vertex of the pyramid. The altitude of the pyramid is the perpendicular distance from the vertex to the base. A pyramid is called triangular, quadrangular, etc., accord- ing to the shape of .the base. A. regular pyramid is one whose base is a regular polygon MECHANICAL DRAWING. 17 and whose vertex lies in the perpendicular erected at the center of the base. A truncated pyramid is the portion of a pyramid included between the base and a plane not parallel to the base. A frustum of a pyramid is the solid included between the base and a plane parallel to the base. The altitude of a frustum of a pyramid is the perpendicular distance between the bases. CYLINDERS. A cylindrical surface is a curved surface generated by the motion of a straight line which touches a curve and continues parallel to itself. A cylinder is a solid bounded by a cylindrical surface and two parallel planes intersecting this surface. The parallel faces are called bases. CYLINDER. RIGHT CYLINDER -UIHEl) CYLINDER. The altitude of a cylinder is the perpendicular distance between the bases. A circular cylinder is a cylinder whose base is a circle. A right cylinder or a cylinder of revolution is a cylinder gen- erated by the revolution of a rectangle about one side as an axis. A prism whose base is a regular polygon may be inscribed in or circumscribed about a circular cylinder. The cylindrical area is call the lateral area. The total area is the area of the bases added to the lateral area. CONES. A conical surface is, a curved surface generated by the motion of a straight line, one point of which is fixed and the end On ends of which move in a curve. 48 MECHANICAL DRAWING. A cone is a solid bounded by a conical surface and a plane which cuts the conical surface. The plane is called the base and the curved surface the lateral area. . The vertex is the fixed point. The altitude of a cone is the perpendicular distance from the vertex ^to the base.. An element of a cone is a straight line from the vertex to the perimeter of the base. A circular cone is a cone whose base is a circle. RIGHT CIRCULAR CONE. FRUSTUM OF CONE. A right circular cone or cone of revolution is a cone whose axis is perpendicular to the base. It may be generated by the revolution of a right triangle about otoe of the perpendicular sides as an axis. A frustum of a cone is the solid included between the base and a plane parallel to the base. TANGENT PLANE. Tho altitude of a frustum of a cone is the perpendicular distance between the bases. SPHERES. A sphere is a solid bounded by a curved surface, every point of which is equally distant from a point within called the center. The radius of a sphere is a straight line drawn from the 264 MECHANICAL DRAWING. center to the surface. The diameter is a straight line drawn through the center and having its extremities in the surface. A sphere may be generated by the revolution of a semi-circle about its diameter as an axis. An inscribed polyedron is a polyedron whose vertices lie in the surface of the sphere. An circumscribed polyedron is a polyedron whose faces are tangent to a sphere. A great circle is the intersection of the spherical surface ana a plane passing through thj center of a sphere. A small circle is the intersection of the spherical surface and a plane which does not pass through the center. A sphere is tangent to a plane when the plane touches the surface in only one point. A plane perpendicular to the extremity of a radius is tangent to the sphere. CONIC SECTIONS. If a plane intersects a cone the geometrical figures thus formed are called conic sections. A plane perpendicular to the base and passing through the vertex of a right circular cone forms an isosceles triangle. If the plane is parallel to the base the intersection of the plane and conical surface will be the circum- ference of a circle. Fig. 1. Fig. 2. Fig. 3. Fig. 4. Ellipse. The ellipse is a curve formed by the intersection of a plane and a cone, the plane being oblique to the axis but not cutting the base. If a plane is passed through a cone as shown in Fig. 1 or through a cylinder as shown in Fig 2, the curve of intersection will be an ellipse. An ellipse may be denned as being a curve generated by a point moving in a plane, the sum oj the distances of the point to two fixed points being always constant. The two fixed points are called the foci and lie on the 265 50 MECHANICAL DRAWING. longest line that can be drawn in the ellipse. One of these points is called a focus. The longest line that can be drawn in an ellipse is called the major axis and the shortest line, passing through the center, is called the minor axis. The minor axis is perpendicular to the middle point of the major axis and the point of intersection is called the center An ellipse may be constructed if the major and minor axes are given or if the' foci and one axis are known. PAKABOLA. Parabola. The parabola is a curve formed by the inter- section of a cona and a plane parallel to' an element as shown in Fig. 3. The curve is not a closed curve. The branches approach parallelism. A parabola may be defined as l>eing a curve every point of which is equally distant from a line and a point. The point is called the focus and the given line the directrix. The line perpendicular to the directrix and passing through the focus is the axis. The intersection of the axis and the curve is the vertex. Hyperbola. This curve is formed by the intersection of a plane and a cone, the plane being parallel to the axis of the cone as shown in Fig. 4. Like the parabola, the curve is not a closed curve ; the branches constantly diverge. An hyperbola is defined as being a plane curve such that the difference of the distances from any point in the curve to two fixed points is etjual to a given distance. HYPERBOLA. 266 MECHANICAL DRAWING. 51 The two fixed points are the foci and the line passing through them is the transverse axis. Rectangular Hyperbola. The form of hyperbola most used in Mechanical Engineering is called the rectangular hyperbola because it is drawn with reference to rectangular co-ordinates. This curve is constructed as follows : In Fig. 5, O X and O Y are the two co-ordinates drawn at right angles to each other. These lines are also called axes or asymptotes. Assume A to be a known point on the curve. In drawing this curve M for the theoretical indicator card, this point A is the point of cut-off. Draw A C parallel to O X and A D perpendicular to O X. Now mark off any convenient points on A C such as E, F, G, and H ; and through these points draw EE', FF', GG', and HH' perpendicular to O X. Connect E, F, G, H anoLC with O. Through the points of inter- section of the oblique lines and the vertical line A D draw the horizontal lines LL', MM', NN', PP' and QQ'. The first point on the curve is the assumed point A, the second point is R, the intersection of LL' and EE'. The third is the intersection S of MM' and FF'; the fourth is the intersection T of NN' and GG'. The other points are found in the same way. In this curve the products of the co-ordinates of all points are equal. Thus LR X RE' = MS X SF'= NT X TG'. ODONTOIDAL CURVES. The outlines of the teeth of gears must be drawn accurately because the smoothness of running depends upon the shape of the teeth. The two classes of curves generally employed in drawing gear teeth are the cycloidal and involute. Cycloid. The cycloid is a curve generated by a point on the circumference of a circle which rolls on a straight line tangent to the circle. The rolling circle is called the describing or generating circle 52 MECHANICAL DRAWING. and the point, the describing or generating point. The tangent along which the circle rolls is called the director. In order that the curve may be a true cycloid the circle must roll without any slipping. 7XNGENT OR D/RECTOR Epicycloid. If the generating circle rolls upon the outride of an arc or circle, called the director circle, the curve thus gener- ated is called an epicycloid. The method of drawing this curve is the same as that for the cycloid. Hypocycloid. In case the generating circle rolls upon the inside of an arc or circle, the curve thus generated is called the hypocycloid. The circle upon which the generating circle rolls is called the director circle. If the generating circle has a diameter equal to the radius of the director circle the hypocycloid becomes a straight line. Involute. If a thread or fine wire is wound around a cylinder or circle and then unwound, the end will describe a curve called an involute. The involute may be defined as being a curve generated by a point in a tangent rolling on a circle known as the base circle. The construction of the ellipse, parabola, hyperbola and odontoidal curves will be taken up in detail with the plates. MECHANICAL DRAWING. 53 PLATE IV. Pencilling. The horizontal and vertical center lines and the border lines for PLATE IV should te laid out in the same manner as were those for PLATE I. There are to be six figures on this plate and to facilitate the laying out of the work, the fol- lowing lines should be drawn : measure off 2^ inches on botli sides of the vertical center line and through these points draw vertical lines as shown in dot and dash lines on PLATE IV. In these six spaces the six figures are to be drawn, the student placing them in the centers of the spaces so that they will present a good appearance. In locating the figures, they should be placed a little above the center so that there will be sufficient space below to number the problem. The figures of the problems should first be drawn lightly in pencil and after the entire plate is completed the lines should be inked. In pencilling, all intersections must be formed with great care as the accuracy of the results depends upon the pencilling. Keep the pencil points in good order at all times and draw lines exactly through intersections. GEOMETRICAL PROBLEMS. The following problems are of great importance to the mechanical draughtsman. The student should solve them with care ; he should not do them blindly, but should understand them so that he can apply the principles in later work. PROBLEM I. To Bisect a Given Straight Line. Draw the horizontal straight line A C about 3 inches long. With the extremity A as a center and any convenient radius (about 2 inches) describe arcs above and below the line A C. With the other extremity C as a center and with the same radius draw short arcs above and below A C intersecting the first arcs at D and E. The radius of these arcs must be greater than one-half the length of the line in order that they may intersect. Now draw the straight line D E passing through the intersections D and E. This line cuts the line A C at F which is the middle point. AF = FC 54 MECHANICAL DRAWING. Proof. Since the points D and E are. equally distant from A and C a straight line drawn through them is perpendicular to A C at its middle point F. PROBLEM 2. To Construct an Angle Equal to a Given Angle. Draw the line O C about 2 inches long and the line O A of about the same length. The angle formed by these lines may be any convenient size (about 45 degrees is suitable). This angle A O C is the given angle. Now draw F G a horizontal line about 2^ inches long and let F the left-hand extremity be the vertex of the angle to be constructed. With O as a center and any convenient radius (about 11 inches) describe the arc L M cutting both O A and OC. With F as a center and the same radius draw the indefinite arc O Q. Now set the compass so that the distance between the pencil and the needle point is equal to the chord L M. With Q as a center and a radius equal to L M draw an arc cutting the arc O Q at P. Through F and P draw the straight line F E. The angle E F G is the required angle since it is equal to A O C. Proof. Since the chords of the arcs L M and P Q are equal the arcs are equal. The angles are equal because with equal radii equal arcs are intercepted by equal angles. PROBLEM 3. To Draw Through a Given Point a Line Parallel to a Given Line. First Method. Draw the horizontal straight line A C about 3J inches long and assume the point P about 1J inches above A C. Through the point P draw an oblique line F E forming any convenient angle with A C. (Make the angle about 60 degrees). Now construct an angle equal to P F C having the vertex at P and one side the line E P. (See problem 2). This may be done as follows : With F as a center and any con- venient radius, describe the arc L M. With the same radius draw the indefinite arc N O using P as the center. With N as a center and a radius equal to the chord L M, draw an arc cutting the arc N O at O. Through the points P and O draw a straight line which will be parallel to A C. r" N r G G " 1 U '' O U MECHANICAL DRAWING. 55 Proof. If two straight lines are cut by a third making the corresponding angles equal, the lines are parallel. PROBLEM 4. To Draw Through a Given Point a Line Parallel to a Given Line. Second Method. Draw the straight line A C about 3* inches long and assume the point P about 1 inches above A C. With ? as a center and any convenient radius (about 2 inches) draw the indefinite arc E D cutting the line A C. Now with the same radius and with D as a center, draw an arc P Q. Set the com- pass so that the distance between the needle point and the pencil is equal to the chord P Q. With D as a center and a radius equal to P Q,' describe an arc cutting the arc E D at H. A line drawn through P and H will be parallel to A C. Proof. Draw the line Q H. Since the arcs P Q and H D are equal and have the same radii, the angles P H Q and H Q D are equal. Two lines are parallel if the alternate interior angles are equal. PROBLEM 5. To Draw a Perpendicular to a Line from a Point in the Line. First Method. When the point is near the middle of the line. Draw the horizontal line A C about 3^ inches long and assume the point P near the middle of the line. With P as a center and any convenient radius (about 1 J inches) draw two arcs cutting the line A C at E and F. Now with E and F as centers and any convenient radius (about 2J inches) describe arcs inter- secting at O. The line O P will be"perpendicular to A C at P. Proof. The points P and O are equally distant from E and F. Hence a line drawn through them is perpendicular to the middle point of E F which is P. PROBLEM 6. To Draw a Perpendicular to a Line from a Point in the Line. Second Method. When the point is near the end of the line. Draw the line A C about 3 inches long. Assume the given point P to be about | inch from the end A. With any point D as a center and a radius equal to D P, describe an arc, cutting A C at E. Through E and D draw the diameter E O. A line from O to P is perpendicular to A C at P. 273 50 MECHANICAL DRAWING. Proof. The angle O P E is inscribed in a semi-circle ; hence it is a right angle, and the sides O P and P E are perpendicular to each other. After completing these figures draw pencil lines for the lettering. The words PL A TE TV" and the date and name should be placed in the border, as in preceding plates. To letter the words u Problem 1," " Problem 2," etc,, draw horizontal lines | inch above the horizontal center line and the lower border line. Draw another line ^ inch above, to limit the height of the P, b and I. Draw a third line |- inch above the lower line as a guide line for the tops of the small letters. Inking. In inking PLATE JVthe figures should be inked first. The line A C of Problem 1 should be a full line as it is the given line ; the arcs and line I) E, being construction lines should be dotted. In Problem 2, the sides of the angles should be full lines. Make the chord L M and the arcs dotted, since as before, they are construction lines. In Problem 3, the line A C is the given line and P O is the line drawn parallel to it. As E F and the arcs do not form a part of the problem but are merely construction lines, drawn as an aid in locating P O, they should be dotted. In Problems 4, 5 and 6, the assumed lines and those found by means of the construction lines should be full lines. The arcs and construction lines should ^ dotted. In Problem 6, the entire circumference need not be Inked, only that part is necessary that is used in the problem. The inked arc should however be of sufficient length to pass through the points O, P and E. After inking the figures, the border lines should be inked with a heavy line as before. Also, the words "PLATE IV" and the date and the student's name. Under each problem the words "Problem 1," "Problem 2," etc., should be inked; lower case let- ters being used as shown. PLATE V. Pencilling. In laying out the border lines and centre lines follow the directions given for PLATE IV. The dot and dash lines should be drawn in the same manner as there are to be six problems on this plate. 274 MECHANICAL DRAWING. 57 PROBLEM 7. To Draw a Perpendicular to a Line from a Point without the Line. Draw the horizontal straight line A C about 3| inches long. Assume the point P about li inches above the line. With P as a center and any convenient radius (about 2 inches) describe an arc cutting A C at E and F. The radius of this arc must always be such that it will cut A C in two points; the nearer the points E and F are to A and C, the greater will be the accuracy of the work. Now with E and F as centers and any convenient radius (about 2^ inches) draw the arcs intersecting below A C at T. A line through the points P and T will be perpendicular to A C. In case there is not room below A C to draw the arcs, they may be drawn intersecting above the line as shown at N. When- ever convenient, draw the arcs below A C for greater accuracy. Proof. Since P and T are equally distant from E and F, the line P T is perpendicular to A C. PROBLEM 8. To Bisect a Given Angle. First Method. When the sides intersect. Draw the lines O C and O A forming any angle (from 45 to 60 degrees). These lines should be about 3 inches long. With O as a center and any convenient radius (about 2 inches) draw an arc intersecting the sides of the angle at E and F. With E and F as centers and a radius of 1| or 1| inches, describe short arcs intersecting at I. A line O D, drawn through the points O and I, bisects the angle. In solving this problem the arc E F should not be too near the vertex if accuracy is desired. Proof. The central angles A O D and DOC are equal because the arc E F is bisected by the line O D. The point I is equally distant from E and F. PROBLEM 9. To Bisect a Given Angle, Second Method. When the lines do not intersect. Draw the lines A C and E F about 4 inches long and in 'the positions as shown on PLATE V. Draw A' C' and E' F' parallel to A C and E F and at such equal distances from them that they will intersect at O. Now bisect the angle C' O F' by 58 MECHANICAL DRAWING. the method of Problem 8. Draw the arc G H and with G and H as centers draw the arcs intersecting at R. The line O R bisects the angle. Proof. Since A' C' is parallel to A C and E' F' parallel to E F, the angle C' O F' is equal to the angle formed by the lines A C and E F. Hence as O R bisects angle C' O F' it also bisects the angle formed by the lines A C and E F. PROBLEM 10. To Divide a Given Line into any Number of Equal Parts. Let A C, about 3|^ inches long, be the given line. Let us divide it into 7 equal parts. Draw the line A J at least 4 inches long, forming any convenient angle with A C. On A J lay off, by means of the dividers or scale, points D, E, F, G, etc., each ^ inch apart. If dividers are used the spaces need not be exactly ^ inch. Draw the line J C and through the points D, E, F, G, etc., draw lines parallel to J C. These parallels will divide the line A C into 7 equal parts. Proof. If a series of parallel lines, cutting two straight lines, intercept equal distances on one of these lines, they also intercept equal distances on the other. PROBLEM 11. To Construct a Triangle having given the Three Sides. Draw the three sides as follows : A C, 2| inches long. E F, llf inches long. M N, 2^ inches long. Draw R S equal in length to A C. With R as a center and a radius equal to E F describe an arc. With S as a center and a radius equal to M N draw an arc cutting the arc previously drawn, at T. Connect T with R and S to form the triangle. PROBLEM 12. To Construct a Triangle having given One Side and the Two Adjacent Angles. Draw the line M N 3J inches long and draw two angles A O D and E F G. Make the angle A O D about 30 degrees and E F G about 60 degrees. Draw R S equal in length to M N and at R construct an 276 r G < LJ \ (L u \ ' x''' \ \ \ i V fc /' / X \ / / u / / < \ " L J_ MECHANICAL DRAWING. 59 angle equal to A O D. At S construct an angle equal to E F G by the method used in Problem 2. PLATE V shows the neies- sary arcs. Produce the sides of the angles thus constructed they meet at T. The triangle R TS will be the required triangle. After drawing these six figures in pencil, draw the pencil lines for the lettering. The lines for the words -PLATE V," date and name, should be pencilled as explained on page <>0 The words "Problem 7," "Problem 8," etc., are lettered as for PLATE IV. Inking. In inking PLATE V, the same principles should be followed as stated with PLATE IV. The student should apply these principles and not make certain lines dotted just because they are shown dotted in PLATE V. After inking the figures, the border lines should be inked and the lettering inked as already explained in connection with previous plates. PLATE VI. Pencilling. Lay out this plate in the same manner as the two preceding plates. PROBLEM 13. To describe an Arc or Circumference through Three Given Points not in the same straight line. Locate the three points A, B and C. Let the distance between A and B be about 2 inches and the distance between A and C be about 2| inches. Connect A and B and A and C. Erect perpendiculars to the middle points of \ B and A C. This may be done as explained with Problem 1. With A and B as centers and a radius of about 1| inches, describe the arcs inter- secting at I and J. With A and C as centers and with a radius of about 1| inches draw the arcs, intersecting at E and F. Now draw light pencil lines connecting the intersections I and J and E and F. These lines will intersect at O. With O as a center and a radius equal to the distance O A, describe the circumference'passing through A, B and C. Proof. The point O is equally distant from A, B and C, since it lies in the perpendiculars to the middle points of A B and 279 60 MECHANICAL DRAWING. A C. Hence the circumference will pass through A, B and C. PROBLEM 14. To inscribe a Circle in a given Triangle. Draw the triangle L M N of any convenient size. M N may be made 3^ inches, L M, 2| inches, and L N, 3J inches. Bisect the angles M L N and L M N. The bisectors M I and L J may be drawn by the method used in Problem 8. Describe the arcs A C and E F, having centers at L and M respectively. The arcs intersecting at I and J are drawn as already explained. The bisectors of the angles intersect at O, which is the center of the inscribed circle. The radius of the circle is equal to the perpen- dicular distance from O to one of the sides. Proof. The point of intersection of the bisectors of the angles of a triangle is equally distant from the sides. PROBLEM 15. To inscribe a Regular Pentagon in a given Circle. With O as a center and a radius of about 1 inches, describe the given circle. With the T square and triangles draw the cen- ter lines A C and E F. These lines should be perpendicular to each other and pass through O. Bisect one of the radii, such as O C, and with this point H as a center and a radius H E, describe the arc E P. This arc cuts the diameter A C at P. With E as a center and a radius E P, draw arcs cutting the circumference at L and Q. With the same radius and a center at L, draw the arc, cutting the circumference at M. To find the point N, use either M or Q as a center and the distance E P as a radius. The pentagon is completed by drawing the chords E L, L M, M N, N Q and Q E. PROBLEM 16. To inscribe a Regular Hexagon in a given Circle. With O as a center and a radius of 1| inches draw the given circle. With the T square draw the diameter A D. With D as a center, and a radius equal to O D, describe arcs cutting the circumference at C and E. Now with C and E as centers and the same radius, draw the arcs, cutting the circumference at B and F. Draw the hexagon by joining the points thus formed. To inscribe a regular hexagon in a circle mark off chords O O equal in length to the radius. r MECHANICAL DRAWING. 01 To inscribe an equilateral triangle in a circle the same method may be used. The triangle is formed by joining the opposite vertices of the hexagon. Proof. The triangle O C D is an equilateral triangle by construction. Then the angle C O D is one-third of two right angles and one-sixth of four right angles. Hence arc C D is one- sixth of the circumference and the chord is a side of a regular hexagon. PROBLEM 17. To draw a line Tangent to a Circle at a given point on the circumference. With O as a center and a radius of about 1^ inches draw the given circle. Assume some point P on the circumference Join the point P with the center O and through P draw a line F P perpendicular to P O. This may be done in any one of several methods. Since P is the extremity of O P the method given in Problem 6 of PLATE IV, may be used. Produce P O to Q. With any center C, and a radius C P draw an arc or circumference passing through P. Draw E F a diameter of the circle whose center is C and through F and P draw the tangent. Proof. A line perpendicular to a radius at its extremity is tangent to the circle. PROBLEM 18. To draw a line Tangent to a Circle from a point outside the circle. With O as a center and a radius of about 1 inch draw the given circle. Assume P some point outside of the circle about 2J inches from the center of the circle. Draw a straight line passing through P and O. Bisect P O and with the middle point F as a center describe the circle passing through P and O. Draw a line through P and the intersection of the two circum- ferences C. The line P C is tangent to the given circle. Simi- larly P E is tangent to the circle. Proof. The angle P C O is inscribed in a semi-circle and hence is a right angle. Since P C O is a right angle P C is per- pendicular to C O. The perpendicular to a radius at its extremity is tangent to the circumference. Inking. In inking PLATE VI the same method should be 62 MECHANICAL DRAWING. followed as in previous plates. The name and address should be lettered in inclined Gothic capitals as before. PLATE VII. Pencilling. PLATE VII should be laid out in the same manner as previous plates. Six problems on the ellipse, spiral, parabola and hyperbola are to be constructed in the six spaces. PROBLEM 19. To draw an Ellipse when the Axes are given. Draw the lines L M and C D about 3^ and 2^ inches long respectively. Let C D be perpendicular to M N at its middle point P. Make C P = P D. These two lines are the axes. With C as a center and a radius equal to one-half the major axis or equal to L P, draw the arc, cutting the major axis at E and F. These two points are the foci. Now mark off any convenient distances on P M, such as A, B and G. With E as a center and a radius equal to L A, draw arcs above and below L M. With F as a center, and a radius equal to A M describe short arcs cutting those already drawn as shown at N. With E as a center and a radius equal to L B draw arcs above and below L M as before. With F as a center and ?- radius equal to B M, draw arcs intersecting those already drawn as shown at O. The point P and others are found by repeating the process. The student is advised to find at least 12 points on the curve 6 above and 6 below L M. These 12 points with L, C, M and D will enable the student to draw the curve. After locating these points, a free hand curve passing through them should be sketched. PROBLEM 20. To draw an Ellipse when the two Axes are given. Second Method. Draw the two axes A B and P Q in the same manner as for Problem 19. With O as a center and a radius equal to one-half the major axis, describe the circumference A C D E F B. Similarly with the same center and a radius equal to one-half the minor axis, describe a circle. Draw any radii such as O C, O D, O E, O F, etc., cutting both circumferences. These radii may be drawn with the 60 and 45 degree triangles. At the L J MECHANICAL DRAWING. c.:; pom* of mtersection of the radii with the Lug. circle C D E and **"*! lines and from the intersection of the radii with circle C', D', E', and F', draw lor^ntal lines intersect eS As in Problem 19, a free hand curve should be sketched pass- ing through these points. About five points in each quadrant will be sufficient. PROBLEM 21. To draw an EI ,ip se by means of . 1 rammel. As in the two preceding problems, draw the major and minor axes, U V and X Y. Take a slip of paper having a straight edge and mark off C B equal to one-half the major axis, and D B one-half the minor axis. Place the slip of paper in various positions keeping the point D on the major axis and the point C on the minor axis. If this is done the point B will mark various points on the curve. Find as many points as necessary and sketeh the curve. PROBLEM 22. To draw a Spiral of one turn in a circle. Draw a circle with the center at O and a radius of 1 ' inches. Mark off on the radius O A, distances of one-eighth inch. As 3 A is 1J inches long there will be 12 of these distances. Draw circles through these points. Now draw radii O B, O C, O D etc. each 30 degrees apart (use the 30 degree triangle). This will divide the circle into 12 equal parts. The curve starts at the center O. The next point is the intersection of the line O B and the first circle. The third point is the mtersection of O C and the second circle. The fourth point is the intersection of O D and the third circle. Other points are found in the same way. Sketch in pencil the curve passing through these points. PROBLEM 23. To draw a Parabola when the Abscissa and Ordinate are given. Draw the straight line A B about three inches long. This line is the axis or as it is sometimes called the abscissa. At A and B draw lines perpendicular to A B. Also with the T square draw E C and F D, 1| inches above and below A B. Let A be 287 64 MECHANICAL DRAWING. the vertex of the parabola. Divide A E into any number of equal parts and divide E C into the same number of equal parts. Through the points of division, R, S, T, U and V, draw horizontal lines and connect L, M, N, O and P, with A. The intersections of the horizontal lines with the oblique lines are points on the curve. For instance, the intersection of A L and the line V is one point and the intersection of A M and the line U is another. The lower part of the curve A D is drawn in the same manner. PROBLEM 24. To draw a Hyperbola when the abscissa E X, the ordinate A E and the diameter X Y are given. Draw E F about 3 inches long and mark the point X, 1 inch from E and the point Y, 1 inch from X With the triangle and T square, draw the rectangles A B D C and O P Q R such that A B is 1 inch in length and A C, 3 inches in length. Divide A E into any number of equal parts and A B into the same num- ber of equal parts. Draw L X, M X and N X ; also connect T, U and V with Y. The first point on the curve is the intersection A ; the next is the intersection of T Y and L X ; the third the intersection of U Y and M X. The remaining points are found in the same manner. The curve X C and the right-hand curve P Y Q are found by repeating the process. Inking. In inking the figures on this plate, use the French or irregular curve and make full lines for the curves and their axes. The construction lines should be dotted. Ink in all the construction lines used in finding one-half of a curve, and in Problems 19, 20, 23 and 24 leave all construction lines in pencil except those inked. In Problems 21 and 22 erase all construction lines not inked. The trammel used in Problem 21 may be drawn in the position as shown, or it may be drawn outside of the ellipse in any convenient place. The same lettering should be done on this plate as on previous plates. PLATE VIII. Pencilling. In laying out Plate VIII, draw the border lines and horizontal and vertical center lines as in previous plates, to divide the plate into four spaces for the four problems. MECHANICAL DRAWING. a Cycloid when the diameter zontal lines. Now with the dividers Lt'so' thn between the points is equal to the chord of the arc C I) nruk off- ' p on the line A B -- " CO' T P er P e dicu1 ^ to the center line This center line is drawn through the point O' with the T square and is the line of centers of the generating circle as it rolls along the line A B. Now with the intersections Q, R S I, etc of these verticals with the center line as centers describe arcs of circles as shown. The points on the curve are the inter- sections of these arcs and the horizontal lines drawn through the points C, D, E, F, etc. Thus the intersection of the arc whose ter ,s Q and the horizontal line through C is a point I on the curve. Similarly, the intersection of the arc whose center is K and the horizontal line through D is another point J on the curve The remaining points, as well as those on the right-hand side are found in the same manner. To obtain great accuracy in 'this eurve, the circle should be divided into a large number of equal parts, because the greater the number of divisions the less the error due to the difference in length of a chord and its arc. PROBLEM 26. To construct an Epicycloid when the di- ameter of the generating circle and the diameter of the director circle are given. The epicycloid and hypocycloid may be drawn in the same manner as the cycloid if arcs of circles are used in place of the horizontal lines. With O as a center and a radius of f inch describe a circle. Draw the diameter E F of this circle and pro- duce E F to G such that the line F G is 2f inches long. With G as a center and a radius of 2f inches describe the arc A B of the director circle. With the same center G, draw the arc P Q which will be the path of the center of the generating circle as it rolls along the arc A fc, Now divide the generating circle into 66 MECHANICAL DRAWING. any number of equal parts (twelve for instance) and through the points of division H, I, L, M, and N, draw arcs having G as a center. With the dividers set so that the distance between the points is equal to the chord H I, mark off distances on the director circle A F B. Through these points of division R, S, T, U, etc., draw radii intersecting the arc P Q in the points R', S', T', etc., and with these points as centers describe arcs of circles as in Problem 25. The intersections of these arcs with the arcs already drawn through the points H, I, L, M, etc., are points on the curve. Thus the intersection of the circle whose center is R' with the arc drawn through the point H is a point upon the curve. Also the arc whose center is S' with the arc drawn through the point I is another point on the curve. The remaining points are found by repeating this process. PROBLEM 27. To draw an Hypocycloid when the diam- eter of the generating circle and the radius of the director circle are given. With O as a center and a radius of 4 inches describe the arc E F, which is the arc of the director circle. Now with the same center and a radius of 3^ inches, describe the arc A B, which is the line of centers of the generating circle as it rolls on the director circle. With O' as a center and a radius of | inch describe the generating circle. As before, divide the generating circle into any number of equal parts (12 for instance) and with these points of division L, M, N, O, etc., draw arcs having O as a center. Upon the arc E F, lay off distances Q R, R S, S T, etc., equal to the chord Q L. Draw radii from the points R, S, T, etc., to the center of the director circle O and describe arcs of circles having a radius equal to the radius of the generating circle, using the points G, I, J, etc., as centers. As in Problem 26, the inter- sections of the arcs are the points on the curve. By repeating this process, the right-hand portion of the curve may be drawn. PROBLEM 28. To draw the Involute of a circle whn the diameter of the base circle is known. With point O as a center and a radius of 1 inch, describe the base circle. Now divide the circle into any number of equal parts 16 for instance) and connect the points of division with the cen- MECHANICAL DRAWING. 67 ter of the circle by drawing the radii O C, O D, O E, O F, etc., to O B. At the point D, draw a light pencil line perpendicular to the radius O D. This line will be tangent to the circle. Similarly at the points E, F, G, H, etc., draw tangents to the circle. Now set the dividers so that the distance between the points will be equal to the chord of the arc C D, and measure this distance from D akrag the tangent. Beginning with the point E, measure on the tangent a distance equal to two of these chords, from the point F measure on the tangent three divisions, and from the point G measure a distance equal to four divisions on the tangent G P. Similarly, measure distances on the remaining tangents, each time adding the length of the chord. This will give the points Q, R, S and T. Now sketch a light pencil line through the points L, M, N, P, etc., to T. This curve will be the involute of the circle. Inking. The same rules are to be observed in inking PLATE VIII as were followed in the previous plates, that is, the curves should be inked in a full line, using the French or irregular curve. All arcs and lines used in locating the points on one-half of the curve should be inked in dotted lines. The arcs and lines used in locating the points of the other half of the curve may be left in pencil in Problems 25 and 26. In Problem 28, all construction lines should be inked. After completing the problems the same lettering should be done on this plate as on previous plates. CIRCULAR BLUE PRINTING MACHINE, GENERAL ELECTRIC CO. MECHANICAL DRAWING. PAKT III. PROJECTIONS. ORTHOGRAPHIC PROJECTION. Orthographic Projection is the art of representing objects of three dimensions by views on two planes at right angles to each other, in such a way that the forms and positions may be completely determined. The two planes are called planes of projection or co-ordinate planes, one being vertical and the other horizontal, as shown in Fig. 1. These planes are sometimes designated V and H respectively. The intersection of V and H is known as the ground line G L. The view or projection of the figure on the plane gives the same appearance to the eye placed in a certain position that the object itself does. This position of the eye is at an infinite dist- ance from the plane so that the rays from it to points of a limited object are all perpendicular to the plane. Evidently then the view of a point of the object is on the plane and in the ray through the point Fig. 1. and the eye or where this perpendicular to the plane pierces it. Let a, Fig. 1, be a point in space, draw a perpendicular from a to V. Where this line strikes the vertical plane, the projection of a is found, namely at v . Then drop a perpendicular from a to the horizontal plane striking it at h , which is the horizontal projection of the point. Drop a perpendicular from # v to H; this will intersect G L at o and be parallel and equal to the line a h . In the same way draw a perpendicular from h to V, this also will intersect G L at o and will be parallel and equal to a a v . In other words, the perpendicular to G L from the projection of a point on either plane equals the distance of the point from the other plane. B in Fig. 1, shows a line in space. B v is its V projection, and B h Copyright, 1908, by American School of Correspondence. MECHANICAL DRAWING its H projection, these being determined by finding views of points at its ends and connecting the points. Instead of horizontal projection and vertical projection, the terms plan and elevation are commonly used. Suppose a cube, one inch on a side, to be placed as in Fig. 2, with the top face horizontal and the front face parallel to the vertical plane. Then the plan will be a one-inch square, and the elevation also 'a one-inch square. In general the plan is a repre- sentation of the top of the object, and the elevation a view of the front. The plan then is a top view, and the elevation a front view. V r 5 e v |H . M Fig. 16. tically up to the heights of the same points in the first elevation. All the other points are found in the same way as point No. 1. Fig. 17. Three positions of a rectangular prism are shown in Fig. 17. In the first view, the prism stands on its hase. its axis therefore 80 MECHANICAL DRAWING. is parallel to the vertical 'plane. In the second position, the. axis ie still parallel to V and one corner of the base is on the horizontal plane. The prism has been turned as if on the line I* l r as an axis, so that the inclination of all the faces of the prism to the vertical plane remains the same as before. That is, if in the first figure the side A B C D makes an angle of 30 with the vertical, the same side in the second position still makes 30 with the ver- . 16. tical plane. Hence the elevation of No. 2 is the same shape and size us in the first case. The plan is found by projecting the corner? down from the elevation to meet horizontal lines projected across from the corresponding points in the first plan. The third posi- tion shows the prism with all its faces and edges making the same angles with the horizontal as in the second position, but with the plan at a different angle with the ground line. The plan then is the same shape and size as in No. 2, and the elevation is found by projecting up to the same heights as shown in the proceeding elevation. This principle may be applied to any solid, whether bounded by plane surfaces or curved. This principle as far as it relates to heights, is the same that was used for profile views. An end view is sometimes necessary before the plan or elevation of an object can be drawn. Suppose that in Fig. 18 we wish to draw the plan and elevation of a tri- angular prism 3" long, the end of which is an equilateral triangle MECHANICAL DRAWING. si U" on each side. The prism is lying on one of its three faces on H, and inclined toward the vertical plane at an angle of 30. We are able to draw the plan at once, because the width will be 1| inches, 'and the top edge will be projected half way between the other two. The length of the prism will also be shown. Before we can draw the elevation, we must find the height of the top edge. This height, however, must be equal to the altitude of the triangle forming the end of Fig. 19. the prism. All that is necessary, then, is to construct an equilat- eral triangle li" on each side, and measure its altitude. A very convenient way to do this is shown in the figure by laying one end of the prism down on H. A similar construction is shown in Fig. 19, but with one face of the prism on V instead of on H. In all the work thus far the plan has been drawn below and the elevation above. This order is sometimes inverted and the plan put above the elevation, but the plan still remains a top view no matter where placed, so that after some practice.it makes but little difference to the draughtsman which method is employed. SHADE LINES. It is often the case in machine drawing that certain lines or edges are made heavier than others. These heavy lines are called shade lines, and are used to improve the appearance *of the draw- ing, and also to make clearer in some cases the shape of the object. The shade lines are not put on at random, but according to some system. Several systems are in use, but only that one which seems most consistent will be described. The shade lines are lines or edges separating light faces from dark ones, assuming the light always to come in a direction parallel to the dotted diagonal of the cube shown in Fig. 20. The direction of the light, then, may be represented on H by a line at 45 running SI' MECHANICAL DRAWING. backward to the right and on V by a 45 line sloping downward and to the right. Considering the cube in Fig. 20, if the light comes in the direction indicated, it is evident that the front, left- hand side and top will be lightj and the bottom, back and right- hand side dark. On the plan, then, the shade lines will be the back edge 1 2 and the right-hand edge 2 3, because these edges are between light faces and dark ones. On the elevation, since the front is light, and the right-hand side and bottom dark, the edges 3 7 and 8 7 are shaded. As the direction of the light is represented on the plan by 45 lines and on the elevation also by 45 lines, 1 2 \ \ \ \ \ \ \ \ 8 1 5 " 5 / / X ^ X / X 4 S 3 8 /I / Fig. 20. we may use the 45 triangle with the T-square to determine the light and dark surfaces, and hence the shade lines. If the object stands on the horizontal plane, the 45 triangle is used on the plan, as shown in Fig. 21, but if the length is perpen- dicular to the vertical plane, the 45 triangle is used on the eleva- tion, as shown in Fig. 22. This is another way of saying that the 45 triangle is used on that projection of the object which shows the end. By applying the triangle in this way we determine the light and dark surfaces, and then put the shade lines between them. Dotted lines, however, are never shaded, so if a line which is between a light and a dark surface is invisible it is not MECHANICAL DRAWING. shaded. In Fig. 21 the plan shows the end of the solid, hence the 45 triangle is used in the direction indicated by the arrows. This shows that the light strikes the left-hand face, but not the back or the right-hand. The top is known to be light with- Fig. 21. Fig. 22. out the triangle, as the light comes downward, so the shade edges on the plan are the back and right-hand. On the elevation two faces of the prism are visible ; one is light, the other dark, hence the edge between is shaded. The left-hand edge, being between a light face and a dark one is a shade line. The right-hand face is dark, the top of the prism is light, hence the upper edge of this face is a shade line. The right-hand edge is not shaded, because by referring to the plan, it is seen to be between two dark surfaces. In shading a cylinder or a cone the same rule is fol lowed, the only difference being that as the surface is curved, the- light is tangent, so an element instead of an edge marks the separation of the dark from the light, and is not shaded. The elements of a cylinder or cone should never be shaded, but the bases may. In Fig. 23, Nos. 3 and 4, the student should carefully notice the difference between the shading of the cone and cylinder. 84 MECHANICAL DRAWING. If in No. 4 the cone were inverted, the opposite half of the base would be shaded, for then the base would be light, whereas it is now dark. In Nos. 7 and 8 the shade lines of a cylinder and a circular hole are contrasted. In No. 7 it is clear that the light would strike inside on the further side of the hole, commencing half way where the 45 lines 1234 678 Fig. 23. are tangent. The other half of the inner surface would be dark, hence the position of the shade line. The shade line then enables us to tell at a glance whether a circle represents a hub or boss, or depression or hole. Fig. 24 represents plan, elevation and profile view of a square prism. Here as before, the view showing the end is the one used to determine the light and dark surfaces, and then the shade lines put in accordingly. MECHANICAL DRAWING. ^ In putting on the shade lines, the extra width of line is put inside the figure, not outside. In shading circles, the shade line is made of varying width, as shown in the figures. The method of obtaining this effect by the compass is to keep the same radius, but to change the center slightly in a direction parallel to the rays of light, as shown at A and B in No. 2 of Fig. 24. No. 2. Fig. 24. INTERSECTION AND DEVELOPHENT. If one surface meets another at some angle, an intersection is produced. Either surface may be plane, or curved. If both are plane, the intersection is a straight line ; if one is curved, the intersection is a curve, except in a few special cases ; and if both are curved, the intersection is usually curved. In the latter case, the entire curve does not always lie in the same planes. If all points of any curve lie in the same plane, it is called a plane curve. A plane intersecting a curved surface must ahvays give either a plane curve or a straight line. In Fig. 25 a square pyramid is cut by a plane A parallel to the horizontal. This plane cuts from the pyramid a four-sided figure, the four corners of which will be the points where A cuts the four slanting edges of the solid. The plane intersects edge o I at point 4^ in elevation. This point must be found in plan vertically below on MECHANICAL DRAWING. the horizontal projection of line o b, that is, at point 4^. Edge e is directly in front of o 5, so is shown in elevation as the same line, and plane A intersects o e at point 1" in elevation, found in plan at 1A Points 3 and 2 are obtained in the same way. The intersection is shown in plan as the square 1234, which is also its true size as it is parallel to the horizontal plane. In a similar way the sections are found in Figs. 26 and 27. It will be seen that in these three cases where the planes are parallel to the bases, the sections are of the same shape as the bases, and have their sides parallel to the edges of the bases. It is an invariable rule that when such a solid is cut by a plane parallel to its base, the section is a figure of the same shape as the base. If then in Fig. 28 a right cone is intersected by a plane parallel to the base the section must be a circle, the center of which in plan coincides with the apex.. The radius must equal o d. In Figs. ii9 and 30 the cutting plane is not parallel to the base, hence the intersection will not be of the same shape as the base. The sections are found, however, in exactly the same manner as in the previous figures, by projecting the points where the plane intersects the edges in elevation on to the other view of the same line. INTERSECTION OF PLANES WITH CONES OR CYLINDERS. Sections cut by a plane from a cone have already been de- fined as conic sections. These sections may be either of the fol- lowing: two straight lines, circle, ellipse, parabola, hyperbola. All except the parabola and hyperbola may also be cut from a cylinder. Methods have previously been given tor constructing the Fig. 25. MECHANICAL DRAWING. 87 Fig. 26. Fig. 27. ' \\ Fig. 28. Fig. 29. Fig. 30. ss MECHANICAL DRAWING. ellipse, parabola and hyperbola without projections; it will now be shown that they may be obtained as actual intersections. In Fig. 31 the plane cuts the cone obliquely. To find points on the curve in plan take a series of horizontal planes Fig. 31 x y z etc., between points d is imagined moved over to the position e? d\ parallel to c d*>. The lines 1 1, 2 2, 3 3 on the plan show the true width of the ellipse, as these lines are parallel to H, but are projected closer together than their actual distances. In elevation these lines are shown as the points 1, 2, 3. at their true distance apart. Hence if the ellipse is revolved aro'inu. its axis c f? c , the distances 1 1, 2 2, 3 3 will appear perpendicular to ctZ", and the true size of the figure be shown. This construction is made on the left, where 1' 1', 2' 2' and 3' 3' are equal in length to 11, 2 2, 3 3 on the plan. In Fig. 32 a plane cuts a cylinder obliquely. This is a simpler case, as the horizontal projection of .the curve coincides with the base of the cylinder. To obtain the true size of the section, which is an ellipse, any number of points are assumed on the plan and projected up on the cutting plane, at 1, 2, 3, etc. 90 MECHANICAL DRAWING. The lines drawn through these points perpendicular to 1 7 are made equal in length to the corresponding distances 2' 2', 3' 3' etc., on the plan, because 2' 2' is the true width of curve at 2. If a cone is intersected by a plane which is parallel to only one of the elements, as in Fig. 33, the lesulting curve is the parabola, the construc- tion of which is exactly simi- lar to that for the ellipse as given in Fig. 31. If the intersecting plane is parallel to more than one element, or is parallel to the axis of the cone, a hyperbola is produced. In Fig. 34, the vertical plane A is parallel to the axis of the cone. In this instance the curve when found will appear in its true size, as plane A is parallel to the vertical. Observe that the highest point of the curve is found by drawing the circle X on the plan tangent to the given plane. One of the points where this circle crosses the diameter is projected up to the contour element of the cone, and the horizontal plane X drawn. Intermediate planes Y, Z, etc., are chosen, and corresponding circles drawn in plan. The points where these circles are crossed by the plane A are points on the curve, and these points are projected up to the elevation on the planes Y, Z, etc. DEVELOPflENTS. The development of a surface is the true size and shape ot the surface extended or spread out on a plane. If the surface to be developed is of such a character that it may be flattened out MECHANICAL DRAWING. 91 without tearing or folding, we obtain an exact development, as in case of a cone or cylinder, prism or pyramid. If this cannot be done, as with the sphere, the development is only approximate. In order to find the development of the rectangular prism in Fig 35, the back face, 1 2 7 6, is supposed to be placed in contact Fig. 33. with some plane, then the prism turned on the edge 2 7 until the side 2 3 8 7 is in contact with the same plane, then this continued until all four faces have been placed on the same plane. The rectangles 1 4 3 2 and 6 7 8 5 are for the top and bottom respec- tively. The development then is the exact size and shape of a covering for the prism. If a rectangular hole is cut through the prism, the openings in the front and back faces will be shown in the development in the centers of the two broad faces. The development of a right prism, then, consists of as many 92 MECHANICAL DRAWING. rectangles joined together as the prism has sides, these rectangles being the exact size of the faces of the prism, and in addition two polygons the exact size of the bases. It will be found helpful in developing a solid to number or letter all of the corners on the projections, then designate each face when developed in the same way as in the figure. If a cone be placed on its side on a plane surface, one element will rest on the surface. If now the cone be rolled on the plane, the vertex remaining stationary, until the same ele- ment is in contact again, tha space rolled over will represent the development of the convex surface of the cone. A, Fig. 30, is a cone cut by a Fig. 34. plane parallel to the base. In B, let the vertex of the cone be placed at V, and one element of the cone coincide with V A I. The length of this element is taken from the elevation A, of either contour element. All of the elements of the cone are of the same length, so when the cone is rolled each point of the base as it touches the plane will be at the same distance from the vertex. From this it follows that the development of the base will be the arc of a circle of radius equal to the length of an element. To find the length of this arc which is equal to the distance around the base, divide the plan of the circumference of the base into any number of equal parts, & twelve, then MECHANICAL DRAWING. 93 with the length of one of these parts as radius, lay off twelve spaces, 1....13, join 1 and 13 with V, and the sector is the development of the cone from vertex to base. To represent on the development Fig. 35. the circle cut by the section plane, take as radius the length of the element from the vertex to. D, and with V as center describe Fig. 36. an arc. The development of the frustum of the cone will be the portion of the circular ring. This of course does not include the '.'1 MECHANICAL DRAWING. development of the bases, which would he simply two circles the same sizes as shown in plan. A and B, Fig. 37, represent the plan and elevation of a regular triangular pyramid and its development. If face C is placed on the plane its true size will be shown at C in the devel- opment. The true length of the base of triangle C is shown in the plan. The slanting edges, however, not being parallel to the vertical, are not shown in elevation in their true length. It be- comes necessary then, to find the true length of one of these edges as shown in Fig. 6, after which the triangle may be irawn in its full size at C in the development. As the pyramid is regular, three equal triangles as shown developed at C, D and E, together with the base F, constitute the development. If a right circular cylinder is to be developed, or rolled upon a plane, the elements, being parallel, will appear as parallel lines, Pig. 87. and the base, being perpendicular to the elements, will develop as a straight line perpendicular to the elements. The width of the development will be the distance around the cylinder, or the cir- cumference of the base. The base of the cylinder in Fig. 38, is divided into twelve equal parts, 123, etc. Commencing at point 1 on the development these twelve equal spaces are laid along the straight line, giving the development of the base of the cylin- der, and the total width. To find the development of the curve cut by the oblique plane, draw in elevation the elements corre- sponding to the various divisions of the base, and note the points MECHANICAL DRAWING. 95 where they intersect the oblique plane. As we roll the cylinder beginning at point 1, the successive elements 1, 12, 11, etc., will appear at equal distances apart, and equal in length to the lengths of the same elements in elevation. Thus point number 10 on the development of the curve is found by projecting horizontally across from 10 in elevation. It will be seen that the curve is symmetri- cal, the half on the left of 7 being similar to that on the right. The development of any curve whatever on the surface of the cylinder may be found in the same manner. The principle of cylinder development is used in laying out elbow joints, pipe ends cut off obliquely, etc. In Fig. 39 is shown plan and elevation of a three-piece elbow and collar, and develop- ments of the four pieces. In order to construct the various parts making up the joint, it is necessary to know what shape and size must be marked out on the flat sheet metal so that when cut out and rolled up the three pieces will form cylinders with the ends fitting together as required. Knowing the kind of elbow desired, we first draw the plan and elevation, and from these make the developments. Let the lengths of the three pieces A, B and C be the same on the upper outside contour of the elbow, the piece B at an angle of 45; the joint between A and B bisects the angle between the two lengths, and in the same way the joint between B and C. The lengths A and C will then be the same, 96 MECHANICAL DRAWING. and one pattern will answer for both. The development of A is made exactly as just explained for Fig. 38, and this is also the development of C. It should be borne in mind that in developing a cylinder we must always have a base at right angles to the elements, and if the cylinder as given does not have such a base, it becomes neces- sary to cut the cylinder by a plane perpendicular to the elements, and use the intersection as a base. This point must be clearly understood in order to proceed intelligently. A section at right angles to the elements is the only section which will unroll in a Fig. 30. straight line, and is therefore the section from which we must work in developing other sections. As B has neither end at right angles to its length, the plane X is drawn at the middle and per- pendicular to the length. B is the same diameter pipe as C and A, so the aection cut by X will be a circle of the same diameter as the base of A, and its development is shown at X. From the points where the elements drawn_on the elevation of A meet the joint between A and B, elements are drawn on B, MECHANICAL DRAWING. '.'7 which are equally spaced around B the same as on A. The spaces then laid off along X are the same as given on the plan of A. Commencing with the left-hand element in B, the length of the upper element between X and the top corner of the elbow is laid off above X, giving the first point in the development of the end of B fitting with C. The lengths of the other elements in the elevation of B are measured in the same way and laid off from X. The development of the other end of the piece B is laid off below X, using the same distances, since X is half way be- tween the ends. The development of the collar is simply the de. velopment of the frus- tum of a cone, which has already been explained, Fig. 36. The joint be- tween B and C is shown in plan as an ellipse, the construction of which the student should be able to understand from a study of the figure. The intersection of a rectangular prism and pyramid is shown in Fig. 40. The base b c de of the pyramid is shown dotted in plan, as it is hidden by the prism. All four edges of the pyramid pass through the top of the prism, 1, 2, 3, 4. As the top of the prism is a horizontal plane, the edges of the pyramid are shown passing through the top in elevation at x* g fa i. These four points might be projected to the plan on the four edges of the pyramid; but it is unnecessary to project more than one, since the general principle applies here that if a cone, pyramid, prism or Cylinder be cut by a plane parallel to the base, the section is a figure parallel and similar to the base. The one point x* is there- fore projected down to a b in plan, giving x\ and with this aj Fig. 40. MECHANICAL DRAWING. one corner, the square xh g jl i h kh is drawn, its sides parallel to' the edges of the base. This square is the intersection of the pyramid with the top of the prism. The intersection of the pyramid with the bottom of the prism is found in like manner, by taking the point where one edge of the pyramid as a b passes through the bottom of the prism shown in elevation as point W, projecting down to w>* on ah /*, and drawing the square mh nh oh ph parallel to the base of the pyramid. These two squares constitute the entire intersection of the two solids, the pyramid going through the bottom and coming out at the top of the prism. As much of the slanting edges of the 10 7 Fig. 41. pyramid as are above the prism will be seen in plan, appearing as the diagonals of the small square, and the rest of the pyramid, being below the top surface of the prism, will be dotted in plan. Fig. 41 is the development of the rectangular prism, show- ing the openings in the top and bottom surfaces through which the pyramid passed. The development of the top and bottom, back and front faces will be four rectangles joined together, the same sizes as the respective faces. Commencing with the bottom face 5678, next would come the back face 6127, then the top, etc. The rectangles at the ends of the top face 1 2 3 4 are the ends of the prism. These might have been joined on any other MECHANICAL DRAWING. 99 face as well. Now find the development of the square in the bottom 5678. As the size will be the same as in projection, it only re- mains to determine its position. This position, however, will have the same relation to the sides of the rectangle as in the plan. The center of the square in this case is in the center of the face. To transfer the diagonals of the square to the development, extend them in plan to intersect the edges of the prism in points 9, 10, 11 and 12. Take the distance from 5 to 9 along the edge 5 6, and lay it on the development from 5 along 5 6, giving point 9. Point 10 located in the same way and connected with 9, gives the position of one diagonal. The other diagonal is obtained in a similar way, then the square constructed on these diagonals. The same method is used for locating the small square on the top face. If the intersection of a cylinder and prism is to be found, we may either obtain the points where elements of the cylinder pierce the prism, or where edges and lines parallel to edges on the sur- face of the prism cut the cylinder. A series of parallel planes may also be taken cutting curves from the cylinder and straight lines from the prism ; the intersec- tions give points on the intersection of the two solids. Fig. 42 represents a triangular prism intersecting a cylinder. The axis of the prism is parallel to V and inclined to H. Starting with the size and shape of the base, this is laid off at a, b h c h , and the altitude of the triangle taken and laid off at a v c v in elevation, making right angles with the inclination of the axis to H. The plan of the prism is then constructed. To find the intersection of the two solids, lines are drawn on the surface of the prism parallel to the length and the points where these lines and the edges pierce the cylinder are obtained and joined, giving the curve. The top edge of the prism goes into the top of the cylinder. This point will be shown in elevation, smce the top of the cylinder is a plane parallel to H and perpendicular to V, and therefore projected on V as a straight line. The upper edge, then, is found to pass into the top of the cylinder at point o, o v and o h . The intersection of the two upper faces of the prism with the top of the cylinder will be straight lines drawn from point o and will be shown in plan. If we can find where another line of the surface o a h 14 pierces the upper base of the cylinder, this point joined 100 MECHANICAL DRAWING. ivith o will determine the intersection of this face with the top of the cylinder. A surface may always be produced, if necessary, to find an intersection. Edge a b pierces the plane of the top of the cylinder at point d, seen in elevation ; therefore the line joining this point with o is the intersection of one upper face of the prism with the upper MECHANICAL DRAWING. 101 base of the cylinder. The only part of this line needed, of course, is within the actual limits of the base, that is o 9. The intersec- tion o 8 of the other top face is found by tine same method. On the convex surface of the cylinder there will be three curves, one for each face of the prism. Points b and 9 on the upper base of the cylinder, will be where the curves for the two upper faces will begin. The point d is found on the revolved position of the base at d r and d { b is divided into the equal parts d { e r e { /j, etc., which revolve back to d h , e h ,f h and , directly above d\ e h , etc., or may be found by taking from the revolved position of the base, the perpendiculars from <7, ih f h y h hence their vertical projections coincide. Points ;?, w, k\ and z ( are formed by projecting across from n h m h k h and I' 1 . The convex surface of the cylinder is perpendicular to H, so the points where the lines on the.prism pierce it will be projected on plan as the points where these lines cross the circle, 14, 13,12, 11 3. The vertical projections of these points are found on the corresponding lines in elevation, and the curves drawn through. The curve 3, 4.. ..8 must be dotted, as it is on the back of the cylinder. The under face of the prism, which ends with the line b c, is perpendicular to the vertical plane, so the curve of intersec tion will be projected on V as a straight line. Point 14 is one end of this curve. 3 the other end, and the curve is projected in elevation as the straight line from 14 to the point where the lower edge of the prism crosses the contour element of the cylinder. Fig. 43 gives the development of the right-hand half of the cylinder, beginning with number 1. As previously explained, the distance between the elements is shown in the plan, as 1 2, 2 3, 3 4 and so on. These spaces are laid off in the development along a straight line representing the development of the base, and from these points the elements are drawn perpendicularly. The lengths of the elements in the development from the base to the curve are exactly the same as on the elevation, as the 102 MECHANICAL DRAWING. elevation gives the true lengths. If then the development of the base is laid off along the same straight line as the vertical projec- tion of the base, the points in elevation may be projected across with the T-square to the corresponding elements in the develop- ment. The points on the curve cut by the under face of the prism are on the same elements as the other curves, and their vertical projections are on the under edge of the prism, hence these points are projected across for the development of the lower curve. In Fig. 44 is given the development of the prism from the right-hand end as far as the intersection with the cylinder, begin- 14 Fig. 44. ning at the left with the top edge a 0, the straight line a b e a being the development of the base. As this must be the actual distance around the base, the length is taken from the true size of the base, a, b h c h . The parallel lines drawn on the surfaces of the prism must appear on the development their true distances apart, hence the distances a, lfl p i ^ii \>\ i ij>J ,04 !ir xl x a 00 Pi: * IN^J! , I '' H M Pi . o UX^i JDLJ IdP!! s MECHANICAL DRAWING 12.3 the unit of measurement, as shown. The use of this letter is not advocated for general work, although if made merely in outline the effect is pleasing. The styles of numbers corresponding with the alphabets of capitals given here, are also inserted. When a fraction, such as 2| is to be made, the proportion should be about as shown. For small letters, usually called lower-case letters, abcdefghijklmn opqrstuvwxyz Fig. 89. abcdefgh/jk/mr? opqrs Fig. 90. abcdefghijklmn op qr s tuvwxyz Fig. 91. the height may be made about two-thirds that of the capitals. This proportion, however, varies in special cases. The principal lower-case letters in general use among drafts- men are shown in Figs. 89, 90, 91 and 92. The Gothic letters shown in Figs. 89 and 90 are much easier to make than the Roman letters in Figs. 91 and 92. These letters, however, do not 124 MECHANICAL DRAWING. 350 MECHANICAL DRAWING give as finished an appearance as the Roman. As has already been stated in Mechanical Drawing, Part I, the inclined letter is easier to make because slight errors are not so apparent. One of the most important points to be remembered in letter- ing is the spacing. If the letters are finely executed but poorly spaced, the effect is not good. To space letters correctly and rapidly, requires considerable experience; and rules are of little value on account of the many combinations in which letters are abcdefghijkLmn opqrstuvwxyz Fig. 92. found. A few directions, however, may be found helpful. For instance, take, the word TECHNICALITY, Fig. 93. If all the spaces were made equal, the space between the L and the I would appear to be too great, and the same would apply to the space between the I and the T. The space between the H and the N and that between the N and the I would be insufficient. In general, when the vertical side of one letter is followed by the verti- cal side of another, as in H E, H B, I R, etc., the maximum space TECHNICALITY Fig. 93. should be allowed. Where T and A come together the least space is given, for in this case the top of the T frequently extends over the bottom of the A. In general, the spacing should be such that a uniform appearance is obtained. For the distances between words in a sentence, a space of about \\ the width of the average letter may be used. The space, however, depends largely upor desired effect 351 126 MECHANICAL DRAWING For large titles, such as those placed on charts, maps, and some large working drawings, the letters should be penciled before inking. If the height is made equal to the width considerable time and labor will be saved in laying out the work. This is especially true with such Gothic letters as O, Q, C, etc., as these letters may then be made with compasses. If the letters are of sufficient size, the outlines may be drawn with the ruling pen or compasses, and the spaces between filled in with a fine brush. The titles for working drawings are generally placed in the lower right-hand corner. Usual a draftsman has his choice of B Block Letters. letters, mainly because after he has become used to making one style he can do it rapidly and accurately. However, in some draft-, ing rooms the head draftsman decides what lettering shall be used. In making these titles, the different alphabets are selected to give the best results without spending too much time. In most work the letters are made in straight lines, although we frequently find a portion of the title lettered on an arc of a circle. In Fig. 94 is shown a title having the words CONNECTING ROD lettered on an arc of a circle. To do this work requires considerable patience and practice. First draw the vertical center 352 MECHANICAL DRAWING 127 line as shown at C in Fig. 94. Then draw horizontal lines for the horizontal letters. The radii of the arcs depend upon the general arrangement of the entire title, and this is a matter of taste. The difference between the arcs should equal the height of the letters. After the arc is drawn, the letters should be sketched in pencil to find their approximate positions. After this is done, draw radial lines from the center of the letters to the center of the arcs. X /\J FOR BEAM ENGINE ' SCALE 3 INCHES == 1 FOOT PORTLAND COMPANY'S WORKS JULY 1O, 1894. Pig. 04. These lines will be the centers of the letters, as shown at A, B, D and E. The vertical lines of the letters should not radiate from the center of the arc, but should be parallel to the center lines already drawn; otherwise the letters will appear distorted. Thus, in the letter N the two verticals are parallel to the line A. same applies to the other letters in the alphabet 353 128 MECHANICAL DRAWING Tracing. Having finished the pencil drawing, the next c tep is the inking. In some offices the pencil drawing is made on a thin, tough paper, called board paper, and the inking is done over the pencil drawing, in the manner with which the student is already familiar. It is more common to do the inking on thin, trans, parent cloth, called tracing cloth, which is prepared for the pur- pose. This tracing cloth is made of various kinds, the kind in ordinary use being what is known as " dull back," that is, one side is finished and the other side is left dull. Either side may be used to draw upon, but most draftsmen prefer the dull side. If a drawing is to be traced it is a good plan to use a 311 or 4H pencil, so that the lines may be easily seen through the cloth. The tracing cloth is stretched smoothly over the pencil draw- ing and a little powdered chalk rubbed over it with a dry cloth, to remove the slight amount of grease or oil from the surface and make it take the ink better. The dust must be carefully brushed or wiped off with a soft cloth, after the rubbing, or it will inter- fere with the inking. The drawing is then made in ink on the tracing cloth, after the same general rules as for inking the paper, but care must be taken to draw the ink lines exactly over the pencil lines which are on the paper underneath, and which should be just heavy enough to be easily seen through the tracing cloth. The ink lines should be firm and fully as heavy as for ordinary work. In tracing, it is better to complete one view at a time, because if parts of several views are traced and the drawing left for a day or two, the cloth is liable to stretch and warp so that it will be difficult to complete the views and make the new lines fit those already drawn and at the same time conform to the pencil lines under- neath. For this reason it is well, when possible, to complete a view before leaving the drawing for any length cf time, although of course on viewc in which there is a good deal of work this cannot always be done. In this case the draftsman must manipu- late his tracing cloth and instruments to make the lines fit as best he can. A skillful draftsman will have no trouble from this source, but the beginner may at first find difficulty. Inking on tracing cloth will be found by the beginner to be quite different from inking on the paper to which he has been accustomed, and he will doubtless make many blots and think ai 354 MECHANICAL DRAWING 120 first that it is hard to make a tracing. After a little practice, nowever, he will find that the tracing cloth is very satisfactory and that a good drawing can be made on it quite as easily as on paper. The necessity for making erasures should be avoided, as far as possible, but when an erasure must be made a good ink rubber or typewriter eraser may be used. If the erased line is to have ink placed on it, such as a line crossing, it is better to use a soft rubber eraser. All moisture should be kept from the cloth. Blue Printing, The tracing, of course, cannot be Bent into the shop for the workmen to use, as it would soon become soiled and in time destroyed, so that it is necessary to have some cheap and rapid means of making copies from it. These copies are made by the process of blue printing in which the tracing is used in a manner similar to the use made of a negative in photography. Almost all drafting rooms have a frame for the purpose of making blue prints. These frames are made in many styles, some simple, some elaborate. A simple and efficient form is a flat sur- face usually of wood, covered with padding of soft material, such as felting. To this is hinged the cover, which consists of a frame similar to a picture frame, in which is set a piece of clear glass. The whole is either mounted on a track or on some sort of a swinging arm, so that it may readily be run in and out of a window. The print is made on paper prepared for the purpose by having one of its surfaces coated with chemicals which are sensi- tive to sunlight. This coated paper, or blue-print paper, as it is called, is laid on the padded surface of the frame with its coated side uppermost; the tracing laid over it right side up, and the glass pressed down firmly and fastened in place. Springs are frequently used to keep the paper, tracing, etc., against the glass. With some frames it is more convenient to turn them over and remove the backs. In such cases the tracing is laid against the glass, face down; the coated paper is then placed on it with the coated side against the tracing cloth. The sun is allowed to shine upon the drawing for a few minutes, then the blue-print paper is taken out and thoroughly washed in clean water for several minutes and' hung up to dry. 355 130 MECHANICAL DRAWING If the paper has been recently prepared and the exposure properly timed, the coated surface of the paper will now be of a clear, deep blue color, except where it was covered by the ink lines, where it will be perfectly white. The action has been this: Before the paper was exposed to the light the coating was of a pale yellow color, and if it had then been put in water the coating, would have all washed off, leaving the paper white. In other words, before being exposed to the sunlight the coating was soluble. The light penetrated the trans- parent tracing cloth and acted upon the chemicals of the coating, changing their nature so that they became insoluble; that is, when put in water, the coating, instead of being washed off, merely turned blue. The light could not penetrate the ink with which the lines, figures, etc., were drawn, consequently the coating under these was not acted upon and it washed off when put in water, leaving a white copy of the ink drawing on a blue background. If running water cannot be used, the paper must be washed in a sufficient number of changes until the water is clear. It is a good plan to arrange a tank having an overflow, so that the water may remain at a depth of about or 8 inches. The length of time to which a print should be exposed to the light depends upon the quality and freshness of the paper, the chemicals used and the brightness of the light. Some paper is prepared so that an exposure of one minute, or even less, in bright sunlight, will give a good print and the time ranges from this to twenty minutes or more, according to the proportions of ths various chemicals in the coating. If the full strength of the sun- light does not strike the paper, as, for instance, if clouds partly cover the sun, the time of exposure must be lengthened. Assembly Drawing. We have followed through the process of making a detail drawing from the sketches to the blue print ready for the workmen. Such a detail drawing or set of drawings shows the form and size of each piece, but does not show how the pieces go together and gives no idea of the machine as a whole. Consequently, a general drawing or assembly drawing must be made, which will show these things. Usually two or more views are necessary, the number depending upon the complexity of the machine. Very often a cross-section through some part of the MECHANICAL DRAWING 131 machine, chosen so as to give the best general idea with the least amount of work, will make the drawing clearer. The number of dimensions required on an assembly drawing depends largely upon the kind of machine. It is usually best to give the important over-all dimensions and the distance between the principal center lines. Care must be taken that the over-all dimensions agree with the sum of the dimensions of the various details. For example, suppose three pieces are bolted together, the thickness of the pieces according to the detail drawing, being one inch, two inches, and five and one-half inches respectively; the sum of these three dimensions is eight and one-half inches and the dimensions from outside on the assembly drawing, if given at all, must agree with this. It is a good plan to add these over-all dimensions, as it serves as a check and relieves the mechanic of the necessity of adding fractions. FORMULA FOR BLUE=PRINT SOLUTION. Dissolve thoroughly and filter. Red Prussiate of potash 2^ ounces, A ' Water 1 pint. Ammonio-Citrate of iron 4 ounces, B - Water 1 P in t Use equal parts of A and B. FORnULA FOR BLACK PRINTS Negatives. White lines on blue ground; prepare the paper with Ammonio-Citrate of iron 40 grains, Water ! ounce ' After printing wash in water. Positives. Black lines on white ground; prepare the paper Iron perchloride 616 grains, Oxalic Acid 308 grains, Water Hounces. (Gallic Acid 1 ounce Develop in ^Citric Acid l ounce (Alum 8ounce& Use 1J ounces of developer to one gallon of water. Paper is fully exposed when it has changed from yellow to white. 357 132 MECHANICAL DRAWING PLATES. PLATE IX. The plates of this Instruction Paper should be laid out at the same size as the plates in Parts I and II. The center lines and border lines should also be drawn as described. First draw two ground lines across the sheet, 3 inches below the upper border line and 3 inches above the lower border line. The first problem on each ground line is to be placed 1 inch from the left border line; and spaces of about 1 inch should be left between tbe figures. Isolated points are indicated by a small cross X, and projections of lines are to be drawn full unless invisible. All construction lines should be fine dotted lines. Given and required lines should be drawn full. Problems on Upper Ground Line: 1. Locate both projections of a point on the horizontal plane 1 inch from the vertical plane. 2. Draw the projections of a line 2 inches long which is parallel to the vertical plane and which makes an angle of 45 degrees with the horizontal plane and slants upward to the right. The line should be 1 inch from the vertical plane and the lower end % inch above the horizontal. 3 Draw the projections of a line 1 inches long which is parallel to both planes. 1 inch above the horizontal, and | inch from the vertical. 4. Draw the plan and elevation of a line 2 incnes long which is parallel to H and makes an angle of 30 degrees with V. Let the right-hand end of the line be the end nearer V, ^ inch from V. The line to be 1 inch above H. 5. Draw the plan and elevation of a line 1 inches long which is perpendicular to the horizon till plane and 1 inch from the vertical. Lower end of line is ^ inch above H. 6. Draw the projections of a line 1 inch long which is perpendicular to the vertical plane and 1 inches above the horizontal. The end of the lino nearer V, or the back end, is inch from V. 358 X ( X t MECHANICAL DRAWING 133 7. Draw two projections which shall represent a line oblique to both planes. NOTE. Leave 1 inch between this figure and the right-hand border line. Problems on Lower Ground Line: 8. Draw the projections of two parallel lines each 1^ inches long. The lines are to be parallel to the vertical plane and to make angles of 60 degrees with the horizontal. The lower end of each lino is \ inch above H. The right-hand end of the right-hand line is to be 2| inches from the left-hand margin. 9. Draw the projections of two parallel lines each 2 inches long. Both lines to be parallel to the horizontal and to make an angle of 30 degrees with the vertical. The lower line to be | inch above H, and one end of one line to be against V. 10. Draw the projections of two intersecting lines. One 2 inches long to be parallel to both planes, 1 inch above H, and | inch from the vertical; and the other to be oblique to both planes and of any desired length. 11. Draw plan and elevation of a prism 1 inch square and 1^ inches long. The prism to have one side on the horizontal plane, and its long edges to be perpendicular to V. The back end of the prism is \ inch from the vertical plane. 12. Draw plan and elevation of a prism the same size as given above, but with the long edges parallel to both planes, the lower face of the prism to be parallel to H and \ inch above it, The back face to be \ inch from V. PLATE X. The ground line is to be in the middle of the sheet, and the location and dimensions of the figures are to be as given, The Qrst figure shows a rectangular block with a rectangular hole cut through from front to back. The other two figures represent the same block in different positions. The second figure is the end or profile projection of the block. The same face is on H in all three positions. Be careful not to omit the shade lines. The figures given on the plate for dimensions, etc., are to be used but not repeated on the plate by the stuient. 134 MECHANICAL DRAWING PLATE XI. Three ground lines are to be used on this plate, two at the left 4 inches long and 3 inches from top and bottom margin lines; and one at the right, half way between the top and bottom margins, 9 inches long. The figures 1, 2, 3 and 4 are examples for finding the true lengths of the lines. Begin No. 1 finch from the border, the vertical projection If inches long, one end on the ground line and inclined at 30. The horizontal projection has one end \ incl from V, and the other \\ inches from V. Find the true length of the line by completing the construction commenced by swinging the arc, as shown in the figure. Locate the left-hand end of No. 2 3 inches from the border, 1 inch above H, and | inch from V. Extend the vertical projection to the ground line at an angle of 45, and make the horizontal pro- jection at 30. Complete the construction for true length as commenced in the figure. In Figs. 3 and 4, the true lengths are to be found by complet- ing the revolutions indicated. The left-hand end of Fig. 3 is | inch from the margin, \\ inches from V, and 1| inches above H. The horizontal projection makes an angle of 60 and extends to the ground line, and the vertical projection is inclined at 45. The fourth figure is 3 inches from the border, and represents a line in a profile plane connecting points a and 1. a is 1 J inches above H and inch from V; and I is inch above H and 1^ inches from V. The figures for the middle ground line represent a pentagonal pyramid in three positions. The first position is the pyramid with the axis vertical, and the base f inch above the horizontal. The height of the pyramid is 2^ inches, and the diameter of the circle circumscribed about the base is 2$ inches. The center of the circle is 6 inches from the left margin and If inches from V. Spaces between figures to be f inch. In the second figure the pyramid has been revolved about the right-hand corner of the base as an axis, through an angle of 15. The axis of the pyramid, shown dotted, is therefore at 75. The method of obtaining 75 and 15 with the triangles was shown in 364 MECHANICAL DKAWING Part I. From the way in which the pyramid has been revolved, all angles with V must remain the same as in the first position, hence the vertical projection will be the same shape and size as before. All points of the pyramid remain the same distance from V. The points on the plan are found on T-square lines through the corners of the first plan and directly beneath the points in elevation. In the third position the pyramid has been swung around, about a vertical line through the apex as axis, through 30. The angle with the horizontal plane remains the same; consequently the plan is the same size and shape as in the Fig. 96. second position, but at a different angle with the ground line. Heights of all points of the pyramid have not changed this time, and hence are projected across from the second elevation. Shade lines are to be put on between the light and dark surfaces as determined by the 45 triangle. PLATE XII. Developments. On this plate draw the developments of a truncated octagonal prism, and of a truncated pyramid having a square base. The arrangement on the plate is left to the student; but we should suggest that the truncated prism and its development be place 136 MECHANICAL DKAW1NG the left, and that the development of the truncated pyramid be placed under the development of the prism; the truncated pyramid may be placed at the right. The prism and its development are shown in Fig. 96. The prism is 3 inches high, and the base is inscribed in a circle 2 inches in diameter. The plane forming the truncated prism is passed as indicated, the distance A B being 1 inch. Ink a suffi- cient number of construction lines to show clearly the method of finding the development. The pyramid and its development are shown in Fig. 97. Each side of the square base is 2 inches, and the altitude is 3 inches. A Pig. 97. The plane forming the truncated pyramid is passed in such a position that A B equals If inches, and A C equals 2 inches. In this figure the development may be drawn in any convenient position, but in the case of the prism it is better to draw the development as shown. Indicate clearly the construction by inking the construction lines. PLATE XIII. Isometric and Oblique Projection. Draw the oblique projection of a portable closet. The angle to be used is 45. Make the height 3- inches, the depth 1 inches, and the width 3 inches. See Fig. 98. The width of the closet 868 MECHANICAL DRAWING 137 - r - -i .. 1 *-,- = C tvi ? * ^ C j ^ . i j i" ri 3 i Fig. 98. centrally in the front of the closet, the bottom edge at the height of the floor of the closet, the hinges of the door to be placed on the left-hand side. In the oblique drawing, show the door opened at an angle of 90 degrees. The thickness of the material of the closet, door, and floor is ^ inch. The door should be hung so that when closed it will be flush with the front of the closet. Make the isometric drawing of the flight of steps and end walls as shown by the end view in Fig. 99. The lower right-hand corner is to be located 2 inches from the lower, and 5 inches from the Fig - " right-hand, margin. The base of the end wall is 3 inches Jong, and the height is 2 inches. Beginning from the back of the wall, the top is horizontal for f inch, the remainder of the outline being comix>sed of arcs of circles whose radii and centers are given 138 MECHANICAL DRAWING in the figure. The thickness of the end wall is f inch, and both ends are alike. There are to be five steps; each rise is to be | inch, and each tread \ inch, except that of the top step, which is I inch. The first step is located f inch back from the corner of the wall. The end view of the wall should be constructed on a separate sheet of paper, from the dimensions given, the points on the curve being located by horizontal co-ordinates from the vertical edge of the wall, and then these co-ordinates transferred to the isometric drawing. After the isometric of one curved edge has been made, the others can be readily found from this. The width of the steps inside the walls is 3 inches. PLATE XIV. Free-hand Lettering. On account of the importance of free-hand lettering, the student should practice it at every opportunity. For additional practice, and to show the improvement made since completing Part I, lay out Plate XIV in the same manner as Plate I, and letter all four rectangles. Use the same letters and words as in the lower light-hand rectangle of Plate I. PLATE XV. Lettering. First lay out Plate XV in the same manner as previous plates. After drawing the vertical center line, draw light pencil lines as guide lines for the letters. The height of each line of letters is shown on the reproduced plate. The distance be- tween the letters should be ^ inch in every case. The spacing of the letters is left to the student. He may facilitate his work by lettering the words on a separate piece of paper, and finding the center by measurement or by doubling the paper into two equal parts. The styles of letters shown on the reproduced plate should be used 374 fcl IS ^ ! -.1-f.Ti ry; r.-H -.EH[ ' O o O H O ^ U ^^^^ ^^i^^ LJ if/} Q ^3 CC z J ^ O S3 tn ^^ ^ o o ^ OCX Q) W i 2 HOJ FT 1 L^^J HS gl S O < u REVIEW QUESTIONS. PRACTICAL TEST QUESTIONS. In the foregoing sections, of this Cyclopedia numerous illustrative examples are worked out in ill m order to show the application of the various methods and principles. Accompanying these are examples for practice which will aid the reader in fixing the principles in mind. In the following pages are given a large number of test questions and problems which afford a valu- able means of testing the reader's knowledge of the subjects treated. They will be found excellent prac- tice for those preparing for College, Civil Service, or Engineer's License. In some cases numerical answers are given as a further aid in this work. REVIEW QUESTIONS ON THE SCJBJKCT OF PLANK SURVEYING 1. Explain by a diagram how to erect (with the tape alone) a line at right angles to a given line. 2. The sides of a triangular field are 820,432 and 529 feet. Find the area of the field in acres, rods and square rods. 3. Find the area of a triangle \vhose sides are 31, 40 and 55 rods. 4. Given in Fig. 14, C B = 2.85 chains, C D = 3.67 chains, C S = C L = 0.52 chains and L S = 0.75 chains. Cal- culate the area of the triangle B C D. 5. A certain line is known to be 530 feet in length, but when measured with a certain tape is found to be 533i feet in length. Determine the true length of the tape. 6. A certain field is measured with a Gunter's chain and is found to contain 5.75 acres. It is afterwards discovered that the chain is ^ of a foot too long. Find the true area of the field. 7. If a line as measured is found to be 432| feet in length and it is afterwards discovered that the tape is too short by | of a foot, what is the true length of the line? 8. A level bubble has a radius of 150 feet and its scale has 10 spaces in an inch. What error in leveling will result at a distance of 275 feet when the level bubble is li spaces out of level? 9. At a distance of 150 feet, two rod readings were 3.704 and 3.745 and the bubble moved over 1 inch. Determine the radius of the bubble tube. 10. What error in leveling will result at a distance of 123 feet if the bubble is 2| spaces out of level, the scale of which has 7 spaces in an inch, the radius being 176 feet? PLANE SURVEYING 11. If the difference of rod readings is 0.03 foot and the bubble having a radius of 50 feet moves over 0.015 foot, deter- mine the distance from the instrument to the rod. 12. Given the distances measured along the straight line A B in Fig. A. Page 22, to be 35, 40, 55,45, 50, 70, 30, 45 and 25 feet. The offsets beginning with at A are 40, 35, 0,-55, -65, -30, 0, 25 and feet at B. Determine the area included between the line A B and the broken line A C D E F B. 1'3. An angle is 5 degrees and 1 minute and the distance 823 feet. What is the length of the subtended arc? 14. In your own words, describe the Gunter's chain, and its advantages in land surveying. Describe the engineer's chain and state what errors are liable to occur in using either chain for measuring lines. 15. In your own words, descril>e the differeut kinds of tapes and explain fully the advantages of the band tape over the chain for measuring the lengths of lines. 10. In your own words, describe the wye level, accompany- ing the description with sketches whenever necessary. 17. Define the following terms: Line of collimation; instrn- O mental parallax; spherical aberration; chromatic aberration. 18. In your own words, state the adjustments of the wye level /// tJu'u' ordt-r. 1 ( .>. In your own words, describe hew to make the tests for the several adjustments of the wye level. 20. In your own words, describe with sketches, how to make several adjustments of the wye level, in tJieir ord<>r. 21. Define the following terms: Back-sight; fore-sight; height of instrument; datum plain; bench-mark; p^g; elevation. 22. In your own words, descril>e how to "set" the wye level and the field operations of leveling. 23. Assuming your own rod readings, prepare a system of level notes; calculating heights of instrument and elevations. 24. In your own words, describe the Dumpy level and ex- plain how it differs from the wye level. 25. State and explain fully the adjustments of the Dumpy level /// their order. REVIEW QUESTIONS ON THE S IT B .1 K O T O V PLANE SURVEYING ART II 1. Give a detailed description of the transit. 2. Give the reasons for each of the adjustments of the transit. Draw a diagram in each case. 3. Describe fully in their order how to test the adjustments of the transit. 4. Describe fully in their order how to make the several adjustments of the transit. 5. Define the following: Latitude of a station; longitude of a line; double longitude; departure. 6. Explain the use of double longitudes. Draw a figure, and explain how latitudes and departures may be gotten from the length and bearing of a line. 7. In order to find the difference in height of two peaks M and N, a base line A B was laid off 5,000 feet long; and the hori- zontal angles BAM = 120 30', BAN = 49 15', ABM = 40 35', and ABN = 95 07', were read. At A, the angle of elevation of M was 17 19', and the angle of elevation of N was 18 45'. Com- pute the difference in the height of the two peaks. Ans. K. is 226.59 feet above N. 8. Draw a figure, and deduce a general rule for the double longitude of any course in terms of thedouble longitude anddeparture of the preceding course. 9. Explain what is meant by balancing a survey. 10. Describe fully how the latitudes and departures of a series of courses may be balanced, and how to determine the balanced length and bearing of each course. PLANE SURVEYING 11. Let c + / = 0.87 feet, *' = 4.65 feet, and / - 3 32'. Compute the horizontal distance to the rod and the difference in elevation. What error results when c + / is neglected? 12. In order to find the direction and distance between two points R and Q, the following lines are run. RA, S 87 37', W. 930.57 feet ; J J5, W 621.03 feet; BQ, S 88 15' West, 82.78 feet. Compute the bearing and length of RQ, and locate the point where it crosses A B, with reference to A. 13. In order to find the direction and distance of a point, the following lines are run: AC X 42 15' E 714.5 feet. CB X 1 8' E 210.5 feet. Compute the distance and bearing of the two points. Draw a diagram before starting the solution. 14. Draw a diagram simi- lar to the hands of a clock, and explain what is meant by an azimuth. 15. If the limb of a transit is divided into 20 minute spaces, show how the vernier must be made in order to read 1 minute; also how to read 20 seconds, (live diagrams of these verniers. 16. Explain the relation between azimuth and bearings of lines in the four quadrants. Determine the azimuths of the courses in Problem 3, page 88. 17. Compute the area of Fig. A, taking the azimuth of BC as 00'. Also taking the azimuth of A B as 90 (M)'. 270 Ki>r. A. .4 H = 800 foot. H C = 500 foot. C I) = 200 foot. I) K = 100 foot. K F = 000 foot. FA = 700 foot. A = 58 14' B = 120 00' C = 12o 00' D = 200 00' E = 83 34' F = 133 12' Axs. Area = 11 acres 1 rood 1.62 rods. 18. Describe fully the various steps involved in determining the area of an enclosed field from latitudes and departures, and state the rule for calculating the area. 19. A polygon of six sides has the following interior angles: PLANE SURVEYING A = 5824';# - 121 30'; C = 127 45'; E = 9519';F - 133 2'. The azimuth AB is 00 0'. Find the azimuth of each of the other sides. 20. Give a complete description of the surveyor's compass. Explain fully how to use the compass in the field in determining areas. 21. Given the latitude of one end of a line as + 2,804.4, its longitude as + 4,661.3, its length as 797.2 feet, and its azimuth as 115 44' 28". Compute the lati- tude and longitude of the other end. Draw a figure before start- ing the solution. Ans. Latitude = 2,458.2 feet. Longitude = 5,379.4 feet. 22. Describe fully the field work of running a traverse. 23. Explain fully the field work of laying out angles. 24. In Problem 2, Page 88, the bearing of the first course is changed to S. 15 E. Calculate Fig. JJ. the changed bearings of the other courses. 25. Describe fully how to test and make the adjustments of the compass in their order. 26. Explain fully how to lay out a true meridian from Polaris. 27. Explain the application of, and describe fully how to carry out, the "peg" adjustment for the transit. 28. Define and fully explain the terms: Daily, Annual, and Secular Variation. 29. Describe the stadia and de- duce the formulae for its use. 30. How many feet are rep- resented by one inch on a scale of Tlj L ) ? How many acres are represented by one square inch on PLANE SURVEYING Ans $ 83;1 ' fl ' et S " I 398.6 acres. 31. Compute the area of Fig. B, which represents a recent survey of a farm, from the following data: A B = 317.8 feet; B F - 284.3 feet; F A = 250.5 feet; F C - 512.7 feet; F D = 510.0 feet; DEF = 90 00'; E F D = 69 45'; DF C = 61 12'; C FB = 49 30'; Ans. Area = 5 acres 103 rods 81 sq. feet. 32. I )escrit>e fully the field work of carrying out a stadia survey over un- even ground. 33. A triangle ARC lias sides with the follow- ing lengths and azimuths : A H, I = 312 feet; / - 45 degrees. H C, 1 = 540.4 feet; Z = 135 degrees. C A, I = 624.0 feet; Z = 285 degrees. Compute the latitude differ- ences and longitude differences and the double longitudes for each course. 34. The bearing of A C (Fig. C) is N 8.J E, that of AD is N 46 E. Find the value of the angles C A D and D A E. 35. Find the angle ARC (Fig. D) when the bearing of A R is 42 E, and that of R C is S 29J E. What will R A be when the first bearing is re- versed? Fig. K i^EVIEW QUESTIONS THK S U B J K er cent. Explain fully how this would be done with the gradienter. 17. In Fig. Ill given BE' = 375 feet; angle CBE' = 11225 ( ; angle CE'B = 52"' 16'. Plot the figure accurately to a scale of 100 feet to the inch. Calculate the length of BC and if etake B is numbered 180 + 36, determine the number of stake C. 18. In your own words describe the plane-table and its uses; its advantages and disadvantages. 19. Explain fully the organization of a topographic party using the transit and stadia, and explain the method of keeping the field -notes. INDEX The page numbers of this volume will be found at the bottom of the pages; the numbers at the top refer only to the section. Aberration Abney hand-level Acute angle, definition of Acute-angled triangle, definition of Agonic lines Altitude of a star, definition of Altitude of triangle, definition of Angles, measurement of Annual variation Assembly drawing Azimuth B Base measurement apparatus for errors in tape-stretcher Base of triangle, definition of Beam compasses Bearings, to change Bench-mark, definition of Black prints, formula for Blue-print solution, formula for Blue printing Boston rod Bow pen Broken line, definition of Capital letters Central angle, definition of Chaining on slopes Chord, definition of Chromatic aberration Circles, definition of age Page Clinometer 39 55 Collimation, line of 54 39 Compass 76 256 adjustment 79 256 magnetic needle 77 76 sights 78 148 tangent scale 78 257 use of 80 260 Compasses, drawing 220 75 Cone, definition of 264 356 Conic sections, definition of 205 97 Cross-section rod 49 Cross-sectioning 69 Cube, definition of 262 173 Curved line, definition of 255 174 Cycloid, definition of 267 174 Cylinder, definition of 263 178 D 257 229 Deflection angles 119 87 Departures 88 66 Diurnal variation 75 357 Dividers 223 357 Division of land 99 355 Drawing board 2 1 3 49 Drawing instruments and materials 211 224 beam compasses 229 255 lx>ard 213 bow pen 224 bow pencil 224 232 compasses 220 260 dividers 223 18 drawing pen 224 259 erasers 214 55 ink 22G 259 irregular curve 228 Note Fa numbers see foot of pages. II INDEX Page Pae Drawing instruments and materials Geometrical definitions paper 211 pyramids 262 pencils 214 quadrilaterals 257 protractor 227 sol ids 261 scales 227 spheres 264 T-square 215 surfaces 256 thumb tacks 213 triangles 256 triangles 217 Gothic letters 230 Drawing paper 211 Gradienter 141 Drawing pen 224 Gurley binocular hand level 60 Dumpy-level 58 E H Ellipse, definition of 265 Hand-level 60 Engineer's transit 111 Abney 39 Epicycloid, definition of 268 Locke 37 Equiangular triangle 257 Horizontal line, definition of 255 Equilateral triangle 256 Hyperbola, definition of 206 Hypocycloid, definition of 268 F 1 Farm surveying 85 balancing t he survey 90 India ink 226 bearings, to change 87 Inscril>ed angle, definition of 260 calculating the content 91 Inscriljed polygon, defln tion of 260 field notes 86 Intersecting lines, definition of 255 intersections, method of 85 Intersection and development 311-328 latitudes and departures 88 Involute, definition of 268 progression, method of 85 Irregular curve 228 proofs of accuracy 86 Isogonic lines 76 radiation method of 85 Isometric projection 329 supplying omissions 96 Isosceles triangle 257 Field notes, keeping of 30 Field work of measuring areas 26 L Frustum of a pyramid, definition of 263 Land measure 14 Latitude difference 88 G Latitude of a place, deflni'ion of 148 Geodetic surveying 11 Latitudes and departures 88 Geometrical definitions testing a survey by 89 angles 256 Lettering 229. 346 circles 259 capital 232 cones 263 Gothic 230 conic sections 265 lower-case 232 cylinders 263 Roman 230 lines 255 \A-\t-\ bubble 34 odontoidal curves 267 Leveling 64 point 255 cross-sectioning 69 polygons 258 profile 67 Note. For pane numbers tee foot of pages. in LyiWy. III Page Leveling instruments care of 51 Odontoidal curves, definition of Page 63 epicycloid dumpy-level hand level 58 60 hypocycloid involute 268 208 "setting up" spirit level wye level 61 60 51 Off-sets and tie-lines Orthographic projection 268 27 295 Leveling rod 41 P Line, definition of 255 Parabola, definition of 266 Line of collimation 54 Parallel lines, definition of 255 Line shading 344 Parallelogram, definition of Lines, measurement of 12 Parallelepiped, definition of 20l' Locke's hand level 37 Peg, plug, or turning point, defin it ion of oo Longitude difference 88 Perpendicular lines, definition of 255 Longitude of a point 88 Plane figure, definition of 250 Lower-case letters 232 Plane surveying 1 1 -209 M Plane-table 182, 201 PI us-siglit definition of 06 Magnetic declination 75 Point, definition of Magnetic needle 77 Polyedron, definition of 201 Major axis 260 Polygon, definition of 25s Mechanical drawing 211-374 Prism, definition of 261 blue printing 355 Prismatic compass 79 geometrical definitions 255 Profile leveling 67 geometrical problems 269-293 Profile plane 301 instruments and materials 211 Project ion intersection and development 311-328 isometric 329 lettering 229, 346 oblique 341 line shading 344 orthographic 295 plates 233-252, 269-293, 358-374 third plane of 301 projections 295 Protractor 227 shade lines 307 Pyramid, definition of 262 tracing 354 Q Meridian 10'-* Meridian plane Quadrilateral, definition of 257 Minor axis 266 H Minus-sight, definition of 66 Radiating triangulation 207 N Radius, definition of 259 New York rod 47 Ranging poles 50 Rectangle, definition of 258 O Rectangular hyperbola 267 Obtuse angle, definition of 256 Relocation 82 Obtuse-angled triangle 256 Resurveys 97 Oblique lines, definition of 255 Rhomboid, definition of 258 Oblique projections 341 Rhombus, definition of 258 Odontoidal curves, definition of 267 Right-angled triangle, definition of 256 cycloid 267 Right angles, definition of 256 Note. For page numbers see foot of pages. IV INDEX Page Page Hod Surveying Boston 49 relocation 82 cross-section 49 resurveys 97 leveling 41 stadia 127 New York 47 tape 16 Roman capitals 230 topographical 189 transit 104 S traversing 121 Scales 227 trian Kill at ion 203 Secant, definition of 259 vernier 31 Sector, definition of 260 Surveyor's transit 111 Secular variation 75 Shade lines 307 T Solar transit 149 T-square 215 adjustments of 151 Tables use of 153 base measurement 173 Sphere, definition of 204 field surveying data 96 Spherical alxsrration 55 land measure 14 Spirit level 00 latitude coefficients 156 Square, definition of 258 mean refraction at various altitudes 155 Stadia 127 I H.I. iris data 138 use of, in field 132 refraction correction 158 Stadia rods 134 Tachymeter 111 Steel tapes 175 Tangent, definition of 259 Straight line, definition of 255 Tape 10 Surface, definition of 250 Tape-stretcher 178 Surveying 11-209 Telemeter 134 azimuth 7 Theodolite 111 base measurement 173 Third plane of projection 301 Boston rod 49 Thumb tacks 213 clinometer 39 Topographical surveying 189 compass 70 equipment 191. 192. 190 cross-section rod 49 field operations 190. 199 farm 85 method of pnx-edure 192 geodetic 11 organization of party 200 gradienter 141 photography 202 Gunter's chain 13 plane table and stadia 201 hand level 37 transit and stadia 199 level bubble 34 Tracing 354 leveling instruments 51 Transit 104 leveling rod 41 adjust ment 112 measurement of lines 12 engineer's 11 1 meridian 102 to "set up" 117 New York rrxl 47 solar 149 plane 11 surveyor's 111 plane-table 182 Transit-theodolite 111 ranging poles 50 Trapezium, definition of 257 Note. For pni/e numbers ace ft lot of pages. INDEX Trapezoid, definition of Traversing checking the traverse keeping notes Triangles, definition of Triangulation adjusting triangle angles, measuring base line, measuring radiating True meridian Note. For page numbers see foot of pages. Page Page 257 Truncated prism, definition of 262 , 188 U 124 U. S. public land surveys 168 256 V 203 Variation 75 209 Vernier 31 205 Vertical line, definition of 255 204 W 207 Wye level 51 102 adjustments 55 University of California SOUTHERN REGION/ JAL LIBRARY FACILITY 405 Hilgard Avenue, Los Angeles, CA 90024-1388 Return this material to the library from which it was borrowed. SRLF QL OCT18 1993 UC SOUTHERN REGIONAL LIBRARY F bgineeri** Library TA 145 000688177 5