UG 47o.S73"°""""'""*>"^''""T' 
 ^'illiMiMimmn '"POflraphy, map readi 
 
 3 1924 014 519 882 
 
 fi>tate CoUese of l^griculture 
 
 Sit Cornell Winibtteits 
 
 3t\)ua. a. s. 
 
 Uibrarp 
 
Cornell University 
 Library 
 
 The original of this bool< is in 
 the Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924014519882 
 
TRAINING MANUAL 
 
 TOPOGRAPHY, MAP READING 
 AND RECONNAISSANCE 
 
 Prepared by 
 
 MAJOR GEORGE R. SPALDING 
 
 CORPS OF ENGINEERS 
 
 • - 
 
 Under the direction of the Chief of Engineers, U. S. Army 
 
 
 1 I 
 - 1 • ( .■ I 
 X » 
 
 itliiM^^''^ 
 
 WASHINGTON 
 
 GOVERNMENT PRINTING OFFICE 
 
 1918 
 
Wah Depaktment 
 
 Document No. 695 
 
 Office of the Adjutant General 
 
TRAINING MANUAL IN TOPOGRAPHY, MAP READING. 
 AND RECONNAISSANCE. 
 
 LESSON I. 
 
 TOPOGEAPHY. 
 
 1. So important is tEe inflTience of "the lay of the land" upon mili- 
 tary operations that every officer must thoroughly acquaint himself 
 9,t the earliest practicable moment with the topography of the area 
 I withia which Ms responsibihties lie. 
 
 ' 2. A mere knowledge of the main roads, of the direction and dis- 
 , tances to near-by towns is by no means sufficient. He must know 
 I where the plains are, where the hiDs rise, how the streams nm. He 
 must have a mental picture, a iird's-eye view, impressed on his brain 
 of the main physical and military features of Ms territory. 
 
 EROSION. 
 
 S. The faculty for grasping the topography of an area as a whole is 
 rarely a natural one. " To the beginner" in the study of topography 
 "every hitt is an isolated feature — the elevations arid depressions of 
 grouTbd present to Mm noihtmg hut irregularity and confusion." 
 
 4. As a matter of fact, there is a considerable degree of regularity 
 and system in our present-day ground forms. Whatever may have 
 been the irregularity and confusion of the original surface of the earth, 
 practically aJl of the ground forms which we Tenme to-day are the resuU 
 cf erosion. 
 
 5. It has been estimated that the average elevation of the land 
 surface has been reduced 7,000 feet by" erosion. Wiuds and waves, 
 glaciers, chemical action, frost, and plant growth have all had their 
 part in the breaking up and wearing down of the origiual rocks, but 
 by far the larger portion of the ground farms which exist to-day have 
 been carved out of the older formations by the action of running 
 water. 
 
 , MASTER LINES OF TOPOGRAPHY. 
 
 6. Not all the water in heavy rains can seep into the soil; much of 
 it runs off. At one point a little stream begins; as it flows downward 
 another joins it; soon several tmite into a fair-sized creek, which 
 rushes along carrying iis burden of soil into the main valley, whence 
 the river may carry it on toward the sea. This simple process con- 
 tinued for ages has sufficed to carve out our deepest valleys. It is con- 
 tinuing to deepen them year by year, and more and more to wear 
 down the ridges between the valleys. 
 
 7. If our remaining ground forms are the rrault of such a process, 
 it is manifest that the drainage lines of an area, together with the 
 ridge lines, form a system of master lines which, once traced out and 
 studied, will give a grasp of the main features of the topography 
 that can be had in no other way. 
 
 3 
 
4 TOPOGEAPHY, MAP BEADING, AND KECONNAISSANCE. 
 
 CONTOURS. 
 
 8. The main features of tlie topography, however, while of fet 
 importance, are not sufficient for modem military operations. The 
 military leader must also know the relative sizes of the valleys, the 
 relative heights of the ridges, wher§ the steep slopes are, where tne 
 hilk are flat. This information may be indicated upon a map ia 
 several ways, but the best way is by a system of contour lines. 
 
 9. The begimier always has trouble with contours. He has learned 
 perhaps that a contour is a Hne connecting points of equal elevation; 
 that contours do not cross; that a contour line either closes on itself 
 or both ends of it go off the map. But rules such as these, important 
 as they may be in checking up a sketcher's work, do not help the 
 beginner much. He is still hazy as to just what contours are and 
 how they represent ground forms. 1 
 
 10. It is true that a contour is a line joining points of equal eleya- 
 tiqri, , But such a definition is not precise. A contour line is a lins 
 each point of whicJh has the sajne elevation. If one walks along a 
 contour line, he neither goes up nor down hiU, hut always on a level. 
 The gurface of a quiet pond which has no outlet is practically level, i 
 Therefore its shore line is a contour line. If one follows it in. the 
 direction of ihs hinds of a clock he will find that he must turn to the 
 left at every entering valley, walk up the valley until he heads the water 
 hne, cross, the valley, then turn again to the right, following down the 
 other sideofthe valley to get around the poird of the hiU or spur which 
 hes between it and the next valley. His course mil tend to the left 
 at,,every little inflowing drainage line, cross it, and turn again on the 
 other side to avoid leaving the level. We see then that vaUey contours 
 apparently go in pairs, that is, there is always one of the same eleva- ' 
 t%orb on, e.acn side of the vaU^y. They form a sort of a V which opens 
 out in the direction of water flow; point of the V upstream. Simi- 
 larly the spur contours apparently go in pairs. They form a sort of 
 U which opens out to the higher ground up the spiir. (In glacial 
 country the valleys are more nearly U shaped and the ridges sharp.) 
 
 11. The typical contour, th"en, is a curving line, alternately salient 
 and reentrant, a series of rude V's for the valleys and U's for the 
 spurs, the point of the V at the stream crossing, the curve of the U 
 at the ridee crossing. If the extreme points of the V's and U's are 
 determined and these points are connected by a curving line opening 
 out gradually as we go downstream from the points of the v and 
 rounding out to the curve of the U, we get a hne which will roughly 
 represent the contour of the ground for that elevation. In other 
 words, the points of the V's, or the points where the contoTir crosses the 
 stream, and the points of the U 's, or the points where the contonr crosses 
 the line of the watershed between two valleys, are essential control 
 points for the drawing of any contour line. 
 
 12. The single contour which marked the shore line of the pond wiU 
 of course, give no indication oi the shape of the bills which surround 
 the pond. To indicate these ground forms, we must have a nwrriber of 
 contours, each one of which shows the path which a man would fol- 
 low if he walked around the hiU forms on a level line. These level 
 lines shotild be spaced at equal intervals in a vertical direction. In 
 other words, if we take the shore-line contour as the datum contour 
 the next line should show the path which a man would follow if 
 walking on a level line, say, 10 feet higher than the pond, and the 
 
TOPOGEAPHY, MAP EEAOIITG, AND EECONNAISSANCE. 5 
 
 next, 20 feet higher, etc. This 10-foot vertical interval between the 
 level liaes is c^led the contour interval or vertical interval. It may 
 be 1 foot, 6 feet, 10 feet, or 100 feet, or any other number of feet. , 
 
 NoTE.4-Our Field Service Regulations provide that the contour interval shomld 
 be 10 feet for a military sketch made on a scale of 6 inches to the mile, 20 feet for a 
 sketch on a scale of 3 inches to the mile, and, in general, the proper contour interval 
 for any scale isto be found by dividing 60 by the number of inches on the scale which 
 represents a mile on the ground. This system of scales is known as the United States 
 Army_ Normal System. The British Amy have a similar system, using 50 as the 
 base instead of 60. In small-scale mapping these systems are usually not adhered 
 to. The United States Geological Siurvey use an interval of 20 feet for their 1/62500 
 (about an inch to the mile) sheets, reducing this to 10 or even B feet in very flat 
 coimtry and increasing it for mountainous covrntry. 
 
 CRITICAL POINTS. 
 
 13. It is manifestly impracticable in mapping to locate with 
 instruments in the field the points where eaai corvtour crosses each 
 drainage line, nor is this necessary. No map can show every change 
 of form of the ground. We must satisfy ourselves, therefore, with 
 locating the critical points of the master lines of the ground. Such 
 points are the Tieads, the cTianges in direction and the cJianges in slope 
 of the drainage lines; the tops, the changes in direction, and the changes 
 in slope of the ridge lines. It will also be impracticable to deternune 
 
 ■ the elevation of the head of every little gully or the location"and 
 elevation of every change in direction of every drainage lineov of every 
 ridge line. But we must locate the master critical points of the master 
 ridge and stream lines, and the elevation of these points must b6 
 determined as well as the location of them. It is very impdrtant; 
 however, to plat in every gully, though the elevation ofits head be 
 not determined, as these small drainage fines effect the shapes of 
 the hiU forms very decidedly and, furthermore, they are landmarlcs of 
 
 f treat value. When the critical points and the drainage nefhas beeii 
 ocated as above described, one who knows something of the laws of 
 nature and remembers what a contour is can interpolate between the 
 master critical points enough of th'e others to enable him to draw in 
 all the contours. 
 
 INTERPOLATION. 
 
 14. Figure I is a skeleton sketch, giving the complete drainage net 
 and a few of the master critical poirds of a section of the Fort iLeav- 
 env/orth (Kans.) MOitary Eeservation. The main stream fine of 
 the area is shown as a solid fine. The direction of flow is indicated 
 by the arrow near elevation 790 (upper right-hand corner); The 
 elevations of two points on the main stream are given (790 ^nd 850). 
 Smaller tributary drainage fines are indicated in dashed fine. Ele- 
 vations not on stream fines are critical points on the ridge fines. 
 
 15. The proilem is to draw logical contours at 10 feet vertical 
 intervals, with no other data than that given. The first step is to 
 locate the points where desired contours cross the mam stream fine. 
 As there are no faUs between elevation 790 and elevation 850, it is 
 assumed that the slope of the stream between these points is nearly 
 uniform, becoming, however, a fittle steeper as the stream is ascended. 
 Under this assumption the crossing points of contours 800, 810, 820, 
 830, and 840 are at once interpolated by eye on the main stream 
 between 790 and 850. By interpolation between these the eleav- 
 
6 TOPOGEAPHT, MAP EEAD-ING, AND BECOlirNAISSANCE. 
 
 tions of the points where each of the tributary ravines enter the 
 main stream are secured. Between these latter poirds and the heads 
 of the ravines, the elevations of which are given, the points where 
 the contours cross eaCh of the ravines are marked. 
 
 16. Each point shown along the ridge lines was chosen as being a 
 critical point, that is, a point where the slope or direction of the ridge 
 changes. From point to point along the ridge lines, it is assumed 
 tliat the slope is uniform. The points where the intervening contours 
 cross the ridge lines have, theref ore^ been interpolated. The; result 
 of interpolation along stream and ridge lines is shown in Figure II. 
 As a rule the ridges and spurs of eroded hills point in the general 
 
 FIGURE I 
 
 direction of the junctions of the streams which have eroded them. 
 Therefore, interpolations have been made in these directions. 
 
 DRAWING THE CONTOURS. 
 
 17. In sketching in open country, where all the features can 
 be plainly seen, the method of drawing in the contours from 
 this control is to revisit the predetermined critical points and to draw 
 in all the contours which are needed to represent one feature, that is 
 one spur or hiM at a time, using the dramage lines as the limits o^ 
 the umtfeature to be drawn. The smaller irregularities of the ground 
 are brought out by slight changes in contour spacing and contour 
 direction, here and there. (See Fig. III. ) Figure IV isthe completed 
 sketch. It is very good representation of the groimd. 
 
TOPOGEAPHY, MAP BEADING, AND EECONNAISSANCE. 
 
 PEACTICAL WORK. 
 
 18. At this point in his course, the beginner will be little benefited 
 by j>ractice with sketching instruments, or by going out and merely 
 looking at the ground. The instructor should, if practicable, have 
 several countors taped out on the ground with the particular purpose 
 of showing the -class how the contours run up the valleys and around 
 the spurs. He will also require each member of the class to trace 
 Figure I and, by following the text without reference to the other 
 figures, to interpolate the contours, on his tracing. The instructor 
 will point out errors in method; particularly will he see to it that 
 
 873. , ^-^ 
 
 872 X J , / 
 ^70 J ^ f 
 
 330 \ 1; 
 
 870^ >i 
 
 
 £, 
 
 -< 
 
 -^ -v 
 
 850 ^ "^""^ 
 
 ■^ / 
 
 T" 
 
 \ 
 
 1- 
 
 •JrBSO 
 
 
 
 870 
 
 \ 
 
 N. 
 
 /" 
 
 885 ^70 
 
 
 \ -x^80 
 1 
 880 
 
 
 /\ 
 
 892 
 
 
 
 44S "' 
 
 FIGURE II 
 
 interpolation be made along the main stream lines j^rsi and then along 
 the tributary streams and that interpolation is along ridge lines, 
 as the ridges point. 
 
 LESSON n. 
 
 MISTAKES IN DEAWING CONTOURS. 
 
 19. The characteristics of contours most frequently violated are 
 listed in paragraph 20. These characteristics musf be studied in the 
 light of what has ieen said in Lesson I. There is no particular purpose 
 in one's committing these rules to memory. What is necessary is that 
 the beginner learn the reasons why contours have these characteristics. 
 On Figure V notation of contour characteristics are noted by numbers, 
 which refer to subnumbers of paragraph 20. _ It wiU be noted that 
 the main stream is shown much steeper in its lower reaches than 
 upper. This might be so^ but is unusual; therefore should not be 
 
8 
 
 TOPOGRAPHY, MAP REAUING, AND BECONNAISSANCE. 
 
 SO shown unless indicated by data on the skeleton. Contours 
 marked 6 should turn up the stream, lines. Contour marked 1 
 crosses same stream twice. It can not, therefore, be a level line, as 
 the stream has some fall. What is 3, a summit or a depression? 
 Contours marked 11 follow no system whatever* ground shaped by 
 water erosion will not be thus. The one marked. 11a (Hit he a con- 
 tour) indicates that one can walk directly across^ country from one 
 stream to another on a level line. Womd the streams be here if 
 this were possible? Imagine yourself walking on contour marked 
 12; will you always have lower ground on the same side? 
 
 FIGURE III 
 
 CHARACTERISTICS OF CONTOURS. 
 
 20. (1) AH points on any one contour have the same elevation. 
 It is a level line. 
 
 (2) Every contour closes on itself, either within or beyond the limits 
 of the map. In the latter case the ends of the contour Hne will run 
 to the edge of the rnaf. 
 _ (3) A contour which closes within the limits of the map indicates 
 either a summit or a depression. In a depression there wiU usually 
 be found a pond or a lake. If there is no water, the depression must 
 be indicated in some way to differentiate it from a siunmit. A usual 
 method is to hachure the inner side of the depression contour. 
 
 (4) Contours never spht and never cross each other except in the 
 case of an overhanging diff, in which case there must be two dis- 
 tinct intersections. These cases are not common. 
 
 (5) Contours are spaced equally to represent a uniform slope. 
 If the slope is a pjane surface (i. e., if it has no irregularities due to 
 erosion or other cause) the contours are parallel and straight. 
 
TOPOGEAPHY, MAP EEADING, AND EECONNAISSANCE. 
 
 9 
 
 (6) Incrossing a valley tlie contours run uj) tlie valley on one side 
 turning at the stream run back on the other side. In crossing a ridge 
 the contours run to the ridge line and, turning, run back on the other 
 side of the ridge. 
 
 (7) Contours are always at right angles to the lines of steepest 
 slope. They, therefore, cross the stream lines and the ridge lines at 
 right angles. 
 
 (8) The contours are farther apart at the top and hottom of an 
 eroded hill than near the middle, because in these portions the slope 
 is somewhat flatter. 
 
 (9) Contours are usually closer together near the sources of streams — 
 as a stream is usually steeper near its source. This is not always so.: 
 
 FIGURE IV 
 
 A stream may have at its source a very flat collecting basin, a lake, 
 or pond. - 
 
 (10) The larger the stream, the flatter the slope in the usual case. 
 Hence, contours are usually closer together on tributaries than on 
 main streams. 
 
 (11) Bad shaping of contours is usually due to iUogicar interpo- 
 lation between critical points. Remember that the arainuge lines 
 and ridge lines are master lines of your contours. Interpolate along 
 drainage lines first, beginning on the main lines and going to the 
 tributaries. Then search out your ridge lines and intdrpolate along 
 the lines of the ridges. 
 
 (12) If one has difficulty in tracing out a particular contour, he 
 may be helped by imagimng himself to be walking along the con- 
 tour. If he starts out with low land on his right ham,, he wiU always 
 have lowland on his right hand as long as he walks that contour m 
 that direction and vice versa. 
 
10 TOPOGBAPHY, MAP HEADING, AND RECONNAISSANCE. 
 
 PRACTICE WORK. 
 
 21. Figxires VI, VII and VIII are sketches giving the drainage 
 net and critical points of three areas in the vicinity of Fort Leaven- 
 worth, Kans. The instructor wiU require each member of the class 
 to trace these skeletons one at a time and to draw in the contours 
 following the systematic method heretofore described. 
 
 There can be no better training for one who is studying topography, 
 sketching, or map reading. Not xuitil one can complete a skeleton 
 such as these, with great readiness, is he able to take up the reading 
 of or sketching of topography. It is not considered desirable for 
 anyone to take up the study of Lesson III until he has mastered 
 
 MISTAKES IN DRAWING CONTOURS 
 FIGURE V 
 
 "logical contouring." It may be necessary, therefore, to devote an 
 additional lesson period to practice in drawing contours. 
 
 LESSON in. 
 
 SKETCHING. 
 
 22. The instruments issued by the Engineer Department for sketch- 
 ing are based on the nlane-table method. They are simple. Skill 
 in their use can not, of couree, be acquired without practice, but one 
 can begin to use them with very little preHmiaary instruction. 
 
 It is the beginner's lack of a general plan of operations, his lack 
 of appreciation of the essentials, his waste of effort and patience on 
 the nonessentials, that make sketching the Mte noire of a soldier's 
 traiiung, rather than the admitted inaccuracy of hand instruments 
 and his lack of skiU in their use. 
 
TOP'OGKAPHY, MAP BEADING, AND EECONNAISSANCE. 11 
 
 670 
 
 f870 
 
 860 : : 
 
 eiO' 
 
 797 
 
 DRAW lO'FT. CONTOURS ON THE ABOVE SKETCH 
 LOWEST CONTOUR 790 
 
 FIGURE VI 
 
12 
 
 TOPOGRAPHY, MAP BEADING, AND EECONNAISSANCE. 
 
 The following plan of operations, if adhered to, wiU produce good 
 work. It would be well if the instructor could select an area and 
 lead his class around and through it as described, explaining the text 
 to them as they proceed from point to point. The first step to be 
 taken is to prepare a slceleton.oi the area entirely similar to those 
 shown in Figures I, V, "VT, VII, and VIII. The next step is to draw 
 the contours and finish the slcetch in all of its details. 
 
 .917 
 
 960 
 
 962 
 
 930 
 
 878 
 
 >i880 
 
 780 
 
 DRAW 20-FT CONTOURS ON THE ABOVE SKETCH 
 LOWEST CONTOUR 780 
 
 FIGURE VII 
 
 PEEPABATION OP THE SKELETON SKETCH. 
 
 23. We havfr learned from our practice in drawing contours wTiat 
 lines and wJiat points are essential on the skeleton. The problem 
 now is how best can these be located. It would not do, of^course 
 to attempt to traverse out every stream system and every watershed 
 to its minute details. This would take forever, and would be con- 
 trary to the first priuciple of aU mapping, which is to treat the area 
 as a unit; "proceed from the whole to the part." There are several 
 good methods of procedure, but most of those usually explained are 
 for the use of the man who has already become a good sketcher. The 
 
TOPOGEAPHY, MAP BEADING, AND EECONNAISSANCE. 
 
 13 
 
 following method is for the beginner. It may be modified as he 
 becomes more expert. 
 
 24. First, traverse around the area, returning^ to the point of begin- 
 ning. In traversing, locate — 
 
 (a) Every drainage line crossed or which runs along generally 
 parallel to the line of the traverse and note the direction m which 
 the stream flows (or the dry creek drains) . This is absolutely neces- 
 sary. 
 
 (h) Every house or other easily identified feature near the road, 
 which wUl enable the sketcher later to identify himself on the sketch. 
 
 930 
 
 BAOx 
 
 780 
 
 PRAW20-Fr. CONTOURS ON THE ABOVE SKETCH 
 
 LOWEST CONTOUR 78b 
 
 FIGURE VIII 
 
 (c) The high points between drainage lines, i. e., the point on the 
 ridge line where the traverse crosses the ridge line and a point or 
 so on each side of the traverse on the ridge to locate the ridge line 
 as well as may be possible. 
 
 . (d) Determine the elevations of aU of these points. Bo not draw any 
 contours yet. Even the experienced sketcher should be slow to put 
 in contours until he has reconnoitered enough of the area to mn 
 down the drainage system. He m,ust wait until he has seen enough of 
 
14 TOPOGRAPHY, MAP BEADING, AND RECONNAISSANCE, 
 
 the country to Tcnow how the lig features lie. The little ones will take 
 care of themselves later as the work progresses. 
 
 25. Second, adjust the traverse. When in the circuit traverse the 
 sketcher has reached a point from which the point of beginning can 
 be seen, it will be found that when the instrument is oriented the 
 ray to the point of beginning on the ground will not pass directly 
 through the point of beginning as plotted on the paper. This is to 
 be expected. K the error in direction is not out more than a degree, 
 let it go at that for the time being and pace the distance to the point 
 of b^ioning. Lay this distance off on the ray as actually drawn. 
 
 There wiU he an error of closure. If there has been no opportunity 
 to check orientation or distance in the circuit, it will be good work 
 if the closing error is not greater than 5 per cent of the entire length 
 of the traverse. Providing the error in direction cannot be accoimted 
 for by local attraction at any pointy nor the earor in distance by incor- 
 rect plotting of any course, aisffihtie the error by moving points of 
 traverse. 
 
 26. It wiU not be necessary to go into refinements in this adjust- 
 ment. Move the last point up to the point of beginning; the next 
 to last point but one in the same direction but a lesser distance, etc. 
 Be sure in doing this that no two courses have a plotted angle between 
 ihem which differs noticeably from the actual angle on the ground; other- 
 wise your sTcetch wiM mislead some one. The sketch must have no 
 Tnisleadinf errors. It wiH have errors, but they must not be those 
 which will mislead. 
 
 27. Third, traverse througli the area. The sketcher has now been 
 aVound the area. He has located all the water that flows into the 
 area and aU that fiows out. He has tied down his drainage system 
 pretty weU. He mil have noted the relative size of the streams or 
 drainage lines and their elevp,tions and this will give him a good 
 idea as to which ones are the jnain drainage lines cf the areas. He 
 must now find out how these drain/me lines connect up inside his 
 area. In open country he may be able to locate the stream junc- 
 tions in the interior by intersections from the traverse. Ordinarily, 
 however, these stream junctions are^iidden by tree growth or folds 
 of the ground. It never does in sketching (at least for a beginner 
 and seldom for an experienced man) to guess how the drainage runs 
 through the area. TJt^e is no royal food to stream junctions. 
 
 If the sketch is to b© a truthful report of a topographic reconnais- 
 sance, the sketcher must make the reconnaissance. Let him traverse 
 into the iTderior aaAfind out what is there. 
 
 Here is where the had misleading mistaTces of the sTcetcJi made hy an 
 untrained Toan are apt to he made. Contrary to all advice he has 
 probably attempted to contour as he went along. He has drawn 
 in a number of ground forms as they appeared to him as he went 
 along on his circuit traverse. These forms are ustiaUy exaggerated 
 in size. Every hiU looks hig when one is on top of %t. His forms 
 therefore project into his area much farther than tbey should. When 
 he gets inside, he must either rub them aU out or change his draiaage 
 lines, which he finds by reconnaissance to be thus ana so to so and 
 thus to correspond to his badly built ground. The result is worse than 
 useless ; it wiB deceive the user of the sketch. 
 
 28. The manner in which the drainage connects up inside the area 
 can only he found hy goin^ into the area and contouring must wait on 
 
TOPOGBAPHY, MAP EEABING, AND BECOSTNAISSANCE. 15 
 
 this information. In entering the area follow a road, or trail (the 
 location of such features are needed anyway) and the sketcher 
 traverses as he goes. One traverse across the area may not b© enough; 
 usually will not be enough, though if the Kne followed is well con- 
 sidered, short offset traverses from the cross traverse will serve to 
 locate the main drainage lines of h small area (less than a square mile). 
 
 In these cross traverses it is sometimes best to follow the main 
 vaUey and sometimes best to foUow the ridges. The knowledge of 
 the area as gained by the circuit traverse will help determine which 
 is the easiest route to traverse to get the stream functions and ridge 
 line points needed. But any line at aU will do if one ia careful to 
 run as accurate a traverse as he can and remembers what he is looJc- 
 ingjor. He must never run a haphazard traverse and be satisfied 
 with the information found along it. 
 
 He must get off the liae by offsets if necessary to get vital informa- 
 tion such as stream junctions and stream heads or spur directions. 
 
 The beginner should never leave his instrument on the line of 
 traverse and wander off to see what he can see. If he does he will 
 plot it wrong when he gets back. He m,ustpick up the instrwnmt 
 and take it with him, traversing as he goes. He may find a gold riaine 
 in the way of an outlook which wiH open up the entire countiysiSe 
 to his view and enable him to check tack and insure his orientation 
 and distance. 
 
 29. Fonrtb, adjust. Now the student has been ar<mn4 his area 
 and has been irmde and searched out and located his streaihjmnctvims 
 and ridge lines. He has a skeleton much like the ones hei Has con- 
 toured by interpolation. But unless he has been adjusting his ele- 
 vations all the time he wiH have some very peculiar levd data. 
 He will find that a stream which is 200 feet high on entering his area 
 and 150 feet h^h as it leaves, has a most unaccountable elevation 
 of 148 feet in the interior of his area. He wiH also have found on 
 his first circuit that the elevations did not close and he wiH have ad- 
 justed this by dropping back. In other words, as he arrives on his 
 circuit at the point where he began, he will find that he is 20 feet 
 above the proper elevation. This is not bad if his circuit has been 
 IfHig. It is to be expected with a clinometer. Of course, he accepts 
 the elevation of the point, of beginning as correct; changes the last 
 station by 15 feet, the next to the last but one by 10 feet, the next 
 to the last but two by 5 feet, and leaves the rest as they are, or he 
 may carry the distribution of the error down a little finer by taking 
 in more stations and making the change more gradually, but any 
 reasonable adjustment is all right, providing, of course, he main- 
 tains eZ«w»^*ons«;i<Am sight of each other relattvely correct. 
 
 He must not in his adjustment make a stream dee^ where it is 
 shallow or vice versa nw must he change a hollow to a nse. Similarly 
 he must adjust his cross lines. When he crosses a stream line, two 
 points of which he has alreadj located in his circuit traverse, he must 
 know that he should get an elevation of his crossing point correspond- 
 ing to the relative distances to the two points already determined 
 (unless there is in betwe^i some feature such as a waterfall or rapids 
 or a dam . These things will discover themselves if the sketcher is on 
 the lookout, for them). 
 
16 TOPOGRAPHY, MAP BEADING; AND RECONNAISSANCE. 
 
 30. The reconnaissance sketcher has few, if any, points located 
 for him with precision. He must, therefore, use all of his knowledge 
 of ground, of water flow, of railroad grades, etc., to keep his work 
 logical and consistent. He must know that large rivers do not fall 
 more than a few iaches to the mUe (unless fuU of rapids) ; that within 
 the area hkely to be covered by a reconnaissance the fall of a large river 
 will be so little as to be not readily ascertainable with reconnaissance 
 instruments. He must know that any water-carrying drainage line 
 (that is one carrying water the year around) is apt to have a lesser 
 fall than a dry guUy in the same vicinity and his acg^uaintance with 
 the region being sketched wiU (if he observes) give him some idea as 
 to the approximate fall per mile. He must know that as a reason- 
 ably true proposition tributaries have a greater fall than main wator 
 lines. His work must be continually required to answer such tests. 
 If it does not he must be ready to give the reason. 
 
 CONTOURING THE SKELETON. 
 
 31. It is assumed that the student has now a logically correct 
 skeleton of the drainage system and ridge lines. He must, of course, 
 be sure that he has in all the roads and trails of military importance. 
 He must not show these connecting up and running through unless 
 he knows they connect up or run through and where they connect 
 up and run. He must always remember that he is making arecon- 
 naissance and reporting the result of his reconnaissance, not filling 
 up inches on his sheet with conjectures. He must leave a blank if 
 he does not Tcnow. The student should now take this skeleton and 
 contour it without going back over the country. (I hear the objec- 
 tion, "I thought you should not draw what you do not see or can 
 not remember. I thought one should never sketch in contours except 
 in the face of facts.") Well enough, for the man who can do this, 
 but the reconnaissance sketcher must learn. He can not follow the 
 methods of the experts at once; By requiring him to contour his 
 sketch, the data of which is incomplete, as of course it will be with the 
 beginner, he wiU find out what essential data he has left out. He 
 will find, that he forgot to get the elevation of a hUl, or the direction 
 line of a ridge or a stream junction • he will have forgotten whether 
 a certain stream had banks 10 feet high or 50 feet high; he will find 
 that he left out the direction of the flow of a certaia stream, and for 
 the life of him wiU not be able to remember which Way it did flow. 
 All of this will teach him to put down his data in sufficient fuUness 
 so that he will be able to contour correctly from the data on his sheet 
 at least as correctly as the time spent on his reconnaissance will 
 justify. 
 
 32. After the beginner has done a number of area sketches in this 
 way he wiU have arrived at the time when he can sketch in form lines 
 as he goes along (hghtly, because they wiU be later rubbed out) and 
 later can, after he has covered enough of the area to find the trend of 
 the big features, complete his work as he goes along. But all of this 
 presupposes that he has become so expert in his traversing that he 
 wiU need to do little adjusting and that his eye for ground has devel- 
 oped to the point where he can see the forms and knows how to 
 j-epresent them on paper. Even then he will be tripped up occa- 
 sionally and be obliged to rub out much work if he begms to contour 
 
TOPOGRAPHY, MAP EEADING, AND BECONNAISSANCE. 17 
 
 too early. One can contour as lie goes along if he is working with 
 accurate instuments and is sure that no adjustment will be needed 
 later, but when working with reconnaissance instruments he is very 
 apt to get iato trouble. 
 
 LESSON IV. 
 
 33. In Lesson III there was described as fully as possible the 
 procedure to be followed by a beginner in executing an area sketch. 
 This procedure was given before the instruments and the technique 
 of their use were described for two reasons: 
 
 First. It was felt that many of the officers and men under instruc- 
 tion will have had previous experience ia the use of sketching instru- 
 ments and that, therefore, they should proceed in the third lesson 
 period as outlined. 
 
 Second. As the main purpose of this manual is not to teach the 
 use of any particular set of instruments, but rather to teach the prin- 
 ciples of topography, it was believed that the dry detail about instru- 
 ments, scales, etc., might well wait until the class had absorbed a 
 proper idea of what a contoured sketch is and what the procedure 
 m learning to make one should be. 
 
 FIELD SKETCHING INSTRUMENTS AND THEIE USE. 
 
 34. The Engineer Department has designed a standard field 
 sketching outfit, based solely upon the flane table method of sketching. 
 The entire outfit packs conveniently into a carrying case. The sup- 
 plies which are expendable consist of pencUs, erasers, sketching 
 paper, and celluloid. The celluloid is for use in the raia. The per- 
 manent equipment consists of a small plane table with folding 
 wooden tripod, a service clinometer, a pace tally, an alidade, pencu 
 pocket, which pins onto the service coat, and a holder for carrying 
 timirig pad in. mounted work. 
 
 THE ALmADE. 
 
 35. The alidade supplied is used for three purposes: (1) To deter- 
 mine differences in elevation from the scales of slopes provided; (2) 
 for use as a sighting vane; (3) for use as a measuring scale, there 
 being provided in addition to the scale of laches, blank spaces for 
 pasting on individual scales of double paces, etc, 
 
 SCALE OF STRIDES. 
 
 36. Before proceeding to sketch it is of course essential for each 
 sketcher to secure or construct a working scale. If the work is to 
 be done dismounted the scale should be in strides (double paces). 
 To prepare a scale of your strides for a sketch which is to be made 
 6 inches equal 1 mUe, proceed as follows: 
 
 Measure oflf a ratiag course of 1 mUe, f mile, or ^ mUe. The longer 
 the course the better. It should be over average ground (up and 
 down hiU) similar to that which is to be sketched and should be 
 carefully measured. Walk this course both ways several times, 
 counting strides with the pace tally. Take the average for this dis- 
 tance and from it find the number of strides to 1 mile. Say the 
 
 58740°— 18 2 
 
18 TOPOGRAPHY, MAP BEADING, AND BJECONNAISSANCE. 
 
 number of strides for 1 mile is 1,048. The scale of the sketch is to 
 be 6 inclies equals 1 mile, therefore 6 inches on the map will represent 
 1,048 of your strides. If you divide 6 by 1,048 you will have the 
 scale length for one stride. It is too small a division to be useful, 
 so multiply by 100 and find length for 100 strides is 0.57 inch. Pages 
 69 and 70 show working scales for the ordinary range of topographic 
 sketches. 
 
 With a scale of equal parts (one reading inches and tenths will do) 
 lay off as many divisions as you may desire in your scale. Each of 
 the divisions will represent 100 of your strides. Divide the left 
 division into 10 equal parts with your eye. Each of these wUl then 
 be 10 of your strides. No smaller distance can be well measured off 
 on your sketch. 
 
 SLOPE SCALES. 
 
 37. The alidade, as has been mentioned, in addition to the blank 
 spaces for the individual stride scales, is provided with slope scales, 
 (These are sometimes called inaf distance scales, which is confusing.) 
 These scales are used in sketching, in connection with the clinometer, 
 to find the differences in elevation between two points, the vertical 
 angle from one to the other having been read by the clinometer. 
 
 The slope scales give results which are sufficiently accurate for 
 sketching purposes. They are not reliable for angles greater than 15° 
 
 Each interval on the slope scale represents the distance on the sketch 
 (scale 6 inches to the mile) corresponding to a rise of 10 feet for the 
 slope ^ven. Working on a 6-inch scale, therefore, the method of 
 determining the difference in elevation between two points is as 
 follows: From the point of known elevation, read the slope to the 
 point the elevation of which is to be determined. Say, it is 3°. Pace 
 to the second point to find the distance and plot the distance. Look 
 on your alidade for the slope scale of 3°. (The figure representing 
 the slope is engraved at the midpoint of the interval for that slope.) 
 Apply the slope interval for 3° to the plotted distance and find out 
 how many times this interval is contained in the plotted distance. 
 Multiply this number and fraction (there will usually be a fraction, 
 which must, of course, be estimated) by 10 feet and the result is the 
 difference in elevation. Siinilarly, for other clinometer readingjs, 
 take the slope interval corresponding to the clinometer reading and 
 find out how many times it is contained in the plotted distance, and 
 then multiply by 10 feet. 
 
 38. This same slope scale may be used for any scale, providing 
 that instead of multiplying the number of intervals and fractions 
 thereof by 10, we multiply hj the rise for that scale. The rise for 
 any given scale is found by dividing 60 by the number of inches which 
 represents the mile. For instance, the rise for a 3 inch to the mile 
 scale is 60/3, or 20 feet; for 4 inches to the mile scale it is 60/4 or 15 
 feet; for | inch to the mile scale, 60/^, or 120 feet. The method is 
 the same, the slope scale is the same, no matter what the scale. The 
 only thing that cnanges is the rise per interval, and this rise per inter- 
 val is found as indicated above. 
 
 39. It wiU occur to the instructor or to some of the sketchers that 
 to hunt around the alidade for the slope scale corresponding to a 
 particular clinometer reading is rather a nuisance, and so it is. 
 
TOPOGEAPHT, MAP BEADING, AND BECONNAISSANCE. 19 
 
 This diffictdty may be obviated by constructing " a scale of differ- 
 ences" and pasting it on the blank space of the ahdade not used for 
 the stride scale. It is constructed, marked, and used as follows: 
 
 On a suitable piece of paper lay off a line; on this liae lay off 
 seTeral intervals equal to the 1-de^ee slope scale interval. These - 
 are about 0.65 inch each. Mark the division points, 10, 20, 30, 40, 
 etc. Divide each interval into 10 parts (the 10-degree slope iaterval 
 will do this about correctly), cut and paste on ahdade. If working 
 on a scale of 6 inches to the mile, this difference scale is used as fol- 
 Ibiire: Read the vertical angle with the clinometer. Pace the dis- 
 tance. Plot the distance. Say the clinometer read 1° exactly; 
 apply the "scale of differences" to the plotted distance between the 
 points and read off directly the difference in elevation. If the 
 chnometer read 2^°, apply the scale as before, but multiply the dif- 
 ference in elevation as secured directly (which is that for a 1° angle) 
 by 2J° to secure the difference in elevation for a 2J° vertical angle, 
 lliis same "scale of differences" may be used for any other scale in 
 sketching by simply multiplying the result as above secured by the 
 proper scale factor. The proper scale factor is found by dividing 
 six by the number of inches to the mile in the scale being used! 
 For a 3-inch scale, multiply by 6/3, or 2 ; if worMng on a 2-inch scale, 
 multiply by 6/2, or 3. Such a scheme saves time and worry and is 
 comparable in convenience to using a percentage clinometer and a 
 per cent scale. 
 
 It wiU be seen that the method simply makes universal use of the 
 1-degree interval, instead of a different interval for each vertical 
 angle. This is not too inaccurate for sketching use. 
 
 Ctf coT^rse, if using aijother scale than 6 inehes to the mile for any 
 considerable amount of work, the "scale of differences" can be re- 
 marked to correspond. That is, change the readings on scale by 
 multiplying by 2 for a 3-inch scale, by 3 for a 2-inch scale, etc. 
 
 SERVICE CLINOMETER. • 
 
 40. To use the clinometer, the observer sights the object through 
 the peephole and, at the same time, sees the scale of degrees inside 
 the rim. The red figures are minus (depression angles), the black 
 ones plus (elevation aiigles) readings. The penduliim is allowed to 
 swing freely by shding back the bar and pressing the stop. The 
 reading, is tai^n when the desired point is sighted aJad the pendulum 
 has come to a rest. ITie pendulim should not be stopped to take 
 the reading as this may displace the scale. It is best to rest the 
 instrument or the hands holding it against a tree, fence, post, or other 
 object while taking a reading. Theoretically the reading should be 
 made to a point about the same height above the ground as the 
 observer's eye. Actually this is very difficult to do and it is best to 
 always read to the ground line and then, after ascertaining the differ- 
 ence in elevation, make allowance for the height of the eye above 
 the ground. The clinometer should be tested occasionally as fol- 
 lows: Hold the instrument, stop uppermost, against a post or tree 
 (point marked by a piece of paper pinned to it) and read to another 
 point marked on another tree, 50 or 60 feet distant, and note 
 the reading. Reverse this operation and read from the second 
 mark back to the first one and note reading. For example, the first 
 
20 TOPOGEAPHT, MAP KEADIITG, AND BECONNAISSANCE. 
 
 reading is minus 2°, the readiag back is plus 1°. Add the two 
 readings and divide by 2 and you have the correct reading (1^°) 
 of the slope between the two points. The clinometer then reads 
 ^° too much on minus and J° too little on plus readings. Reading 
 must be corrected accordingly. 
 
 41. The sketching board is a simple board with a magnetic needle 
 set ia a trough in one edge. The needle is 3 inches long and quite 
 sensitive. No plumb bob is provided, but a hole is accurately bored 
 so that a plumo bob can be improvised and use made of the slope 
 scale on the board in case the clinometer is lost. The plate on the 
 back of the board is let in flush, so that the board can be turned freely 
 on the tripod for orientation, and then firmly clamped by a slight turn 
 of the tripod screw. As the tripod is not used in mounted work 
 holes have been bored at the comers of the board for the insertion 
 of a carrying cord if desired. 
 
 TO ORIENT THE BOARD BY COMPASS. 
 
 42. Set up the tripod with board loosely screwed onto it and 
 level the board by eye by moving the tripod Ifegs. Free the 
 needle by turning the cam and then turn the board slowly around 
 until the needle swings from side to side in the trough. Let 
 the needle settle, turning the board as necessary so that the needle 
 when settled Ues directly along the north and south line (the median 
 line of the trough). Without changing the position of the board 
 reach under and tighten up the screw of the tripod. Care must 
 always be taken not to turn the tripod screw too tight, as the 
 threads are likely to be started by rough treatment. T^ie board is 
 now oriented. Draw a line parallel to the needle on the paper and 
 mark the north end with a naif arrow. Above this write M. M. or 
 N". (magnetic meridian or' north). 
 
 TO ORIENT THE BOARD BY BACK SIGHT. 
 
 43. Having plotted Gocated and drawn in) a station and arrived 
 at a point fartner on to which you have sighted and drawn a line, 
 set up the tripod and board, measure ofi the number of strides taken 
 between the two points on yoiu" line and stick a pin in the point 
 found. This is your present position. Place your ruler against the 
 pin and along the line between the two points and turn the board 
 imtil the station you have just come from is sighted. Tighten up the 
 tripod screw without moving the board and you are oriented. 
 Verify this by reading the needle. 
 
 TO LOCATE A POINT BY TRAVERSING. 
 
 44. By this is meant measuring the distance between two points 
 by counting strides, or time, or travel reqiiiced to pass from one to 
 the other. The term traverse is apphed to the route followed by the 
 sketcher in making the sketch. Bemg at point A (whose position is 
 plotted on your sketch), with board oriented, stick a pin m point A 
 on the sketch. Lay alidade alongside the pin and pivot it around 
 until point B (the point to which you are to traverse) is sighted. 
 Verify the position of the needle and then draw a ray toward b! 
 Move to B, counting strides with pace tally, and upon arrival set up 
 
TOPOGEAPHY, MAP BEADING, AND RECOKNAISSANCEi 21 
 
 the tripod, orienting roughly by a back sight on A. (Do not waste 
 any time on this.) Lay a ruler along the ray, zero of scale of strides 
 at A, and lay off on the ray the number of strides that your tally 
 rerister shows was taken between the two points. The point marked 
 isB. 
 
 TO LOCATE A POINT BY INTEKSECTION. 
 
 45. Assume that from point A, with your board oriented, you 
 took a careful sight and drew a ray toward a church a few hundred 
 yards off to the side. After arriving at B and plotting its position, 
 you carefully orient the board. Pivot the ruler around the pin in 
 B until the church is sighted, then, verifying the position of the 
 needle, draw a ray. toward the church. The intersection of this ray 
 and of the one you took from A is the sketch position of the churofi 
 
 TO LOCATE BY RESECTION. 
 
 46. This is determining the sketcher's position (the reverse of inter- 
 section) by orienting the board and drawing rays toward himself 
 from two or more pomts whose positions are aJready determined and 
 plotted. Having previously plotted the position of points AandB, 
 the sketoher comes into his area later, at some pomt from which 
 those two poiats can be seen. His present position is not yet deter- 
 mined. Tb determine it, set up the board and carefully orient by 
 compass, then stick pins in the sketch positions of A and B. Pivot 
 the alidade on the sketch position of A, sight A, and draw a ray. Do 
 the same with B. The intersection of the two lines is your position 
 
 TO EESECT FROM THREE KNOWN POINTS. 
 
 47. In the previous paragraph a method of resection from two 
 known points was given. For this a needle orientation is necessary. 
 At times local attraction may deflect the needle and the three-point 
 resection may be useful. The simplest method is by use of a trans- 
 parent or translucent paper (celluloid wiU do splendidly). The 
 process is as follows: Pm the paper on the board by thumb tacks. 
 Stick a pin in it anywhere at a convenient point to represent your 
 station (pay no attention to what may be on your sketch). Pivoting 
 on the pm, draw, with great care, rays to each of three points which 
 are in view and which also appear on your sketch. Lift the paper 
 off the board. You have on it three rays, which make with each 
 other the proper angles, considered from your position. AU you 
 need now to do is to put the tracing paper over your sketch, and by 
 looking through to your sketch shift the paper until each ray passes 
 throu^ the sketch representative of the object to which this ray was 
 drawn. As soon as each of the rays passes through its proper point 
 (which will take a deal of fussing around the first time) you have it. 
 Prick the point of intersection through to your sketch and the pinhole 
 will give you your plotted position. This has been aU done without 
 orienting the board. If we wish to work from this station, the next 
 step is to orient. This is done by back sighting to one of the three 
 objects. Remember, this is but a cut-and-try method. It may take 
 considerable time, it may help much at times. 
 
22 TOPOGEAPHY, MAP BEADING, AND EECONNAISSANOE. 
 
 LESSON V. 
 
 48. The class by this time has progressed to the point where a 
 great majority know what to look for m maMng an area sketch and 
 Snow how to use all of their sketching instruments. It is now 
 beheved essential, at the risk of some repetition, for the instructor 
 to give a special talk on the technique of the skdcher's traverse, the 
 detail which should be shown on the sketch, and a few hints which 
 may make for dearrvess, 
 
 TECHNIQUE OF A SKETCHER'S TRAVERSE. 
 
 49. Set up the board at the point of beginning, station 1. Free 
 the needle. Swing the board until the needle comes to a rest along 
 the median line of the declinator trough. The board is now oriented. 
 Clamp the board by tightening the screw holding the board to the 
 tripod. The billiard-cloth covering of the tripod nead will hold the 
 board in position. Do not use an excess of force in clamping. Plot 
 a point (use a pin to pivot on) on the paper to represent the point 
 of beginning, tlace tiie alida,de on the board. Pivot it on the pin 
 representing station 1. Sight along the line to be traversed to sta- 
 tion 2. It IS advisable where possible to sight on some well-defined 
 object, as a tree, fence, comer, etc. This is necessary when the 
 course from 1 to 2 is short. If tbe course is a long one, and follows 
 a well-marked road or trail, a sight along the mid Ene or edge of the 
 road will serve. Draw the ray. Pencil should have a chisel point 
 and should be held vertically so that the pencil line will follow edge 
 of ahdade as closely as may be. Always see that needle is along mid 
 line of declinator trough when ray is drawn. Put in local data around 
 station 1. Not necessary to draw rays to all poin'ts to be plotted. 
 This fills Tip the sketch with lines. Point the alidade at object, and 
 having ascertained the distance to it by pacing or by estimatii^ 
 plot the object in its proper place at once. Do Tiot plot any points 
 ahead of station 1 along the line of tro/verse. Draw rays to well-defLned, 
 easily "vdentified distant objects, noting on the ray the object to which 
 drawn. 
 
 A ray to these same objects from another point on the traverse 
 will serve to locate the objects, and once located they will serve as 
 excellent checks upon orientation or may serve to locate the instru- 
 ment by resection when for some reason or other the location is 
 otherwise unknown. The tendency with the beginner is to draw 
 rays to everything in sight. This is to be discouraged, as he loses 
 much tune m. gettmg the rays— usually gets them confused so that 
 all are useless-^and seldom is he able to identify more than a very 
 few of the objects so sighted. Read the clinometer, reading from 
 station 1 to the objects sighted, noting the slope on the ray. Eead 
 the clinometer to station 2 last of alL Most texts tell you to sight 
 at a point at station 2, which is the same distance above the gromid 
 as is your eye in taking the sight. It is much better to pick out a 
 definite point to sight on. In other words, sight at the top of the 
 chimney of a distant house, the lower window sdi of a nearby house 
 the base of a tree, or, failing anything else, the ground line. Now' 
 the slope you have read is the slope from your eye to the point 
 sighted. You know the elevation of your eye. After getting the 
 
TOPOGRAPHY, MAP BEADING, AND KECONNAISSANCE. 23 
 
 difference in elevation between eye and object sighted this difference 
 is added to or subtracted from the elevation of your eye and the 
 result is the elevation of the object sighted. From this the elevation 
 to the ground level at station 2 is found. 
 
 50. Set yoTir pace tally to zero, pick up your equipment, and start 
 off in the direction of station 2, waUdng with your customary step. 
 Do not try to make your steps uniform. Walk naturally. Yonr 
 scale should have been made to read double strides while you were 
 walking natiirally over country similar to that which you are to 
 sketch. Press the pace tally every time your right foot strikes the 
 ground, pace tally in your leit hand. Whenever you reach an object 
 which should appear on your sketch, as a house, bridge, stream line, 
 ridge line, etc., do not take the time to set up and orient unless 
 necessary. This will only be necessary when you wish to get a 
 direction with care, as in the case of the direction of an important 
 stream or ridge line. If the object is a house close to the roadside, 
 it should be plotted in at once without set-up or orientation. 
 
 51. Arrived at station 2, set up. Scale off number of strides 
 shown by your pace tally between stations 1 and 2 on ray already 
 drawn. Mark station 2 by small circle with dot in center; elevation, 
 as soon as found beside the circle. To find elevation of 2. Sight 
 back to 1 with clinometer. If reading is not the same as it was from 
 1 to 2, take the mean (providing of course line of sight is from ground 
 to ground, or allowance has been made for elevation of eye in each 
 case). Now hunt for the proper slope scale for the slope read. Apply 
 the interval to the plotted distance from 1 to 2, to find the number of 
 intervals and fractions thereof contained in the distance. Multiply 
 this number by the rise per interval, 10 feet for 6-inch scale, 20 feet 
 for 3-inch scale, etc. As soon as station 2 is plotted and its elevation 
 foimd, orient board by compass and check by back sight. Then 
 
 Eroceed as before. (The elevations of all points on the traverse 
 etween 1 and 2 should of course be noted as well as the elevation of 
 the side shots taken. These are foimd in the same way as that of 
 station 2.) 
 
 In plotting distances avoid as much as possible scaling off long 
 lines by short plotted distances. In other words, keep track of the 
 number of strides from the beginning point of a long tangent or 
 slightly curving road and locate all stations on the tangent by scaling 
 from the first point. This means that the pace tally is only set back 
 to zero at a marked change of direction. Watch your pace tally to 
 see that it does not fail to register. 
 
 DETAIL AND CLEARNESS. 
 
 52. The principal aim of the course so far has been to instruct the 
 student officer in ground forms and the method of representing them 
 on a naap by contours. Every effort has been made to impress upon 
 him and the instniotor aUke the necessity for viewing an area of 
 ground as a unit; the necessity for searching out the system of the 
 tom)grap7i,y before nainor matters are taken up. 
 
 In the sketchiog which has been done by the class, therefore, 
 every effort has been made to compel each individual to $rei around 
 and through his area so that he might pin down the drainage system if 
 nothing else.- Upon this rather inadequate framework he has been 
 
24 ' TOPOGRAPHY, MAP BEADING, AND EECONNAISSANOE. 
 
 compelled to draw in contours at very small vertical intervals. lu 
 puttmg in the troublesome contour lines lie has very likely rubbed 
 out roads, houses, most of his fences, and all of his trees. The result- 
 ing sketch is apt to be characterized more by its intense blackness 
 than by its clearness and miUtary value. 
 
 A military sketch must show the essential military features of the 
 area and as many of the less important miUtary and other features 
 as the scale of the sketch and the skill of the sketcher make possible. 
 Just what amount of detail should be shown on a sketch in any Par- 
 ticular case is a matter of judgment. As one's skiU. in sketching 
 and training in mihtary tactics proceeds^ his ability to determine 
 these questions for himself will grow. 
 
 53. A sketch to be of value must be clear. The essential features 
 must be brought out with bold strokes. Do not fill up the sketch 
 with rude conventional signs for trees, grass, and com. Do not fill 
 it up with a mass of crowded contour lines. While the normal 
 system requires 10-foot contours for a 6-inch scale, and 20-foot con- 
 tours for a 3-inch scale, this system is not to be loUowed slavishly. 
 If the country is so steep that 10-foot or 20-foot contours will crowd 
 each other, use a larger mterval. 
 
 Break contours at the railroads and roads, so that these important 
 lines of communication will stand out. Break them at cliffs and 
 Write the words "chflf 60 feet high." Break them at streams and 
 write "stream 60 feet wide, banks 40 feet high, unfordable except 
 as indicated." Omit no important landmarlcs. 
 
 54. The user of the sketch must be able to locate himseK on the 
 sketch and for that reason landmarks must appear upon the sketch 
 at frequent intervals (if they exist in fact). A tumbled-down shack 
 in a sparsely-settled region, a group of cottonwood trees iu a treeless 
 waste, a waterhole in a semiarid country, a group of large buildings 
 of a pubHo or semipubho institution in the vicinity of a town wul 
 serve to help the user locate himseK. A stone bridge may name a 
 battlefield. 
 
 55. The conventional signs as adopted for use by all United States 
 Government departments, as well as the special sketching signs given 
 in the Field Service Regulations , follow paragraph 155. They must be 
 learned. 
 
 56. At least three lesson periods must now be spent in sketching. 
 
 LESSON vm. 
 
 MAP READING IN THE FIELD. 
 
 67. Map reading is essentially the reverse of map making. One 
 who has been practiced in sketching by the plane-table method 
 should have no difficulty in reading and using a large scale sketch of 
 a small area. The essential features stand out almost at a glance. 
 But a map of a large area of ground can not be read at a glance. It 
 must be measured and studied in much the same way as the ground 
 itself would have to be measured and studied before we become 
 famihar with the country represented. 
 
 Scale. — ^As in making a sketch we need a "worTcing scale," so in 
 studying a map we need a "reading scale." It is jiot sufficient ia 
 reading a map for mihtary pm-poses to glance at the scale given on 
 
TOPOGRAPHY, MAP KEADISTG, AND BECONSTAISSANOE. 25 
 
 the map and then to glance at two points and guess at the distance 
 between them. The units of distance most used in our service are 
 the yard and the mile. It will be useful, therefore, to prepare our 
 reading scale in yards or mUes. Pages 67 and 68 give scales for the 
 usual range of topographic maps. 
 
 58. Examples. — To prepare a reading scale of yards for a map on a 
 scale of 1:80,000. Here 1 yard on the ground is represented by 
 
 gQ QQQ of a yard on the map, 1,000 yards on the ground by ' r.r.(^ = 
 
 1 ' ' 
 
 gQ of a yard or 36/80 = 0.45 of an inch. Lay off a line 4.5 inches long 
 
 and divide it into 10 parts and you have your scale. If a smaller 
 unit than 1,000 yards is desired, divide the left imit into 10 equal 
 parts. 
 
 59. To prepare a reading scale of miles for a map on a scale of 
 
 1 :100,000. Here 1 mUe on the ground is represented by -i nn »Qr> of 
 
 a mUe on the map. One mile is 5,280 feet, or 63,360 inches. There- 
 
 fore the map representation of 1 mUe on the ^otmd is ' or 
 
 0.634 of an inch. Lay off a line 6.34 inches long. It will represent 
 10 miles. Divide it into 10 equal parts and each part will represent 
 1 mUe. 
 
 ORIENTATION AND DIRECTION. 
 
 60. To one who does not know his own position on the map or who 
 does not hold it properly oriented when in use, a map is a hindrance 
 rather than a help. One who is responsible for guimng troops by a 
 map must keep his position on it by TJonstant reference to the map. 
 Too frequently in maneuvers, and even ia actual war, is the map 
 hidden away in a dispatch case. Then, after the column has already 
 gone astray, frantic efforts are made by all concerned to recall the 
 various textbook methods for orienting a map and for locating one's 
 position upon the map. 
 
 PRACTICE WORK IN ORIENTATION. 
 
 61. It is useless to describe methods in a training manual unless 
 the class is actually practiced in them. An indoor reading of the 
 method of making a sketch or of orienting a map will do a beginner 
 little good. 
 
 The following methods of orientation and resection should be tried 
 out in the field by the class under the eye of the instructor: 
 
 First method. — Take a map of the training camp with the magnetic 
 meridian marked upon it. Set up a sketching board on its tripod. 
 Put the map on the board. Shift it until the magnetic meridian on 
 the map becomes parallel to the meridian line of the needle trough. 
 Pin the map down with thumb tacks. Orient the sketching board 
 and resect from two points in view as described in paragraph 46. 
 
 Second metJiod. — ^Using a compass, but not the sketching board. 
 Lay the map on the ground. Lay the sight line of the compass 
 along the magnetic meridian of the map. Rotate map and compass 
 together imtu needle points north. If sight line of compass has 
 
26 TOPOGEAPHT, MAP HEADING, AND EECONNAISSAIirCE. 
 
 been kept along, magnetic meridian of map, the map is oriented. 
 Your location may be f oimd by resecting as described in paragrapn46. 
 Third meihod. — No compass or no magnetic meridian. True 
 meridian given; position mJmown. Point the hour hand pf your 
 watch (held face upward) at the sun^ if in the Northern Hemisphere. 
 The Hne drawn from pivot to the point midway between the outer 
 end of the hour hand and XII on the dial will point toward the 
 south. Shift your map to correspond. This will give a very rough 
 orientation. You may now be able to identify two or three distant 
 points on the ground and their representations on the map. If you 
 can do so, resect by the method explained in paragraph 46 or para- 
 graph 47. Your resection will not be accurate, but it will serve to 
 aid you in locating yotu^eK. 
 
 62. While the last method is a good instruction method, there is 
 no excuse in campaign for an officer not to have the magnetic me- 
 ridian on his map. if the magnetic meridian is not shown on the 
 map, ascertain the naagnetic declination and draw the meridian on. 
 The magnetic dechnation, it must be remembered, is the angle 
 which .the magnetic needle makes with the true north at the point. 
 If the dechnation is east, it means that the north end of the needle 
 settles to the east of true north. If west, the north end of the needle 
 settles west. If the magnetic decliTiation can not be readily ascer- 
 tained, you can determine it approximately in the following manner: 
 
 DETEBMINATION OF MAGNETIC DBCUNATION. 
 
 63. In the explanation of this method, the term magnetic azimuth 
 as here used is the horizontal angle froin the north point of ihe needle 
 measured clockwise around the circle to the object sighted. To read 
 azimuths correctly, a box or pocket compass should be graduated 
 counier-cloekmse. If yours is not so graduated, better add a rough 
 graduation in the counter-clockwise direction. 
 
 Observe the magnetic azimuth of the sun, a planet, or a bright 
 star at rising and setting on the same day or at setting on one day 
 and rising the next. Add these two azimuths together. Take the 
 difiEerence between this sum and 360°. One-half of this difference 
 is the declination of your compass — east, if the sum of the azimuths 
 is less than 360°; west, if it, is greater. In usin^ this method the 
 observations are best taken when the object is just above the true 
 horizon, or at a gradient of zero. This can usually be done if a high 
 point is chosen lor observation. If this can not be done, be carenil 
 to take both observations with the object at the same gradient (as 
 determined with clinometer). This is Tnost important with the sun. 
 Under the least favorable conditions an inequality of 1° in the 
 gradients at the time of observation on the sun may introduce an 
 error of }4° i^i the result. 
 
 In using a star, choose one which rises nearly east from the point 
 of observation. If this be done the inequahty of a degree in the 
 gradients wiU be immaterial. Both observations need not be made 
 at the same point, but should not be more than 10 miles apart in 
 east and west or north and south directions. (See ako par. 68.) 
 
TOPOGBAPHY, MAP BEAtHnSTG, AND EECONNAISSANCE. 27 
 
 LESSON IX. 
 
 DAT GUIDING BY MAP AND COMPASS. 
 
 64. The following condensed quotation from a recent supplement 
 to the British Manual of Map Reading and Sketching is taken from 
 the International Mflitary Digest, March,, 1916: 
 
 After mastering tlie conventional signs, map reading is only a matter of observation 
 and common seng«, a fact which, leads people to underestimate its difficulty and the need 
 for continual practice. 
 
 When finding the way by road, it is important to consider the time,factor in attemp- 
 ing to identify points of the road in advance, and not place sole reliance upon cross- 
 roads, or side roads, etc., which may have changed since the map was published. 
 The speed of march may be obtained from a speedometer or by a rough approxima- 
 tion of the marching rate of the column. * * * 
 
 Orienting the map by pointing at the sun [a common textboolc method and at 
 times a valuable one] may lead to an error in the direction of as much as 7° or 8°. 
 Therefore, some permanent point, as a church steeple or a straight piece of road, should 
 be sighted on. 
 
 When a point of the country has been identified on the map, time often will be 
 saved by using it as a reference point in referring to others. Such points may be 
 noted as so many degrees from it, or in line with it, or just to the right or left. A pro- 
 tractor will assist in laying off angles and measuring bearings on the map. 
 
 On a small-scale map many details are not shown, but much may be inferred from 
 what is shown. 
 
 In Tught movements practice is essential, and although it is not difficult to march 
 by compass bearing, much assistance can be obtained by the ability to recognize a 
 Jew stars in different parts of the heavens. 
 
 65. After the remarks ^ven above have been discussed^ the 
 instructor should practice the members of the class in following a 
 course along roads and across country from the map. In order 
 that this practice may not be done in a perfunctory manner, it is 
 desirable to require the student officers to traverse a course out- 
 lined by the instructor, and note on the map all features which may 
 be altered. If the map is so nearly correct that this method is 
 impracticable, flags bearing special numbers may be erected along 
 the course and the student officers required to locate the flags on 
 their sheets. It will be particularly good practice if a group of 
 flags representing a trench are so locatea that they can be found only 
 by following a compass bearing across country for a distance of 
 several miles. 
 
 Such practice wiU convince all of the great difficulties in guiding 
 by compass alone. A judicious use of the map and of prominent 
 physical features will simplify the matter in open country. 
 
 Before starting, the course should be carefully protracted onto the 
 map and note taken of the prominent 'physical features through which 
 or near which the line passes. Before starting, identify one of th^e 
 points on the ground and march on it. If you must descend into a 
 valley to lose sight of the feature, take some point in the valley which 
 is in line to march on. Thus, continue from feature to feature. In 
 close country this is not so simple. Here it may be necessary for 
 you to actually plot on your map, or an enlargement of it, your 
 course as you go. Make your courses as long as possible. 
 
28 TOPOGRAPHY, MAP BEADING, AND EECONNAISSANCE, 
 
 LESSON X. 
 
 NIGHT GUIDING BY MAP, STAK, AND COMPASS. 
 
 66. The following is taken from the Artillery Joiimal: 
 
 As an aeronaut in Ladysmith, I had plenty of opportunities of -foreseeing the great 
 power aeronautics would have in warfare in the future, and that most of the effective 
 fighting would be done at night. . ■ ■' 
 
 At liat time the regulations described night operations aa extremely hazardous,^ 
 and warned the commander who undertook such operations that he did so at his own 
 peril and was responsible for the results. Various expedients were suggested to 
 enable the troops to keep their directions, such as that the route should be previously 
 reconnoitered and marked by tins, pieces of paper, and other devices; but how the 
 reconnoitering party was ever to carry out this operation nobody _ has yet been able 
 to understand. We found the colonials never required this artificial help, and could 
 move about on a starlit night as easily as in daylight and as fast as the ■ nature of the 
 ground permitted, * * * and ascertained from various colonials, Basutos, Indians,! 
 and Arabs that they could instinctively read the heavens as a compass, this Imowledge, 
 Laving been transmitted from father to son for generations. * * * 
 
 Although the system was only perfected in June, 1915, soldiers of all ranks have ; 
 begun to realize the simplicity and wonderful utility of beiiig able to read the universal 
 compass J the heavens, and we begin to hear how useful this knowledge has been found I 
 for guiding supporting troops up to the first-line trenches, etc. 
 
 The heavens can not go wrong, and on a starlit night you can rely absolutely upon 
 them to take you to your xiestination, once you grasp me rudiments of the system; 
 you only require to know three or four first magnitude stars, for their exact position 
 IS given for every hour of the night in Marching and Plying without a Compass (Tilney, ' 
 lieutenant Colonel, F. R; 0. S.) 
 
 TO FIND THE STARS. 
 
 67. The Dipper and Cassiopeia. — The star plans given here, and 
 parts of the descriptions, are taken from The Star Pocket Book by 
 Capt. Weatherhead, British Royal Navy. 
 
 Ursa Major (called the Big Dipper), shown in the upper portion 
 of Figure IK, is the most important of the constellations. It is 
 at once the easiest to distinguish, the easiest to find the North Star 
 by, and the best startiing point from which to learn the other stars. 
 The "Poiaters" a and jS point to Polaris (the North Star) at aU 
 times as the Dipper circles the Pole. On the opposite side of Polaris 
 and at about the same distance from it is the conisteUation of Cassio- 
 peia. Its form is that of the letter "W. 
 
 ■The great importance which attaches to Polaris (the North Star) 
 is that it is never more than 2° away from the point where the axis 
 of the earth if extended would pierce the heavens. It therefore ap- 
 pears to the eye to be always in the same place, and it is, except for a 
 maximum variation of about 2°. 
 
 68. As the latitude of any place is equal to the altitude of the Pole 
 when the Dipper and Cassiopeia are on either side of the North Star 
 (east and west), the elevation of the North Star will give a reasonably 
 correct figure for the latitude of the place of observation. When the 
 Dipper and Cassiopeia are above or below the North Star, a compass 
 reading to the North StarwiU give the declination of your compass to 
 within the least reading of your compass. 
 
 69. Arcturus, Spica, Deneiola. — ^The lower haK of Figure IX shows 
 the Big Dipper again. By continuing the sweep of the Dipper Jiandle 
 you wiU find the bright star Arcturus about 30° from the end of the 
 handle. Continue on with the curve and you will find Spica about 
 30° farther on. 
 
TOPOGEAPHY, MAP BEADING^ AND EECONNAISSANCE. 29 
 
 
 i* POLARIS). 
 
 CASSlOPEfA 
 
 
 
 NORTH 
 
 
 
 
 5 € 
 
 
 
 o<et 
 
 / 
 
 / URSA 
 
 7 '' 
 
 MAJOR 
 
 
 \4a. 
 
 / 
 
 / 
 t 
 
 
 1- 
 
 
 ^ARCTURUS 
 
 DENEBOLA^ 
 
 < 
 
 
 \ 
 
 
 UJ 
 
 
 \ 
 \ 
 
 \ 
 
 
 
 
 SPICA-ilf' 
 
 
 SOUTH 
 
 FIGURE IX 
 
30 a?0POGBAPHT, MAP BEADISrO, AND EECONlSrAISSAJTCE, 
 
 ArctuTus has a good marker near it in the semicircle of stars called 
 the Northern Crottm, and Spica has the kite-shaped quadrilateral 
 Corvus. 
 
 Denehola forms with Ardunis and Spica an eqmlateral triangle. 
 
 70. As an additional aid in findmg these first three important 
 single stars, it is pointed out that if a star is to be found at a certain 
 place in the heavens at a certain hour on a certain night, it will be 
 found in that same place the next nyht about four minutes earlier and 
 on the next jTour minutes earlier stilt, or at the end of a month in that 
 same place about two hours earher, and so on for each successive 
 month. Now, the "Pointers" (a and /3 of the Big Dipper) are to be 
 found high on the meridian at midnight on March 7 any year. At this 
 time Deneiola will also be near the meridian (a httle to the east) and 
 in the south. FoUow the sweep of the Dipper handle and high up in 
 the heavens almost to our east will be Arcturus, with Spica stUl on 
 the sweep of the handle in the southeast, completing the eqtiilateral 
 triangle. 
 
 71'. Hold the chart directly above your head, north to north, 
 west to west, etc., and you should have no trouble in finding these 
 stars. Remember the stars are to be found in this portion of the 
 heavens and as stated at 10 o'clock on April 7, at 8 o'clock on May 7, 
 etc. After May 7, therfore, we may expect to find them getting lower 
 and lower in the western heavens as the summer advances until 
 September, when the Dipper is to be found low down imder the 
 Pole; Denebola and Spica will have set before night comes on. 
 
 72. In looking for the stars during the summer months, therefore, 
 face the west and hold the chart in front of you, north to the north and 
 east side up. See table in paragraph 77 for the date on which Arc- 
 turus, Spica, and Denebola transit at midnight. 
 
 73. Vega, Altair, and Deneb. — The upper haK of Figure X shows 
 the Dipper once more and the important stars Vega, Altair, and 
 a Cygnus (also called Deneb). 
 
 Vega is to be found on the meridian at midnight July 1, Altair 
 July 19, and Deneb August 2. They all, therefore, transit within 
 about two hours of each other. There should be no difficulty in 
 finding the T of Cygnus and from it Altair, and then Vega, which is 
 on a Une with Altair and its two accompanying stars. 
 
 In the early summer months, in the hours before midnight, we will 
 find these stars in the eastern heavens. In the fall we wiU look for 
 them in the west. 
 
 74. Perseus, Andromeda, the Great Square, and Pleiades. — ^In line 
 with the "Pointers" on the other side of the North Star, and beyond 
 Cassiopeia about as far as it is beyond the North Star, we find a and /3 
 of Pegasus. These five stars are all on a Une (great circle), and aU 
 cross the meridian together, but, of course, when the Big Dipper is 
 high in the heavens, Pegasus being on the other side of the PoL is 
 out of sight in ordinary latitudes. When Pegasus is high in ' the 
 heavens, the Dipper is below the North Star. It is to be expected 
 then, that a and j8 Pegasi wiU be on the meridian at midnight on 
 September 7, just six months after the "Pointers" made their upper 
 transit at midnight and on the sam^ night as they make their hwer 
 transit at midnight. 
 
TOPOGEAPHY, MAP BBAMNG, AND EECONNAISSANCE, 
 
 31 
 
 The stars on these charts, therefore, are to be looked for in the 
 early hours of the summer evening in the east; high in the heavens 
 in the early fall; and in the west along Christmas time. 
 
 POLARIS* 
 
 ^ 
 
 
 
 
 
 
 1 
 
 ^CfJI^ ALTAIR 
 
 
 
 * POLARIS 
 
 
 4' !L 
 
 CAPEl,L/f -»?M 
 
 :VUhE PLEIADES 
 
 FOMALHAUT ^^ 
 
 *ALDEBARAN 
 
 
 FIGURE X 
 
 75. a and /3 Pegaai point directly at Fomalhaut, which, therefore, 
 also cross^ the meridian at midnight on September 7. These three 
 stars form as perfect a set of pointers to the Pole as do the "Pointers 
 
32 
 
 TOPOGBAPHY, MAP BEADING, AND RECONNAISSANCE. 
 
 of the Dipper and may be of great use, therefore, when the pointers,'' 
 low in the heavens, are obsctixed by clouds. 
 
 76. CapeUa, Aldebaran, Gastw, and Pollux (Fig. XI). — ^Aldebaran 
 is to be found on the meridian at midnight of November 29, CapeUa 
 on December 9, Castor on January 13 and Pollux on January 16. lu 
 the fall months and early mater we "wiR find them ia the east in the 
 hours before midnight and ia the west in the later wiater months. A 
 
 URSA MAJOR 
 
 -nJ^CAPELLA 
 
 Q^>*^4iCAST0R 
 ^ "^POLLUX 
 
 ALDEBARAN 
 
 jIfCAPELLA 
 
 ^ AURIGAE 
 
 * CASTOR 
 • POLLUX 
 
 ■^ALDEBARAN 
 
 BELLATRIX 
 BETELGUESE* o1^/^J^^ 
 
 PROCYONijt 
 
 ALPHARD 
 
 ff *RIGEL 
 
 n 
 
 ^ /3 MAJORIS 
 
 FIQURE XI 
 
 liae drawn from the pole star perpendicular to the hne from the 
 "Pointers" pa'sses through CapeUa. This line also passes directly 
 through Bigel, a bright star ia the consteUation of Orion. These two 
 stars, therefore, cross the meridian together and form another 
 excellent set of north pointers. 
 
 The group of stars in the constellation of Orion are particularly 
 brilliant, perhaps the most conspicuous of aU star groups in the 
 
TOPOGRAPHY, MAP EEADING, AND KECONNAISSAlifCE. 
 
 33 
 
 •winter heavens. Rigel, the stars in the belt of Orion, Bellatrix, and 
 Betelguese are very near the celestial equator. Betelguese and a 
 Aurigae transit together as do Rigel and Capella. We therefore 
 have in the winter tnis double line runrung from the vicinity of Orion 
 to the North Star. 
 
 Sirius, the brightest star in the heavens, is practically ia line with 
 the belt of Orion and also in line and about equidistant from the belt 
 is Aldebaran. 
 
 77. Date of midnight transit and declination of stars shown in 
 Figures IX-XI. 
 
 Date. 
 
 DecUn^ 
 tlon. 
 
 Star. 
 
 Date. 
 
 Declina. 
 tion. 
 
 Star. 
 
 January 1 . . 
 January 13. 
 January 15. 
 January 16. 
 
 March? 
 
 Marclil9... 
 April 12.... 
 
 April 25 
 
 Jiflyl 
 
 16.5° S. 
 
 5.5° N. 
 28.0° N. 
 
 15.0° N. 
 11.0° S. 
 19.5° N. 
 39.0° N. 
 
 Sirius. 
 
 Castor. 
 
 Procyon. 
 
 Pollux. 
 
 Pointers. 
 
 Denebola. 
 
 Spica. 
 
 Aroturus. 
 
 July 19 
 
 August 2 
 
 Septembers. 
 October 22... 
 November 29. 
 December 9. . 
 December 9-. 
 December 12. 
 December 19. 
 
 8.5° N.. 
 45.0° N.. 
 30.0° S . . 
 23.0° N.. 
 16.0° N.. 
 46.0° N.. 
 
 8.0° S.. 
 
 6.0° N.. 
 
 7.5° N.. 
 
 Altair. 
 
 Deneb. 
 
 Fomahaut. 
 
 Hamel. 
 
 Aldebaran. 
 
 Capejla. 
 
 Bleel. 
 
 Bellatrix. 
 
 Betelguese. 
 
 7^. The time of transit for any other night may be found approxi- 
 mately from the rule that the star transits four minutes earlier each 
 successive jnght, about two hours each successive month, or 24 hoxirs 
 earlier (same date) at the end of the year. 
 
 79. The decHnation of a star is its angle, measured in the meridian 
 from the equator. The latitude of a place on the earth is its angle 
 measured in the meridian from the equator. Both are noted as north 
 if north of equator, or south if south of equator. If the declination 
 of a star is greater than the latitude of a place, it will pass the me- 
 ridian to the north of the place. If less, it will pass the meridian to 
 the south. Of course, if the declination is south and the latitude 
 north, the star wiU always be to the south, and similarly, if the north 
 declination of star is small and the latitude of the place high. 
 
 It is evident from the f oregoiug that by use of the additional sets of 
 north-poiating stars, and with a little practice in studying the times 
 of the year and hours of the nights when certain stars are to be found 
 in the meridian to the north or south of us, we should seldom have any 
 difficulty in knowing approximately at least our points of the compass. 
 
 If headquarters have, as we understand they have in France, tables 
 from which the true azimuth of a star can be predicted for any hour 
 of the day, and if orders based on these tables are to be issued, we 
 must know the prominent stars. To learn them requires observar 
 tion and practice, nothing else. 
 
 LESSON XI. 
 
 USE OF MAPS IN POSITION WARFARE. 
 
 80. The following condensed quotation from an article in the 
 British Jom-nal Eoyal Artillery is taken from the International Mih- 
 tary Digest: 
 
 Now that lai^e-scale maps, such as 1 : 20,000 or 1 : 10,000, are used extensively 
 by batteries in the field, map reading has become an important part of an officer's 
 
 58740°— 18 3 
 
34 TOPOGBAPHT, MAP BEADIN-Q, AND RECONNAISSANCE. 
 
 work. From the map lines of fire are obtained, aiming points picked out, the position 
 of targets identified, and ranges obtained. , 
 
 Three distinct elements must be considered with respect to any pomt on the grouna 
 that is to be identified on the map, viz, (1) direction, (2) distance, and (3) shape ot 
 ground or relative height. It is not safe to decide on a point which appears to tumil 
 two of these conditions without examining as to the third. ^ 
 
 (1) DIKECnON. 
 
 In measuring angles a semicircular celluloid protractor (Steward's) will be found 
 useful and fairly accurate. If several points are to be identified in a given zone, a 
 well-defmed distant reference point must be selected and identified on the map. Using 
 the protractor, measurg the angle between the reference point and the object on the 
 ground, then plot it off on the map. 
 
 (2) DISTANCE. 
 
 The approximate distance may be found in two ways: 
 
 1. By estimation. 
 
 2. By noting whether it is farther off or closer than other points easily identified or 
 
 already Jcnown. 
 As a rule the latter method will be the more satisfactory. The scale of the map 
 should be kept clearly in mind when considering distances to objects. 
 
 (3) SHAPE OP GROUND. 
 
 This condition, though considered laat, is not of least importance, since a carefu 
 study of the contours will fix the position of an object on the map with far more certainty 
 than will an estimated distance. It sometimes helps to examine all the features 
 surrounding an object and then draw a rough plan of what is expected to be found on 
 the map. 
 
 Often a thick, well-defined hedge indicates a road. What looks like, a wood may be 
 only scattered trees. A low ridge or embankment may conceal a hedge, so theirs* hedge 
 visible beyond may be the second hedge shown on the map. 
 
 Watch the smoke of a distant railway train; it not only helps to identify the line 
 of the railroad, but may be useful as a reference point for other objects. Finally, even 
 the good maps may have mistakes, usually in connection with the roads. 
 
 PRACTICAL WORK. 
 
 81 . The ability to read maps as above described can only be 
 acquired by practice in the field. The following practical work 
 shoidd be executed by each member of the class in the field. If the 
 training period permits, this character of the work should be con- 
 tiaued until all are proficient in reading ground from a distance. 
 A ridge in or near the training area will be sdected from which 
 a reasonably good view can be had. The members of the class 
 with their sketching equipment and maps will be deployed along 
 the ridge, so that as nearly as may be they have the same view. 
 Before the lesson hour the instructor wiQ have had placed several 
 conspicuous objects; red or white flags wiU do if nothing else is at 
 hand. A prominent object in the foreground will be designated 
 by the instructor as the reference point, and each student after locating 
 his position on the map and orienting will draw a ray to this poinL 
 He will then be required to sight and draw rays to the severaOags 
 He will then be required to draw a small circle at the poiat on each 
 ray where he beheves the flag to be. He wiU then be required to 
 draw a profile from his map along each of the rays. The drawine 
 of each profile will require about 30 minutes of a beginner's time 
 There should not be so many required of him as to absorb the entire 
 lesson period. 
 
TOPOGRAPHY, MAP EEADIITG, AND EECONNAISSAlirCE. 35 
 
 TO DRAW THE PROFILE. 
 
 82. Look along the ray very carefully and find the lowest contour 
 which the ray crosses and also the highest. Suppose the lowest 
 to be 520 and the highest to be 640. The highest point on 
 your profile wiU then, of course, be 640 feet and the lowest 520. 
 ITie difference will be 120 feet. Suppose the contour interval is 20 
 feet; 20 goes into 120 six times. There are, therefore, six contour 
 intervals between the highest and lowest point. Take a sheet of 
 paper, the long edge of which is long enough to reach from your 
 position on map to the position of nag. With your aUdade as a 
 ruler, draw six lines parallel to the long edge of the paper at equal 
 distances apart and from the edge. This distance jnay be any 
 convenient one, but should not be less than one-half inch, if practi- 
 cable. (A method of drawing the lines parallel to each other, which 
 suggests itself, is to lay off half-inch spaces along the short edges of 
 the paper and connect them.) Now lay the long edge of the paper 
 on the ray. Mark the edge of the paper as elevation 520 (the lowest 
 elevation) and the highest line as 640 ; mark the intermediate ones 
 540, 560, 580, 600 and 620. Now, beginning at your station, look 
 along the ray very carefully, and every time the ray crosses a contour 
 (say the first one met is the 620) run a hght line up to the correspond- 
 ing parallel line (620 in this case) and make a cross there. Do the 
 same at the next contour, and continue until you reach the position 
 of the flag. Now connect the crosses in succession, beginning at the 
 first one^ by straight hues. The result wiU be a proMe or vertical 
 section of the ground between your position and the flag's (providing 
 you located the flag correctly). Now compare your profile with the 
 ground between you and the flag. Does your profile help you to tell 
 whether you located the flag correctly? Did you overestimate the 
 distance and plat the flag's position in a valley beyond the hUl where 
 you really could not see it ? 
 
 83. Draw a straight line on your profile sheet from your position to 
 the -flag's position. Does this line clear all intervening points on the 
 profile ? If not, you could not see the flag where you have plotted it. 
 This latter Httle problem is what is commonly called a visiMlity 
 prohlem. 
 
 84. Each sketcher will now be required to outline on his map the 
 areas (included between the rays to two flags) which are invisible 
 to him. To do this, he will find it desirable to draw a number of 
 profiles. He should draw no more than absolutely necessary, and 
 these for the sole purpose of checking a decision which he has made 
 by comparison of the map with the ground. 
 
 LESSON xn. 
 
 I visiBiLrrY. 
 
 85. A problem frequently arising ia map reading is that of deter- 
 mining what points are visible from a given point. A point is visible 
 when the gradient to it, if rismg, is greater, and if falling, is smaller 
 than the gradient to any intermediate point. 
 
 For this comparison gradients are conveniently represented by 
 the quotient of mstance in feet divided by the difference of elevation 
 in feet. The point will be visible when this quotient is smaller, if 
 
36 
 
 TOPOGEAPHY, MAP BEADING, AND KBOONNAISSANOE. 
 
 rising, and larger if falling, than the quotient for the intermediate 
 point. Thus, to determine whether the Bridge near the French- 
 man's is visible from Atchison Hill or is concealed by intermediate 
 ground, assume the highest point of Atchison Hill to be in the center 
 
 of the 1,040 contour and to have an elevation of 1,050. The dis- 
 tance from this point to the bridge is 5,610 feet, fall 250 feet, quotient 
 22.4. The line of sight from this point to the bridge crosses the 960- 
 foot contour on the flank of Sentmel HiU at 3,060 feet distance, fall 
 90 feet, quotient 34; hence bridge is not visible from Atchison 'Hill 
 
TOPOGBAPHY, MAP EBADUiTG, AND EEOONNAISSAITCE. 37 
 
 since the gradient is falling, and the nearer point has the larger quo- 
 tient. 
 
 WorMng from the bridge the quotient for the whole distance ia 
 22.4, as before, but the gradient is rising. The distance from the 
 bridge to the high point is 2,550 feet, rising; difference of elevation 
 160 feet, quotient 16: hence, as before, the top of Atchison Hill is not 
 visible from the bridge, since the gradient is rising, and the nearer 
 point has the smaller quotient. 
 
 If one gradient is rising and the other falling, no computation is 
 necessary. A point of rising gradient •will hide a farther point of 
 falling gradient, but will not be hidden by a nearer one. 
 
 86. The explanation given above niay be cleared up to some as 
 follows: The point of view on Atchison HUl is at elevation 1,050; the 
 bridge is at elevation 800. -The difference is 250 feet. If you can 
 see the bridge from Atchison HiU your line of sight, as you look at 
 the bridge, will be a straight line which slopes 250 feet in 5,610 feet, 
 or 1 foot in 22.4 feet. Now, a straight line from Atchison Hill to the 
 bridge crosses the 960 contour on Sentinel HjU. The point of crossing 
 is 3,060 feet from the point of view on Atchison Hill. Your line oi 
 sight to the bridge must fall 1 foot in 22.4. Therefore, it would have 
 
 fallen So"! ™ 3,060 feetj or 136.6 feet. One thousand and fifty feet 
 
 less 136.6 feet is 913.4 feet. If you could see the bridge, your line of 
 sight must have fallen to 913.4 feet at Sentinel HiU; but the ground 
 there, as shown by the contour, is 960 feet high, so it interferes. 
 
 87. An instruction test as to whether an intermediate point ob- 
 scures the view between two other points may be made by setting 
 up at three points on the map pencils or other suitable objects having 
 the corresponding elevations marked on them on a convenient 
 assumed scale. Sight, or stretch a thread, along the. pencils. If 
 the middle mark is above the line joining the other two, each of the 
 two extreme points is invisible from the other. If the middle point 
 is below the line, each extreme point is visible from the other. 
 
 LESSON xin. 
 
 RECONNAISSANCES. 
 
 88. Reconnaissance is an examination made for the purpose of 
 gaining information relative to the terrain, the enemy's forces, or for 
 other purposes. 
 
 The subject is of particular interest to all officers. In past wars 
 officers have been called upon for much reconnaissance duty and 
 such wiU xmdoubtedly be the case in any future war. In the Mexican 
 War the close reconnaissance of the enemy's positions by officers 
 contributed in no small measure to the success of our forces. In the 
 Civil War the Engineer officers acting as such in the field, and Engi- 
 neer officers holdSig volunteer commissions in command of troops, 
 were constantly making reconnaissances and putting to use their 
 topographical knowledge in correcting existing maps, reporting, by 
 hasty sketches, the positions occupied by troops, and in guiding 
 troops through close country and seeing that they were posted in 
 accordance with the plans of the higher commanders. 
 
38 TOPOGRAPHY, MAP EEADIITG, AND EECONNAISSANCE. 
 
 89. To perform such duties well and without hesitation, the officer 
 must possess a keen eye for topography and an unfailing sense of 
 orientation; in short he must never get lost, and he should spare no 
 pains to become thoroughly familiar with the large drainage and other 
 important features of the section of coimtry in which the operations 
 are being carried on. 
 
 90. Keconnaissance should not be confused with mappmg. The 
 information obtained by a recoimaissance may be reported verbally 
 or in writing and may or may not be illustrated by a sketch or route 
 map. A reconnaissance is seldom made for the purpose of bringing 
 ia a sketch, but a clear sketch will save much writing and will be of 
 value as a record and further description of the route or country 
 covered. 
 
 91. In any reconnaissance in proximity to the enemy, the time 
 available is a most important element. The information desired by 
 higher authority must be obtained and to that end tioae should not 
 be wasted on unimportant or irrelevant investigations. An officer 
 on this duty must keep his mission constantly in mind and use his 
 best judgment as to what is important and what is not. 
 
 92. The following quotations illustrate some of the various classes 
 of reconnaissances which officers may be called upon to make : 
 
 Beport of Maj. Nathaniel Mlcher, Corps of Engineers, United States Army, Acting 
 Chief Engineer, Headquarters, Army of the Potomac, Engineer Department, 1864: 
 
 Although the enemy has no doubt suffered at times from want of accurate maps, 
 still he has at all times possessed a superior knowledge of the country, and could 
 always obtain reliable guides from among its inhabitants, thus affording him a very 
 great advantage over hia adversary. In order to be able to cope with him with any- 
 thing like equal advantage, it soon became apparent that the difficulties to which 
 reference has been made would not only have to be overcome by gathering material 
 with the onward march of the army, but that the desired information would have to 
 be obtained in anticipation of any move. To accomplish this the officers and assist- 
 ants of my party were kept constantly occupied both day and night: they were not 
 only called upon to prepare the much-needed maps with the detailed corrections, 
 but also in the entire absence of reliable guides to act as such to the different columns, 
 either as they moved along their respective routes of march or while maneuvering 
 for favorable positions previous to an attack. 
 
 On the morning of the 8th, some severe skimashing commenced between the ad- 
 vance of the Fifth Corps and the enemy, showing that the latter was falling back from 
 the wilderness toward Spotsylvania Court House. At break of day I was directed to 
 make a reconnaissance of the country along the Brock Road and parallel to the Po 
 River, to select a good position for the Second Corps to take up in the event of the 
 enemy attempting to strike oiu: flank. 
 
 Early on the 10th the Second Corps was advanced across the Po by ponton bridge 
 Subsequently, by order of the commanding general, I guided Gibbon's and Birney's 
 divisions back again across the river and placed them in position to the rear and right 
 of the Fifth Corps, where they were massed to make a combined assault. Lieut. 
 Mackenzie was on the same day engaged on a recoimaissance to the front of the Sixth 
 Corps, and, in company with General Russell, selected the point of attack so success- 
 fully made by Upton's brigade of that corps. 
 
 On the 1st of June * * *, dunng the day, accompanied by Capt. Gaiespie who 
 had joined the army a few days previous, and several assistants, I directed the ex 
 amination of the country to the southeast of the Old Church Tavern for the pumose of 
 finding several parallel roads over which to move simultaneously diflferent columns 
 
TOPOGRAPHY, MAP BEADING, AND EECONNAISSANOE. 39 
 
 On the morning of the 10th * * *^ Capt. Mendell, accompanied by Lieut. 
 Howell, made a reconnaissance to Windsor Shades, on the Chickahominy, to ascertain 
 the practicability of crossing at that point the supply train, but reported unfavorably. 
 
 On the 24th, accompanied by Capt. Mendell and Lieut. Howell, I made a recon- 
 naissance of the country between the Avery House and the Blackwater Swamp, for 
 the purpose of selecting a line * * * the crossings of the swamp were also care- 
 fully searched, and its character examined in regard to forming an obstacle to the 
 passage of artillery and infantry. * * * On the 29th the Appomattox was also ex- 
 amined in reference to the facilities for bridging it. 
 
 Gen. Sheridan's expedition toward GordonsviUe retinmed on the 30th, and the 
 assistants who accompanied it brought back most valuable topographical information, 
 among other interesting matter a survey ofthe enemy's works at Spotsylvania Court 
 House. This latter enables me to furnish in fuU, and with accuracy, the battle field 
 map of that locality. 
 
 93. From the above, it is seen that reconnaissance made by officers 
 serving with, troops in the field fall naturally under the following 
 headings: 
 
 (a) Reconnaissance of a river crossing. 
 
 (&) Reconnaissance of a defensive position to be occupied by 
 our own troops. 
 
 (c) Reconnaissance of a position occupied by the enemy with 
 
 a view to fi^nding the best point for an attack. 
 
 (d) Reconnaissance of routes of march. 
 
 LESSON XIV. 
 
 RIVER CROSSING. 
 
 94. If a crossing is to be made in the face of the enemy, the loca^- 
 tion selected must fit the tactical situation; that condition being com- 
 plied with, choose the location which wiLL require the least labor and 
 material to render it practicable. 
 
 Fords should not be more thaji 4^ feet for Cavalry, 3^ feet for 
 Infantry, and 2^ feet for guns and wagons. The nature of the 
 stream bottom is most important. It should give good footing and 
 should not scour wnder the action of wheels and hoofs. Almost any 
 ford except one with rock bottom wiU scour under the coEtinued 
 passage of Artillery trains. The velocity of the current and ease of 
 approach and exit must be considered. If work is necessary to 
 prepare the immediate approaches, or roads connecting with the 
 nearest main roads, an estimate of the men, tools, and tune should 
 be made. 
 
 PONTON BRIDGES. 
 
 95. Other things being equal, a point where the stream is narrow 
 should be selected; this facilitates speed of construction. Low, 
 firm hanks and easy approaches are very important. A marshy 
 approach, or a steep bank 10 feet or more in height, may require 
 more time in preparation than the construction of a bridge several 
 hundred feet long. In the Civil War the principal worTc in the con- 
 nection with ponton bridge crossings was the preparation of corduroyed 
 approaches. If the enemy is opposing the passage from the opposite 
 
40 TOPOGBAPHY, MAP BEADING, AND BECONNAISSANCE. 
 
 bank tactical conditions are paramount. A point should be selected 
 where the approach to the bank is concealed hy trees or otherwise, 
 where the iridge vnU he screened from the enemy's artiUery observers, 
 and if possible, where there is httle cover for the enemy's infantry on 
 the far bank. The lower reach or mouth of a tributary often luifiils 
 these conditions. The bank of departure should preferably be the 
 higher of the two, and the site should be such as to permit the con- 
 centration of artmery and rifle fire on all points from which the 
 enemy might oppose the passage. The concave side of the bend ol an 
 alluvial stream often fulfills the above requirements for bank depar- 
 ture, and has the added advantage that most of the approach work 
 will ie on that (the high) side and can he carried on while the hndge is 
 being constructed, fi the farther bank is occupied by _the enemy 
 and his small-arms fire can not be silenced by artillery and small-arm 
 fire, an attack must be made before the construction of the bridge 
 can be begun. This should be dehvered by Infantry ferried across 
 in ponton boats. A sheltered spot for launching and loading the 
 boats is most desirable, but the operation may be effected with fair 
 rapidity on the open hank, particularly under cover of darkness. 
 
 IMPROVISED BRIDGES. 
 
 96. By iinprovised bridges are meant those which are constructed 
 of material collected from the vicinity of the site. On account of the 
 length of time required to coUect the m,aterial and to improvise a suitable 
 bridge out of the misceUamy, such bridges will seldom be built for the 
 advance troops. 
 
 These troops must, when time is pressing, get across the stream by 
 fording or by ferrying, etc. The reconnaissance officer must, there- 
 fore, in every case when time is pressing, and ponton equipment is 
 not at hand, search for a/oailahle fords and for hoats to use 'hnferrying. 
 There should be no delay in transmitting ioformation as to fords or 
 ferries to the troops. 
 
 The crossing of lai^e bodies of troops either by fording or ferrying 
 is, however, very unsatisfactory, and the recoimaissance officer should, 
 after he has reported on fords and ferries, look up material aad site 
 for an improvised bridge, to be constructed by the engineers while the 
 advance troops are crossiag. 
 
 97. The reconnaissance, therefore, wiU have two phases. The first, 
 a search for means available for an immediate crossing. The report 
 on this phase mU he made at once and it may be necessary for the 
 reconnaissance officer toguide the troops to the ford or to supervise 
 the passage by boats. This part of his work completed, he continues 
 the second phase of his reconnaissance. He must now search out and 
 note where the material can he procured and by what routes and trans- 
 portation it can be moved to the site selected. He must then examine 
 the site sufficiently to he able to give to the constructing officer the width 
 of the stream and the character of the bottom. 
 
 To meet emei^encies such as the above it is of prime importance to 
 have ponton material always with the advance troops. With a 
 large command improvised bridges will usually be built for the pur- 
 pose of releasing the ponton material so that the bridge train can 
 |)roceed in advance. 
 
TOPOGRAPHY, MAP BEADING, AND EECONNAISSANCE., 41 
 
 LESSON XV. 
 
 RECONNAISSANCE OF ROUTES OF MARCH. 
 
 98. If time is pressing, sucli a reconnaissance may consist simply 
 of riding the road, keeping oriented by the stih or compass, and making 
 notes or important features either on paper or mentally. If there is 
 sufficient time for better work, a sketch of the route traveled shotild 
 be made. This may be done in any one of the ways described here- 
 after, which best suits the conditions and the personal preference of 
 the officer making the reconnaissance. 
 
 99. A good sketch will show the road followed and the important 
 topographical features as far to either side as they can be drawn from 
 adtual observations. By important features is meant the important 
 drainage hues and the hill forms as controlled by those lines. It 
 must be kept in mind that a road has absolutely no influence on 
 topography. TTie topograpJiy was there first and the road was put in 
 on top of it afterwards. It is a common mistake to assume that, 
 because in carrying approxinxate elevations, readings are better taken 
 on slopes along a road, the contours or form lines cross the road at 
 right angles. The result is a sketch showing a succession of ground 
 waves, ridges, and valleys, all perpendicular to the road, which is of 
 course entirely worthless so far as giving any real information of the 
 country is concerned. 
 
 100. There are but two things absolutely essential in the making 
 of a good road sketch, first, a good traverse, and, second, the loca- 
 tion of the drainage system in its proper relation to that traverse. 
 With this control approximate contours can be drawn by anyone 
 having a knowledge or topographical principles. Ground forms so ob- 
 tained will never be misleadmg, and if drawn by a skiUfuI topographer 
 will give valuable information. Time should not Ijfe spent on plotting 
 uiiimportant details, for in addition to wasting time the really valuable 
 parts of the sketch are thereby obscured. Prominent hjiildings and 
 
 • farmhouses are of value in assisting anyone using the slcetch in locating 
 himself thereon. Wooded areas and orchards are for the same reason 
 important, and may be more so for tactical reasons. It should not 
 be expected that a route sketch wiU show every fence, ditch, or bit 
 of cover that might hide a patrol, and a striving after such detail 
 only befogs the real issue, that of finding where the road goes and 
 how ; and in general what the country adjacent to it is like. 
 
 101. In road slcetching with the issue table, the tripod as a rule 
 is not used, and to get good orientation it will usually be necessary 
 to dismount, as the average horse will not stand still while an obser- 
 vation is being made. Points off the road can be located by inter- 
 section or by direction and estimated distance. Distances are 
 plotted along the road by means of a time scale, if using a horse or 
 a scale reading to the least reading of your speedometer if using an 
 automobile.- The speedometer should first be tested by rating over 
 a known or measiu-ed course, and the horse should be timed at a 
 walk, trot, and gallop over a measured course. 
 
 102. To make a time scale, having rated the horse over a known 
 course: Suppose the scale of your map is 1 : 10,0'00. ^ Suppose your 
 
42 TOPOGBAPHT, MAP READING, AND KECONNAISSANOB. 
 
 course is 8,280 feet long and your horse takes 9 minutes to travel the 
 
 8,280 feet. He, therefore, travels ^^ or 920 feet in one minute. 
 
 Keduce this to inches by multiplying by 12 and we have 11,040 
 
 inches. The distance on the map to represent this wiU be jq^qqq 
 
 or 1.1 inches on the map will represent a minute of the horse's gait. 
 Lay off 1.1 inches several times on a suitable piece of paper. Each 
 division will represent one minute of travel at that gait. 
 
 103. To make a scale for speedometer. — These instruments usually 
 read to tenths of a mile. They may not be accurate. Exm over a 
 measured course at least 10 miles long. Suppose your speedometer 
 shows 9.2 miles; the tenth-mile register is, therefore, inaccurate. 
 Lay off the proper distance 10 miles to scale and divide it iiito 92 
 equal parts. Each one of these parts wiU represent the distance 
 traveled for each tenth register of your speedometer. See para^ 
 graph 217 for explanation of method of dividing a line into a given 
 number of equal parts. 
 
 LESSON XVI. 
 
 COMBINED ROAD SKETCHING. 
 
 104. As soon as the individual sketchers have shown a proficiency 
 in road reconnaissances, at least one exercise should be had in com- 
 bined work. In combined road sketching the work of individual 
 sketchers is in no way different from that heretofore described, but 
 strict precautions must be taken to identify the proper location of 
 each sketcher's work in the combined sketch. To accompHsh this, 
 it is absolutely necessary to mark every crossroad by a distinctive 
 card, or otherwise,* and this distinctive mark must appear on each 
 sketch which starts or terminates at, or includes that point. 
 
 105. A method which has been found to work well in peace times 
 is as foUow^ Suppose it is necessary to recormoiter three roads 
 leading toward the front and generally parallel, also all connectiag 
 roads and aU roads for a distance of one-half mile leading to the 
 right from the right-flank road and to the left from the left-flank road. 
 The problem is primarily one of organization. Three parties should 
 be organized as loUows : 
 
 Party A: 
 
 For right-flank road — 
 
 1 director. 
 
 1 principal sketcher. 
 
 Assistant sketchers, say 3 5 » 
 
 Party B: 
 
 For center road — 
 
 1 director. 
 
 1 principal sketcher. 
 
 Assistant sketchers, say 6 8 
 
 Party C: 
 
 For left-flank road — 
 
 1 director. 
 
 1 principal sketcher. 
 
 Assistant sketchers, say 3 5 
 
 18 
 
TOPOGEAPHY, MAP EEADING, AlTD KEOONISrAISSANOE. 43 
 
 106. Each, director should be supplied with an. aneroid barometer 
 and all three barometers should be set at the same reading on starting 
 out. 
 
 The duty of the director of party A is to give general instructions 
 to his sketchers and to then ride his road, putting up at each cross- 
 road or side road a card having marked thereon an arrow showing 
 the direction to be followed by his principal sketcher, the letter of his 
 party, the number of the cross or side road counting from the start, 
 and the aneroid elevation, as Aj, 850; A., 980, etc. 
 
 The duty of the principal sktecher of party A is to sketch the 
 right-flank road indicated by the cards and to put on his sketch at 
 each cross or side road intersection the letter and number of the card 
 found at that point. Assistant sketchers take the side roads in 
 succession. 
 
 The duty of each assistant sketcher is to mark the starting point 
 of his sketch with the letter and number of the card foimd at that 
 point, then to sketch his road to the flank the distance ordered. He 
 should place his initials on the card before leaving. On completing 
 his work, tmless otherwise ordered, he returns to the right-flank road 
 and sketches the first road he comes to which lias not been initialed 
 by another assistant sketcher. 
 
 The duties of the director of party B are similar to those of the 
 director of party A and those of the principal sketcher of party B 
 are similar to those of party A. The assistant sketchers of party B 
 are sent off to the right and left of the center road as cross or side 
 roads are encoimtered. As above, each must place at the starting 
 point of his sketch the letter and number of the card found at that 
 point. He must follow through and sketch, his road until he finds a 
 card on the right or left flank road and he must place the letter and 
 number of this card at the end of his sketch. If his road does not run 
 through, he must indicate on his sketch that it does not. The duties 
 of tbe director of party C are similar to those of the director of party A. 
 The duties of the principal sketcher and assistants are similar to 
 those of party A, but opposite hand. 
 
 107. If all the sketches are marked as indicated there wiU be no 
 trouble in combining them, although, of course, they wiU not fit 
 exactly. Copies may be made by blue printing from each sketch 
 separately and combining the prints, or by tracing the combined 
 sketch and printing from the tracing. Sketchers must be instructed 
 as to time and place for turning in their work. 
 
 LESSON xvn. 
 
 COMBINED AREA SKETCH. 
 
 108. If any map is available the area to be reconnoitered should 
 be outlined on it and subdivided into as many tasks as there are 
 sketchers, the parts being apportioned according to the time neces- 
 sary to complete them so that all sketchers will be in at about the 
 same time. Each of the parts is assigned to a sketcher, with definite 
 instructions as to the amount and class of work to be done, the scale 
 to be used, which should be the same for all, and the place and time 
 at which the sketch must be turned in. 
 
44 TOPOGRAPHY, MAP READING, AND EEOONNAISSANCB. 
 
 If the avaUable map will give fairly good control it should be 
 enlarged to the scale required for field work and a section of the map 
 given to each sketcher with his instructions. 
 
 The boundaries between sketchers must be physical features which 
 are shown on the available map, such as roads, trails, streams, etc. 
 These boundaries should appear on both adjacent sections of the 
 map as handed to the sketchers. If a sufficient number of copies 
 of the available or base map are not at hand to cut into sections 
 with overlaps, the necessary boundaries or the control for each 
 sketcher may be traced from the base map. 
 
 In the absence of an available map if a combined road sketch be 
 made and satisfactorily combined it will furnish a reasonable satis- 
 factory base map for an area. 
 
 LESSON xvm. 
 
 BRIDGES. 
 
 109. In reconnaissances for routes of march it is essential that spe- 
 cial attention be called to all bridges which are in bad condition or 
 too Hght for ordinary army loads. If guns heavier than the 4.7-inch 
 rifle or the 6-inch howitzer, heavy truc^, etc., are to use the bridges 
 it wiU be desirable to have aU bridges examined with care, and refer- 
 ence should be made to the Engineer Field Manual, chapter on 
 "Bridges." 
 
 Every officer, however, should be able to determine quickly when 
 a bridge is undoubtedly safe for ordinary army loads and when it 
 is undoubtedly unsafe. This he will determine ordinarily by mental 
 comparison with other bridges Tcnown to be safe or unsafe. To 
 develop the faculty for mental comparison he must have practice in 
 examining bridges. The following thuml rules wiU aid him: 
 
 PLANKING. 
 
 110. For ordinary army loads tke planking should be as thick in 
 inches as the stringers are apart in feet. For 4.7-inch guns or 6-inch 
 howitzers it would be well to lay wheel tracks whenever the planking 
 is less than 3 inches thick. 
 
 STRINGERS. 
 
 111. Let h represent the breadth of stringer in inches, d the depth 
 in inches, and L its length in feet between supports. The index, as 
 it may be called, is equal to the breadth in inches multiplied by the 
 square of the depth. Therefore, to find the index, measure the 
 breadth and depth of a stringer in inches and multiply hxdxd. 
 
 Now measure the distance in feet between the supports of the 
 stringers. In truss bridges this is not the distance between the piers. 
 It is the distance between the tranverse beams (usually steel) which 
 carry the roadway. Coimt the stringers. There must be four or 
 more or the thumb rule will not apply. Divide the index by the 
 leThgih in feet between supports. H the quotient is 16 or over, the 
 stnngers are safe for loaded escort wagons or 3-inc7i guns. If the 
 quotient is 30 or over, the stringers are safe for ^-T^-inch gu^. If 
 the length of stringers between supports is not over 15 feet (that is 
 not over twice the distance between axles of the carriages) the 
 
TOPOGRAPHY, MAP BEADING, AND RECONNAISSANCE. 45 
 
 quotient may be reduced to two-thirds of the values given and the 
 stringers will stUl be safe. 
 
 112. Tlhe stringers may be steel I-beams. If they are, measure 
 their defih. Square this for the I-beam index. Divide this index 
 by the length between supports in feet. If the quotient is l^ or more 
 the stringers are safe for 3-iach guns. If the quotient is 3 or more, 
 the stringers are safe for 4.7-inch guns. If the length of stringers 
 between supports is not over 15 feet, these quotients may be reduced 
 to 1 and 2, respectively. 
 
 ROADWAY BEARERS. 
 
 113. The index of wooden roadway bearers divided by the length 
 of the hearer between its supports should be 50 or over; if steel, 5 or 
 over. This rule is based on a spacing between road bearer of 15 
 feet. If the spacing is 20 feet, the quotients should be doubled; 
 between 15 and 20 feet, interpolate. 
 
 It will be understood that the above thumb rules are given as an 
 aid to the nontechnical officer. They are not construction rules. 
 
 LESSON XIX. 
 
 RECONNAISSANCES OF POSmONS. 
 
 114. It is beUeved that the reconnaissance of a position to be 
 taken up by oiir own forces has no place in this Manual. Tactical 
 considerations govern. Something has been said of the high-class 
 work required m securing a hose skeleton of an enemy position in the 
 "position warfare" of to-day. The following quotation is given as a 
 suggestion of a possible method of securing such a skeleton. 
 
 115. Operations in front of Petersburg. — ^Extract from report of Maj. 
 Nathaniel Michler, Corps of Engineers, United States Army, 1864: 
 
 On the 9th. of Jtily, 1864, orders were issued by the commandiag general that "the 
 operations of this army against the intrenched position of the enemy defending Peters- 
 burg will be by regular approaches on the fionts opposed to Gen. Bumside's and 
 Gen. Warren's corps, " and on the following day a plan of cohducting the siege was 
 submitted. 
 
 On learning the plan adopted, I directed my principal assistant, Maj. John E. 
 Weyss, to commence on the 9th an exact triangmation of the front of Petersburg, 
 locating our own line of work as well as that of the enemy, and to take the immediate 
 charge of the surveying party. My assistants, Messrs. Theilkuhl, Schumann, and 
 Jacobsen aided him. The work was extended from the south of the Jerusalem plank 
 road as far north as City Point. By this triangulation, performed under the fire of 
 the enemjf's batteries and sharpshooters, the different spires and certain prominent 
 buildings in Petersburg were accurately located, and halving been kindly fiu-nished 
 by Prof. Bachs, Superintendent of the United States Coast Survey, with a copy of 
 the beautiful map of that city and the Appomatox River prepared a few years ago 
 in his department, I was able to combine the two, and thereby obtain ah exact con- 
 nected map of the locality of our siege operations, covering the whole ground occu- 
 pied by both armies. 
 
 116. Large scale surveys, such as are described above and as have 
 imdoubtedly been made of the various fronts in Europe, wiU ordi- 
 narily be executed by expert surveyors attached to army headquar- 
 ters; but triangulation of a reasonable degree of accuracy and 
 topographical surveys controlled by such character of triangulation 
 or by transit traverses wiU from time to time be required of the 
 divisional Engineers. For this reason Part II has been added to 
 this manual on the use and adjustments of the various instruments 
 
46 TOPOGBAPHY, MAP BEADING, AND RECONNAISSANCE. 
 
 used in surveying for the measurement of angles, distance, and eleva- 
 tion. Tbe following plan of procedure is suggested for a topograph- 
 ical survey of an area too large to permit of the use of reconnaissance 
 methods and yet not large enough to justify geodetic methods. The 
 time allotted to this training course and the personnel of the class wiU 
 determine whether instruction can be given in this part of the manual. 
 
 LESSON XX. 
 
 TOPOGRAPHICAL SURVEY OF AN AREA APPROXIMATELY 100 SQUARE MILES. 
 
 117. In this work, as indeed in aU map work of sketching, it "must 
 be accepted as an axiom that however large or however small may be 
 the area to be surveyed, it must be treated as a whole, and that all 
 over the area a nuinber of carefviRy determined points must be fixed," 
 and these points adjusted among themselves to form an accurate 
 framework on which the less accurate work may be hung. 
 
 T^e accuracy of the entire map will depend upon the accuracy of. 
 the framework or control. The framework may be located by triari- 
 gulation or by transit traverses. A combination of the two will be 
 the ordinary rule. 
 
 TKIANGULATION. 
 
 118. Paragraphs 187 to 189 give the methods to be followed in meas- 
 uring the hase line. It should be remembered that an error in the 
 base line wiH be reproduced in the triangulation. Thus, if the base 
 measurement is xtjW larger than it should be and the farthest point 
 of the triangulation is actually 10 miles from the base, the triangula- 
 tion wiU place the point at a distance of -j^ mile, about 50 feet too 
 far. This would not be serious if the work is to be carried no further 
 
 ANGLE MEASUREMENT. 
 
 119. If possible, aU of the angles of each triangle should be meas- 
 ured. The procedure at each station is as follows: Level carefully 
 with the telescope level and see that the plate levels and otlier adjust- 
 ments are satisfactoiy. For each principal angle: Set the A 
 vernier to read zero and, with the telescope direct, set with the lower 
 motion carefully on the left-hand station. Read and record both 
 verniers. Unclamp above and set on the right-hand station. Eead 
 and record the A vernier for the approximate angle. Find the number 
 of whole times 60 wiU go into this approximate angle and call this M. 
 Then — 
 
 Unclamp below and set on the left-hand station\a , . . 
 
 Unclamp above and set on the right-hand station/ *®°°'^*^ repetition. 
 Unclamp below and set on the left-hand station Irpi, • ■■ . . 
 
 Unclamp above and set on the right-hand station/ ""^^ repetition. 
 
 Plunge the instrument, and without disturbing the vernier setting, 
 Unclamp below and set on the left-hand stationli, 
 Unclamp above and set on the right-hand station/ °^"^'^ repetition. 
 Unclamp below and set on the left-hand station!™ p., . . 
 
 Unclamp above and set on the right-hand station/^^™ repetition. 
 Unclamp below and set on the left-hand stationlq. 
 Unclamp above and set on the right-hand station/ "^ repetition. 
 
 Read and record both verniers. The mean of the "seconds" of the 
 two verniers wiU be taken with the degrees and minutes of the A 
 
TOPOGRAPHY, MAP HEADING, AND EECONNAISSANCB. 47 
 
 vernier as the vernier reading. To this must be added 360° multipHed 
 by M as above determined. This large angle is then divided by 6. 
 This is the value of the angle for one set of readings. A second set 
 shoidd now be taken. In the second set the vernier at the start should 
 be set at 35°. If the values from the observed sets do not check out, 
 the observations are immediately repeated, if they do take the mean. 
 
 120. In reading veiitical angles, level the instrument with the 
 telescope level. Measure and record the HI. Read, direct, the 
 vertical angles. Plunge the telescope and reread the vertical angles. 
 Relevel the instrxunent and repeat the direct and reverse observa- 
 tions. The recorder notes the point observed and the vertical angle 
 for each station, landmark, or flag. 
 
 121. After the angles which pertain to the triangulation system 
 have been read and recorded as above described, a series of "poiat- 
 ings" are taken to all spires and other prominent landmarks in view, 
 record of each object "pointed" being kept in such a manner that 
 there wiU be no diflB.culty in identifying it later, when "pointed" 
 from another station. Tixese pointings are taken as follows: 
 
 Set A vernier at zero, and with the telescope direct, set carefully 
 with the lower motion on any pnTvcipal station. Then, unclamping 
 above, read the angles to the successive landmarks in turn around 
 the horizon, closing and reading in the original zero line. Set B 
 vernier at zero and, with the telescope reversed, again set by lower 
 motion on some principal station. IJnclamp above and read land- 
 marks as before, closing -on the zero hne. In case either reading 
 on the closing line or direct and reverse reading on any landmark 
 do not check within one minute, repeat the entire operation. 
 
 THE STATION ADJUSTMENT. 
 
 121. After all the main angles have been satisfactorily observed, 
 closing the horizon, their sum should equal 360°. The difference 
 between 360° and the sum will be divided equally and appUed as a 
 correction to each angle, so that the resultant simi is 360°. This 
 adjustment is made in the notebook, and the final values checked. 
 
 EEDUCTION TO CENTER. 
 
 122. In triangulation of the character being described, it fre- 
 quently happens that angles are taken to some point, such as a tree, 
 where it is impossible to erect the transit. In such a case the instru- 
 ment is set up at the nearest convenient point, called a satellite station, 
 at a short (fistance from the tree', and the angles taken from this 
 station can be afterwards reduced to the true values at the object 
 itself. 
 
 From the satellite station which we wiU call S, the round of angles 
 wiU be read as usual, taking care to include the tree which we wiU 
 call T. Measure the distance from S to T carefully. Let A and B 
 represent two stations from which T has been observed, A being on 
 the same "hand" of B as is S from T, in this case the left. We 
 measured the angle ASB. The angle desired is ATB. 
 
 (a) ATB = ASB+SAT-SBT. 
 
■18 TOPOGBAPHY, MAP BEADING, AND BECONNAISSANCE , 
 
 To find SAT and SET we need to know the length of AT and BT 
 
 approximately. To determine them approximately, we can plot 
 
 the triangle ASB, lay off ST in direction and distance as observed 
 
 and measure AT and BT. This should be done at a large scale. 
 
 Now — 
 
 ST 
 (6) Sin SAT = Sin TSAj^ and 
 
 ST 
 (c) SinSBT=SinTSBg^* 
 
 From (I) find Sin SAT and from it the angle SAT. ^ , . „„ 
 
 From (c) find SBT, substitute these values in (a), and we find ^ ii^. 
 (See fig. 14.) 
 
 ADJUSTMENT OF ANGLES OF TRIANGLES. 
 
 123. In securing the triangulation control for an area of 100 square 
 miles or so, the purpose of which is primarily the construction of a 
 map on a reasonably small scale, the computations necessary for the 
 adjustment of quadrilaterals are not justified. Each triangle, there- 
 fore, win be adjusted as a single figure as completed and the com- 
 puted sides considered as correct, unless, of course, an actual mistake 
 IS discovered. It should be realized, therefore, that the fewer of 
 these main control triangles we use, the better for the work. If the 
 country is suited to triangulation, we may wish to have a number of 
 small triangles to control the actual taking of the topography, but 
 these sJioula not be made a part of the main triangulation system.' 
 They should be adjusted to it, not with it. Except for station adjust- 
 ment of angles, which is of course desirable, nothing should enter 
 into the adjifstment of any of the main triangles but the angles at 
 the verticals of each triangle. 
 
 124. It is evident, therefore, that if the triangulation net is ex- 
 tended to the sides as well as forward that each vertex located to 
 the side must depend for its adjustment upon the side used for a 
 base for its determination and no interadjustment between adjoining 
 side triangles must be attempted. Do the work carefully and you 
 wiU have confidence in the resulting system of control. 
 
 124. The angles of any one of the main triangles are to be adjusted 
 as follows: Add them together. The difference between the sum and 
 180° is the error. (This should not be greater than 30 seconds and 
 will be less if care has been taken to use rigid points to sight on and 
 Ught conditions have been favorable.) Distribute the error equally 
 (not proportionately) among the three angles so that their sum will 
 be 180°.' 
 
 AZIMUTH. 
 
 125. The initial meridian of the triangulation should be a true 
 north and south line. Azimuths in this work are reckoned from the 
 initial meridian or lines drawn parallel to it; from the south point 
 in the direction S-W-N-E and from 0° to 360°. 
 
TOPOGBAPHT, MAP BEADING, AND EEOONNAISSANCE. 49 
 
 COMPUTATION OF SIDES. 
 
 125. The base line is the only side of the triangle whose length is 
 known. The other sides are to be found from the law of sines, that 
 is — 
 
 Sig A _a a Sia B _, 
 
 SinB~b SinA""* 
 
 Sig A a a Sin C _ 
 
 SinC^c SinA"^- 
 
 Assuming a to be the hase and the angles A, B, and C to have been 
 measured, the calculations are arranged as follows: 
 
 (1) log a (1400.74) =3.1463575. 
 
 (2) colog Sin A (57° 42' 16") =0.0729874.' 
 
 (3) log Sin B (61° 17' 63") =9.9430639. 
 
 (4) log Sin C (60° 59' 51") =9.9418088. 
 Sum of (1) (2) (3) =log b =3.1624088. 
 Sum of (1) (2) (4) =log c =3.1611537. 
 
 COMPUTING THE COORDINATES. 
 
 127. The next step in the computations is to put the results in a' 
 form which permits accurate plotting and which enables a record to 
 be Tceft of the position of each trigonometrical point. The system 
 used will in this work be rectangular coordinates. 
 
 128. If practicable, the origin of coordinates should be so located 
 that the entire area to be surveyed will be north and east of it. The 
 coordinates of all points of the area will then be plus. This is to be 
 desired. 
 
 129. It is to be understood that ia rectangular coordinates the 
 initial meridian, that is, the north and south liae through the origin, 
 is the only true north and south line oj the survey. All azimuths are 
 referred to it and the conve^ence of meridians is neglected. This 
 is allowable in small areas. Through the origin of coordinates, two 
 lines are imagined, one line north and soum, which ia the initial 
 meridian, the other perpendicular to that line at the origin. The 
 location with respect to the origin of any point is estabhshed when 
 its de'^arture (perpendicular distance from the initial meridian) and 
 its latitude (perpendicular distance from the east and west line axis) 
 are l^novm. 
 
 130. The sign of departure when east of the initial meridian is plus. 
 The sign of a latitude when north of the origin of coordinates is plus. 
 It is usual to speak of the initial meridian as the Y axis, and the other 
 as the X axis. Differences in latitudes are, therefore, spoken of as 
 differenpes in Y, and differences in departure as differences in X. 
 
 131. If the coordinates of the point of origin are Tenown or assumed, 
 the coordinates of any other point which is tied to it by angle and 
 distance may be foimd by adding, algebraically, the difference in X 
 and the difference in Y to the coordinate's of the point of origin. The 
 difference in coordinates between the origin and the other point will 
 be obtained as follows : 
 
 Difference in X = distance to pomt x sin azimuth. 
 Difference in Y = distance to point x cos azimuth. 
 
 68740°— 18 4 
 
50 TOPOGEAPHY, MAP BEADING, AND EECONNAISSANCE. 
 
 The coordinates of any new point may similarly be determined from 
 those of any other point whose coordinates are known and to which 
 the unknown point is tied by angle and direction. 
 
 LESSON XXI. 
 
 TRAVERSE CONTROL. 
 
 132. The purpose of the higher or triangulation control just de- 
 scribed is to establish a few points in the area with a considerable 
 degree of precision upon which to hang the control of lower order. 
 If the country does not lend itself to triangulation, these few points 
 may be established by a traverse control. 
 
 133. If a railroad or reasonably level road runs through the area, 
 it is quite possible- to traverse it and measure it by stadia with an 
 accuracy of 1 in 500. If the distance across the area is 10 miles 
 (52,800 feet) the total error in the line would be about 100 feet, 
 probably quite as good a "fixation" as would be made by rapid 
 triangulation. There is a disadvantage, however, in the single-fine 
 traverse for main control in that there is no check on mistakes in 
 distance. 
 
 134. If the triangulation system has estabfished a known point on 
 which to close the traverse, well and good; but if there have been no 
 points established by triangulation, the main traverse must be run as 
 a loop, closing on itself or be remeasured as is a iase line. The follow- 
 ing procedure is preferred for this class of traverse : Through the origin 
 a true north-and-south liae will have been laid out. The first set up is 
 over the point of origin, and instrument is set to read zero pointing to 
 a pin ia the meridian south of the origia. This pin is the orientation 
 station for the first set up. 
 
 135. At the first station. — ^Party arrives at the first known position. 
 
 I. Transit is set up carefully over the station, with tripod screws 
 imclamped. Level approximately; center the instrument accurately, 
 with the plumb-bob just above the mark. Clamp the tripod legs. 
 
 II. Level with telescope level. Lower the needle. Measure and 
 record height of instrument. 
 
 III. Set the "A" vernier to read the azimuth to the orientation 
 point. If the A and B verniers do not agree in minutes, so set "A" 
 that the mean of the minutes on A and B will be the correct azimuth. 
 
 IV. With the lower motion, sight accurately on the orienting sta- 
 tion, clamping securely below. 
 
 V. Read and record magnetic bearing. Eecorder makes note of 
 declination. (Even witt. local attraction, the needle wiU check the 
 front with the back sights from the same station.) 
 
 VI. Unclamp above and take side shots, if any, reading (1) stadia, 
 (2) control vernier "A," (3) vertical angle. In taking these side shots, 
 clamp the afidade lightly. 
 
 Vn. After the last side shot, sight again on the orienting station, 
 checking the vernier reading. Plate levels are observed to see that 
 the instrument has not gotten out of level. 
 
 VIII. The front roinan, having chosen the new station with 
 respect to the conditions of having a good line fixrther on, topo- 
 graphic control and proper distance from the instrument, is signaled 
 *'up. " He presents the edge of his rod vertically over the new mark. 
 
TOPOGRAPHY, MAP READING, AND RECONNAISSANCE. 51 
 
 IX. At the instrument sight carefully with the upper motion on the 
 new station, clamping firmly. Signal the front rodman, who then 
 presents the flat stadia Tertically over the mark. The rod rests on 
 the mark, not on the ground. 
 
 X. Read and record (1) the stadia intercept; then set the horizon- 
 ,tal wire on the stadia number corresponding to the HI. Glance at 
 the plate levels to see if the tastrument is stiU level. Eead and record 
 in order (2) control vernier "A," (3) vertical angle, (4) check ver- 
 nier "B, " (5) magnetic bearing (needle). The recorder repeats all 
 readings as given, notifjdng the observer if the vernier readings or 
 the magnetic reading do not check. (See par. 139.) 
 
 XI. Signal down the front rodman. Raise and clamp the needle. 
 Put telescope in carrying position. Loosen the lower motion. Ob- 
 server with instrument and recorder move to forward station. Rear 
 rodman stays at the station. 
 
 136. At the second station. — 
 
 I. Set up accurately as before, level with telescope level. Un- 
 clamp the needle. Measure and record the HI. 
 
 II. Check the reading of the "B" vernier, which at this station is 
 control. If there has been a slip, reset the "B" vernier at the same 
 reading the "B" vernier had on the last foresight. Signal "up" the 
 rear rodman to present the edge of his rod vertically over the old 
 station. 
 
 III. With the upper motion still clamped and the telescope direct 
 (not "plunged") sight carefuUy with the lower motion on the rear 
 station, clamping firmly. 
 
 IV. Signal the rear rodman to present the flat of his rod vertically 
 over the mark. Read and record the stadia intercepts, the recorder 
 noting the check with the former fore shot. 
 
 V. Set the horizontal wire on the stadia number corresponding to 
 the present HI-, glancing at the plate levels. 
 
 VI. Read and record in order (2) control vernier "B, " (3) vertical 
 angle, (4) check vernier "A," and (5) magnetic bearing. Recorder 
 notes if the verniers check, if the vertical angle checks with former 
 front shot.. If the vertical angle does not check, remeasure the HI, 
 relevel with the telescope level, check the vertical circle adjustment 
 and reread the vertical angle. Take this value as final, the recorder 
 making special remarks to that effect in the notes. 
 
 VII. Signal down the rear rodman. Read the necessary side shots 
 as before, clamping the alidade (upper motion) lightly. The first 
 side shot should be well-defined landmark, preferably, at least 800 
 feet distant as an orientation check. On this shot read and record 
 both verniers and the magnetic bearing. 
 
 VIII. After taking the last side shot, call up the front rodman who 
 has chosen the new forward station and who presents his rod as 
 before. 
 
 IX. Check the azimuth on the orienting point (first side shot), 
 noting that the instrument is still level. 
 
 X. Sight carefully on the forward station, clamping the alidade 
 (upper motion) firmly. Then read and record as before (1) stadia, 
 (2) control vernier ("B" for this station), (3) vertical an^le, (4) 
 check vernier "A," (5) magnetic bearing. The "B" vernier has 
 remained control for this station. 
 
52 
 
 TOPOGRAPHY, MAP BEADING, AND BECONNAISSANOE. 
 
 XI. Clamp, the needle, loosen the lower motion, put the instru- 
 ment in carrying position and move forward again. ^ 
 
 137. At succeeding stations. — ^Proceed as above, 'A and B ^ 
 alternate as the control vernier. In this way, the azimuth is carried 
 forward without introduciug errors of coUimation, and errors due to 
 eccentricity, inclination of the horizontal axis and vertical circlp 
 adjustment, are automatically balanced and corrected. 
 
 138. If, on closing on a triangulation station or closing the trav- 
 erse on point of beginning, the azimuth is not out suflSciently to 
 indicate a mistake, the error in azimuth wiU be divided by the num- 
 ber of stations and distributed, the changes being noted in ink in the 
 note book. It wiU be noted that the control vernier gives the degrees 
 of azimuth. The minutes taken will be the mean of the minutes of 
 the two verniers. 
 
 139. Metlod of recording traverse notes. 
 
 L. A. traverse from A Asan to A Agana. 
 
 Station. 
 
 Stadia. 
 
 Azimuth. 
 
 v. A. -. 
 
 Dift. El. 
 
 From— 
 
 To- 
 
 H. Dist. 
 
 Control. 
 
 Check. 
 
 Elev. 
 
 ©4 
 ©3 
 
 gs 
 03 
 
 0LAZ 
 
 A-A-san 
 
 Mean 
 
 LAI 
 
 AChachao 
 
 525 
 820 
 
 820 
 
 935 
 
 935.5 
 
 936 
 
 347 
 
 348 
 
 349 
 
 135 
 
 134.5 
 
 134 
 
 •162 06 
 320 41 
 
 140 41 
 314 57 
 
 e 1 
 
 332 06 
 140 21 
 
 320 41 
 134 57 
 
 o ; 
 
 -0 01 
 -0 41 
 +0 40 
 +0 40 
 -0 06 
 +0 06 
 +0 07 
 +0 05 
 -0 04 
 -0 04 
 +3 25 
 24 
 -3 24 
 +1 11 
 
 30 
 30 
 30 
 30 
 
 
 ©LA& 
 
 134 57 
 310 28 
 
 134 57 
 130 28 
 
 
 ©LAI 
 
 130 28 
 45 10 
 
 310 28 
 225 10 
 
 
 
 225 10 
 179 32 
 
 45 10 
 259 32 
 
 
 
 
 
 140. The azimuth error distributed, the differences in X and Y 
 are computed as stated in paragraph 131. 
 
 Care must be taken to give the computed latitudes and departures 
 their proper signs. With azimuths reckoned from the south point 
 as zero and in the direction of the movement of the hands of a clock; 
 
 For azimuth between and 90, latitudes are negative and depar- 
 tures are negative. 
 
 For azimuth between 90 and. 180, latitudes are positive and depar- 
 tures are negative. 
 
 For azimuth between 180 and 270, latitudes are positive and 
 departures are positive. 
 
 For azimuth between 270 and 360, latitudes are negative and 
 departures are positive. 
 
 141. If the azimuths and lengths of courses had been exactly 
 determined m the field and no error has been made in computa- 
 tion, the survey would close, that is, the sum of aU the plus latitudes 
 (northmgs) would equal the sum of all the minus latitudes (south- 
 ings) and the sum of all the "eastmgs" would equal the sum of all the 
 "westings." Such exactness is, of course, not attainable and the 
 sum of all the "northings" will not equal the sum of aU the "south- 
 ings," nor the "eastings," the "westings." 
 
TOPOGRAPHY, MAP READING, AND RECONNAISSANCE. 53 
 
 142. The total error in latitude is the difference between the sum of 
 the northings and the stun of the southings. Similarly the total error 
 in departure is the difference between the sum of the eastings and the 
 sum of the westings. 
 
 143. As the errors in latitude and departure are those due to dis- 
 tance alone (the error in azimuth ha-ving oeen distributed before com- 
 putations), the corrections should be appHed according to the lengths 
 of the courses. The rule is: The correction to be apphed to the 
 latitude of any course is to the total amount of the error in latitude 
 as the latitude of that course is to the sum of all the latitudes (with- 
 out regard to algebraic signs). A similar rule detemunes the cor- 
 rection in departure for each course. The corrections to latitudes and 
 departtires of each of the courses having been applied, the total lati- 
 tudes and departures (coordinates) of a station are found by adding 
 algebraically to the latitude or departure of the initial poiat of the 
 traverse, the algebraic sum of all the latitudes or departures of the 
 preceding courses. These coordinates are used for the plotting of 
 the stations. 
 
 144. The main control traverses having been run, computed, and 
 adjusted as above described, the stations of the traverse are plotted 
 ia ink, station numbers and elevations being noted in pencil. This 
 done, a careful tracing of aU the data on the projection sheet is made 
 on vellum, cut to a size to fit the engineer sketching board. Each 
 topographer is given one or more of these field sheets to fill in. When 
 practicable, the areas assigned to a topographer wiU be those over 
 which he and his party have been engaged in running the main trav- 
 erses. Boundaries of these field sheets should be roads, railroad or 
 trails. If this be impracticable, then well defined physical feature, as 
 streams or cutting through brush and timber. 
 
 VERTICAL CONTROL. 
 
 145. Over the main traverse lines it is advisable in most cases 
 to run a line of levels. (Par. 205.) 
 
 SUBSIDLVRY CONTROL. 
 
 146. Each topographer now proceeds to complete the control of 
 his area. His procedure wiU be as foUows: Beginning at a con- 
 venient "fixed" point in his area, he runs, or causes to be run by 
 Ids assistants, flying transit and stadia, compass and notebook, or 
 sketching board traverses; using stadia or aneroid or clinometer 
 for carrying elevations, as may be required by the length and impor- 
 tance of the line and as preferred by the individual. In this manner 
 the topographer searches out the complete network of roads, trails, 
 and streams for the entire field sheet on which he is engaged. This 
 network should be so complete that the contour sketcher will need 
 do no adjusting in his work. This will require that the distances 
 between subsidiary traverse lines or other fixations should not 
 exceed 1 mile on the scale of 1 mile to the inch or in general 1 inch 
 on the map whatever the scale. 
 
 147. It is to be understood, of course, that the subsidiary traverses 
 run by the topographers between points of higher control, etc., 
 must not be plotted on the topographers record sheet until they 
 have been properly adjusted. In long important lines, the adjust- 
 
54 TOPOGEAPHY, MAP EEADING, AND KECONNAISSANCE. 
 
 ment will be made by latitudes and departures, as described for 
 main traverse lines. In the shorter lines, these coinputations are 
 not justified and the graphical methods indicated m figures 12 ana 
 
 13 will be used. , ,. , j. •■ j. 
 
 148. Figure XII shows method of graphical adjustment lor a trav- 
 erse run between two points, the positions of which are already 
 plotted on the record sheet. Proceed as follows: 
 
 Plot the traverse, as actually measured m the field, in plane 
 table work, of course, this is already done. Draw a Jme A ±5 Irom 
 the beginning point to the ending point of the plat. Now from the 
 record sheet m'easure the straight line distance between the record 
 positions of A and B. Measure off this distance from A on Ime A 13, 
 Call the end of the distance h. t. at j? 
 
 Take any convenient point O and draw O A and O B. Now from 
 h draw line parallel to O A. Where this line crosses O B is new 
 position (B') foj B. From B' draw line parapllel to A B. Where 
 this line intersects A O is new position of A. 
 
 To find new position of station 1, draw line from A' parallel to Al 
 and where it intersects 01 is now position (1') of 1. From 1 draw 
 line parallel to 1-2, etc. 
 
 The traverse A', 1', 2'-B' is now adjusted and can be traced off 
 onto record sheet. 
 
 149. Figure XIII shows method when a circuit is run from a known 
 point back to the same point. Plat the traverse as measured in the 
 field. The point A' should be at A, but, due to errors, does not so 
 
 f)lat. On a long straight line O A lay off from O, in succession, the 
 engths of the courses (A-1, 1-2, 2-3, etc., — — — ■ 13-A'). From 
 the end of this fine lay off in any convenient direction the line A B 
 equal to the error in closure A'A. Connect the outer end of this 
 offset line to O. Now from each succeeding station point on the 
 ZoJMT line, draw a line parallel to the offset line A B. 
 
 Returning to the plat of tte traverse, draw through each plotted 
 station a line parallel to the final closure line A'A and in the direc- 
 tion of the closing station. On each line lay off its respective offset 
 length, giving new positions for each station. Connect these new 
 stations and the traverse is adjusted. 
 
 SKETCHmG. 
 
 150. The field sheet which has been thus prepared consists of a 
 network of traverses along roads, trails, streams, and across country 
 with the horizontal detau along these traverses, with stations and 
 easUy identified objects shown ia their proper adjusted locations and 
 with their proper adjusted elevations given. It is the contour 
 sketcher's task to go over the area assigned to him, sketching the 
 draiaage, culture, and forms of relief; generafizing the features to 
 suit the scale and purpose of the map. The course in sketching wiU 
 have taught the sketcher how to sketch rapidly. He must, on this 
 work, follow similar methods, but "watch his step," as the map must 
 answer severe tests. Especial care must be taken to see that aU 
 prominent landmarks are so located that they may be later used as 
 orientation points in sketching or map reading. 
 
TOPOGEAPHY, MAP KEADING, AND EECOKNAISSANCE. 
 
 55 
 
 
 FIOURE XIV 
 
66 TOPOGBAPHY, MAP HEADING, AND RECONNAISSANCE. 
 
 LESSON xxn. 
 
 MONUMENTS, OFFICE RECORDS, AND DRAFTING. 
 
 151. Any properly made survey consists of three distinct parts. 
 These are: (a) Permanent marks or monuments on the ground, (&) 
 complete records of the field work, properly filed, and (c) the pub- 
 hshed sheets or copies issued for use. 
 
 (A) MONUMENTS. 
 
 152. A map represents to scale a certain portion of the earth's 
 surface, everything shown being presumably properly located with 
 
 -respect to everything else, but if it ever becomes necessary to add 
 new matter to the map or if we wish to extend it over a greater area 
 or to incorporate it as part of a larger survey, we must be able to 
 locate the same points both on the map and on the ground. Only a 
 Hmited number of points in any map have their positions with rela- 
 tion to each other determined with great care. In the case of tri- 
 angulation these are the triangulation stations; if the survey was 
 controlled by traverses, they are the traverse stations. If, then, we 
 put in permanent marks at each triangulation station, or, in the case 
 of traverse control, permanent marks at certain traverse stations 
 and also show these stations on the map, the relation between the 
 map and the ground is estabhshed for as long a time as the monu- 
 ments are undisturbed. 
 
 For the same reasons it is also essential to leave permanent level 
 bench marks on the ground and to show their location on the map. 
 
 (B) OFFICE RECORDS. 
 
 The real map is the office record, the maps issued being copies 
 of it. The office records consist of the field-instrument notes, the 
 projection sheets on which the control is plotted, and the field topo- 
 graphical sheets. AH office work should be plotted on double- 
 mounted drawing paper, which is little affected by atmospheric 
 changes and practically uniformly in all directions. These records 
 must be filed so as to be easily accessible at all times. Double 
 mounted paper, or, for well-controlled work, single mounted paper 
 or vellum is used for field' topographical sheets. Celluloid is very 
 useful for field topographical sheets in regions where there is much 
 rain or dew. 
 
 (C) PUBLISHED COPIES. 
 
 Copies issued for use may be .blue prints from a tracing, black 
 prints from a paper negative, or hthographic copies. For any large 
 issue the usual military practice will be to print from zdnc plates. 
 {oee Map reproduction.) 
 
 DRAWING. 
 
 153. The essential requirements of a good topographical drawing 
 are accuracy and clearness. By accuracy is meant a faithful exhibit 
 of measurements and observations made in the field, or of data taken 
 from other maps. Clearness involves absence of confusion or crowd- 
 ing, and neatness in execution. Beauty and pictorial effect are ob- 
 
TOPOGRAPHY, MAP HEADING, AND KECONNAISSANOE. 57 
 
 tainable by skilled draftsmen only, and while always desirable, are 
 rarely necessary. Persons who are not skilled draftsmen should 
 not attempt pictorial effect, as it wiU detract from accuracy and 
 clearness without substituting anything of equal value. 
 
 Avoid unnecessary haste in plotting and drawing. If possible, take 
 time to check carefully all azimuths and distances plotted and be 
 sure they are exact. There should be no approximation on the 
 drawing ooard. Although an observer may have simply guessed a 
 distance to be 550 yards in the absence of other information, the plot- 
 ter should be careful to lay it down at exactly 550 yards. 
 
 Start with clean paper and keep it as clean as possible. In the 
 office, wipe off the instruments before using, especially nilers', scales, 
 and triangles. Dust the drawing carefuUy before beginning work. 
 Dust again when stopping and cover with a cloth or paper. If 
 necessary, dust the drawing and wash the hands occasionally while 
 at work. 
 
 Make all ink lines firm and very black. — ^A drawing to be made in 
 ink is usually drawn first in pencil, and in such cases a very hard 
 pencil (4H or 6H) is best. If the pencil drawing is to be traced, a 
 softer and blacker pencil should be used, but must be kept well 
 pointed. ' 
 
 India ink in stick form gives the best results, but the time required 
 for proper grinding precludes its extensive use in military field work. 
 The prepared india inks in liquid form are ready for use and are 
 satisfactory. They must be kept well corked when not actually 
 filling a pen. If the ink gets thick in the bottle so that it will not 
 run freely from a fresh-filled pen, add a little water. 
 
 Papers. — Manila paper of cream or buff tint, usually called 
 detail paper, is suitable for sketches and drawings which are to be 
 traced or used in the field. Only the better grade stands erasing, 
 and that imperfectly. This paper comes in roUs 36, 42, and 54 
 inches wide. It may be ordered by the pound or yard. 
 
 White drawing paper may be had in roUs or sheets mounted on 
 muslin or unmounted. Whatman's cold-pressed fine-grain is most 
 generally useful. It comes in sheets of names and sizes as follows: 
 Eoyal, 19 by 24 inches; Imperial, 22 by 30 inches; Double Elephant, 
 27 by 40 inches; Antiquarian, 31 by 53 inches. EoU papers are 27 
 to 63 inches wide. 
 
 Sheet papers unmoimted and kept flat are best for field -topo- 
 graphical use. 
 
 If a blot drops on the drawing, take a piece of blotting paper, 
 tear a corner or edge to expose a fresh surface, and hold it in the blot 
 without touching the drawing until the surplus ink is absorbed. 
 Then press a dry blotter firmly on the spot and let it dry thoroughly 
 before attemptmg to erase. A piece of newspaper may be used 
 instead of blotting paper, but should be slightly moistened to hasten 
 the absorption. For a large blot several pieces may be required. 
 
 Erasers for ink are of steel or rubber. A steel eraser or penknife 
 must be very sharp to give good results. An eraser of gritty rubber 
 is most generally used. It is best to use an erasing shield of thin 
 metal or celluloid, which exposes the area to be erased through one 
 of the openings and protects the rest. 
 
58 TOPOGRAPHY, MAP BEADING, AND KECONNAISSANCE. 
 
 Tracing linen is usually dull hack, having one side glazed and the 
 other dull. Erasing can be done on the glazed side -only. The 
 glazed side is used lor ink and the dull side for pencil work. The 
 glazed side requires preparation before use to remove excess of 
 paraflSn, which prevents mk from ruiming well and clogs the pen. 
 Rubbing hard with fresh blotting paper is the simplest method. 
 
 Tracing paper is alike on both sides. It will not erase. Most 
 varieties are less transparent than tracing cloth. 
 
 In tracing, it is helpful to use a dtdl-pouited instrument m the 
 left hand — a stylus or top of a penholder — to press the linen against 
 the drawing at the point where the pen is restmg. 
 
 ENLARGEMENT AND REDUCTION. 
 
 154. The simplest method is by squares. Divide the original into 
 squares of 2 inches or less by hnes drawn parallel to the borders. 
 Divide the paper on which the copy is to be made into squares with 
 sides corresponding to the same distance on the scale of the copy 
 that the side of a square on the original itself does to the scale of 
 the original. If a plotting scale of the original be placed oh the side 
 of a square on the originS and the plotting scale of the copy on the 
 side of a square of the copy the readings should be the same. The 
 square on the copy will be larger if the drawing is to be enlarged 
 and smaller if it is to be reduced. The ratio between the sides of 
 the squares on the origuial and the copy is the ratio of reduction or 
 enlargement. This ratio must not be confused with the ratio of 
 areas of the two maps, which is different and not important. 
 
 Select a square of the original and reproduce its contents in the 
 corresponding square of tjie copy, or take a feature of the original, 
 as a road or stream, and trace its course through several squares. 
 
 Usually the position of a point in a square or on one of the sides 
 can be estimaUd with sufficient accuracy. Important points may 
 be located by measurement of distances from the nearest sides of 
 the squares, using the scale of the map and the scale of the copy, 
 respectively. 
 
 Instead of drawing the squares on the original, they may be drawn 
 on tracing linen or paper laid over it,'' or fine threads may be stretched 
 to form flie squares. Every drawing board shotdd have a scale of 
 inches on each edge marked with fine saw-cuts or with small tacks 
 to facilitate the drawing of squares. 
 
 CHARACTER OP WORK. 
 
 155. All lines must be clear, sharp, and distinct, drawn or printed 
 in jet black, waterproof ink. Colors will not be used except when 
 clearness of representation absolutely demands their use; special 
 pains will be taken to avoid them in drawings intended for publica- 
 tion. So far as practicable, uniformity of size must be mamtained 
 in figures or letters presentiag a particular Mnd of information, such 
 as soundings, elevations, names of proprietors, names of minor 
 towns, names of counties, etc., and figures and letters must be clear, 
 distinct, and readily legible, especisQly on drawings intended for 
 pubHcation, in which case the letters and figures must be made of 
 size lai^e enough to avoid blurring and obscurity when reduced by 
 
TOPOGRAPHY, MAP READING, AND RECONNAISSANCE. 
 
 59 
 
 photolithography. The use of type for letters and figures and of 
 films or rollers for representing typographic features is recommended. 
 
 The title should be placed in the lower right-hand comer, unless 
 use of this space for other matter is absolutely unavoidable. If 
 practicable, the size should not exceed 5 inches by 5 inches. The 
 use of such words as "Map of," "Plan of" is thought to be unneces- 
 sary. The approval of the ofloicer in charge will be placed near, 
 and preferably at the side of the title. At the upper left-hand cor- 
 ner (and outside of the border, if a border is used) the words "War 
 Department" will be placed; and at the upper right-hand comer, 
 in similar position, the wording "Corps of Engineers, U. S. Army," 
 will be placed. When the title is p.ot placed as above, a brief title 
 will be placed outside the border, on lower right-hand comer, to 
 identify the drawing. 
 
 A graphical scale should be shown in all cases, and a statement 
 of the scale may be added when useful. Drawings submitted for 
 publication must be prepared with a view to reduction to the smallest 
 practicable compass, and they must bear a statement in pencil, on 
 the lower margin, of the amoimt of reduction contemplated. AH 
 maps are usually reduced at least one-third. 
 
 The true meridian should be shown on aU maps or charts of land 
 or water areas, if known, and the magnetic declination should also 
 be given. 
 
 In the case of topographic or hydrographic surveys, a brief descrip- 
 tion should be placed upon the map or chart, in tabular form, of the 
 principle triangtdation points or bench marks, including the general 
 location and character of the monuments or bench marks, the 
 coordinates and the elevation. Whenever possible the direction of 
 water flow of all waterways must be indicated by arrowheads point- 
 ing in the direction toward which the water moves. Tidal flow 
 must be indicated by double or multiple arrowheads. 
 
 The following are the standard topographical symbols and styles 
 of lettering to be used on all maps: 
 
 A Arroyo. 
 
 abut . . . -Abutment. " 
 
 Ar Arch. 
 
 b... Brick. 
 
 B. S Blacksmltli shop. 
 
 bot Bottom. 
 
 Br Branch. 
 
 br Bridge. 
 
 C Cape. 
 
 cem Cemetery. 
 
 con. ..--.Concrete. 
 
 cov Covered. 
 
 Cr Creek. 
 
 d Deep. 
 
 cul Culvert. 
 
 D. S Drugstore. 
 
 E East. 
 
 Est Estuary. 
 
 f fordable. 
 
 Ft Fort. 
 
 ABBKEVIATIONS . 
 
 G. S General store. 
 
 gir Girder. 
 
 G.M.... Gristmill. 
 
 I Iron. 
 
 I /.Island. 
 
 Jc Junction. 
 
 k.p King-post. 
 
 L Lake. 
 
 Lat Latitude. 
 
 Ldg Landing. 
 
 L. S. S.. Life-saving station. 
 
 L. H Lighthouse. 
 
 Long Longitude . 
 
 Mt Moimtaiu. 
 
 Mts Mountains. 
 
 N North. 
 
 n. f Not fordable. 
 
 P Pier. 
 
 pk Plank. 
 
 P, O Post office. 
 
 Pt .Point. 
 
 q.p Queen-post. 
 
 B River. 
 
 R. H- - .Roundhouse. 
 
 R. R Railroad. 
 
 S South. 
 
 s ;. Steel. 
 
 S. H SchooUiouse. 
 
 S.M.-- Sawmill. 
 
 Sta Station. 
 
 st Stone. 
 
 str Stream. 
 
 T.G..--Tollgate. 
 
 Tres Trestle. 
 
 tr Truss. 
 
 W.T... Water tank. 
 W. W.. Water Works. 
 
 W West. 
 
 w Wood. 
 
 wd Wide. 
 
60 
 
 TOPOGEAPHY, MAP EEADIKU, AKD KECOISTNAISSANOB. 
 
 Brid^ 
 
 + 
 
 Indicate character and span by abbreviations. 
 
 Example: 
 
 VJiliift. 
 
 40x20 
 
 Meaning wooden kingpost bridie.40feet lon|,20feet wide, 
 and 10 feet above the waten 
 
 Streams — . —^^ 
 
 Indicate character, by abbreviaEon* 
 Example: "'"■i^is^^»V2^ 
 MeanlnlastreamlSfeet wide,8feel<feep.andmt fbrdabfe. 
 House - Church* School house "SH. 
 
 Wooda J^^l^ Orchards irn) Cultivated UndlcUit 
 
 If boundary lines are fences they are 'indicated as such. 
 Brush, crops or Ipasa, important as cover or forage 
 
 TreeSiiaolatecJ 
 cut IOfeetdee[> 
 fill 10 
 
 BruaKcom,! 
 fraas.ete. - I 
 
 Cemetery 
 
 " + 
 
 + 
 
 V%| 
 
 
 Cut and fill- 
 
 
 . Gut 
 
 . 
 
 ' 10' 
 
 . / fill 
 
 
 
 — v-0 ■ 
 
 ■<T— 
 
TOPOGRAPHY, MAP KEADING, AND EECONNAISSANOE. 61 
 
 J&leffraph, Zin» 
 
 Sjjoiel 
 
 „T * rrTTTTTTT 
 
 ,„-.. I I t I i j r 
 
 "l ( l I I 
 
 l | l lHt | l |ll|l| l |l | l jl | 4H I 
 
 ISinfflerThack 'kj i h ; 
 
 \l>oiii>la S^aok L 
 
 RaUroads ilWo Railroads . ! 
 
 I Urban or 
 
 \_Siibiirbaiv 
 
 /J^Claas. .Metaled ... 
 
 Z^Clatx...CoiaUiy Road, ^ 
 fgooJL) 
 
 I S^CUftjOouitttyltoaJ. ::::::: ===== 
 
 St»ep JholinA.... 
 
 V]h«2 ffp fiU/w. 
 
 Jtoad Crossings 
 Gmde- „_. . 
 
 Ahoye' Grade/.- .... 
 
 JSelow Grada 
 
 . I I I I t I I I I I , 1 ., t I t 
 
 I i I I I I J M t II t I 1 I I t ^ 
 
62 TOPOGKAPHY, 'MAP KEADING, AND EECONNAISSANCB. 
 
 -V 
 
 Cify ar VvCLoffa 
 
 B]rp.3625 
 
 Capital @ ■ Counfy Seat @ Ci^ or VOlaga ...» 
 
 SvalcUn^a^ .. i. .- - ,. — 
 
 JnanfftdatioTv StatUm — ■ . 
 
 Flane-tdbUt Statian. _ .._ ; — _ 
 
 Common: Sw^ay Statian ._ — 
 
 'Benohj Mctrh -■ 
 
 JkSnes OTvd, Quarries ^_.,^ 
 
 BOtlNBARY LINES 
 
 StoUe^Zava _^__ _ _ , 
 
 /3« 
 
 « 
 
 Caiaify arPrdvinda - 
 Jbimship. ar£arrio.. 
 S&SeiryatimL _ ■. 
 
 Lettering arv Baifridcay Unas .. 
 
 irEW YORK 
 
 VERMONT 
 
TOPOGEAPHY, MAP BEADING, AND EECONNAISSANCE. 
 
 63 
 
 Medioal Carps „. 
 
 OrAutnoa „_. 
 
 Signal, Corps... 
 
 Engineer Corps ^_ 
 Gun £aiiery...^ 
 
 Mortar Battery --.....-..-^.^...^ 
 
 i&rt TlhiA plcfn to he. 
 
 Redmd>t,A. shcnrtv ifkn/own, 
 
 Cantp L . , . „ 
 
 HaiSe- . — : 
 
 JZ?«!?i€!^*— — i - I.- ....*.— . — 
 
 8 
 ,f5 
 
 T|r-ip 
 
 •H 
 
 AAA 
 
 ....■^ 
 
 OBSXACZ£S 
 3rOT£. WTierv ooloris used, eaxcuto iiie^» in red 
 
 Muf^,. ..,.- ,... ..-....^.....if. '^\'^ 
 
 Viri' irdanffUmeiht , 
 PaUsades.,..^ 
 
 Contact iSnes „ 
 
 Controlled ISnes.. 
 Demolitions.- 
 
 ..^.-, .P^V>) 
 
64 
 
 TOPOGBAPHY, MAP BEADING, AND BECONNAISSANCK. 
 
 JteffimerOal S&adtpj/zrters 
 
 Srigajd0 " 
 
 Dinsiiyn " 
 
 Carpa " - ; 
 
 fftfititiry tnt Une' , , , 
 
 jbuinttry ixv coliamv ~~~^.ji .h— ^ 
 
 Ccwdby izv liner. .•„. 
 
 Gam^Bt coTiortn ._„.i.. 
 
 ..&! 
 
 .2B 
 ■ 4.0*35 
 
 S0^3C 
 
 'Sac 
 
 MouTtled, .Bcfizntry _. .., 
 
 Artillery. . ^ ._ ,., . 
 
 Sertby . „ 
 
 II II 
 
 Sxtppott " « M , .^ _ 
 
 Vagontraai/.^,..— .^.,._ , 
 
 .1' 'I' 1' 
 -i 
 
 A^tdant Generat „...,^„ . 
 
 Quarter-Tnaster' , ,^., .^^._,_ 
 
 Contiiuss^Ty . . —. „ _. 
 
TOPOGRAPHY, MAP READING, AND RECONNAISSANCE. 
 
 65 
 
 Gajg& of Leiters 
 (in, DecimUUmMeirs ) 
 
 ^ Im i i W = 
 
 HKLt UUItlM-l 
 
 -rtf.wnn E 
 
 IIJIMIW I, /. 
 
 !■ iLituvnTTT" 
 
 -h^Ut rt Knf gc 
 
 WKWmiWTT- 
 
 is l.nJi'i^ f^i'./tfvjTPr 
 
 - HHKK'I' Nn I 20 
 
 "AL HAN Y 
 
 22 RfU-.ICy- MTS 
 
 '26 
 
 ^c i^/rrnMAr. r. 
 
 niSTRUVl— 3s 
 
 (-.Al.VKR'r 
 
 jmTSEHL 
 
 VIH(^IN»A 
 
 PACIELC: 
 
 IIHK) 
 
 JTdckness of Utter 4 ofheiffjit. ^ j\ 
 
 Slope> of IMter 3 peats of base- to 8 ofheigTit. i.J 
 
 58740°— 18 5 
 
66 
 
 TOPOGRAPHY, MAP BEADING, AND BECONNAISSANCB. 
 
 HYPSOfiRAPHX 
 
 Mountains, tlaUaus, Lirve^ of Cliffs 
 oTudL CanyoTis (tM^apibal let^ra) 
 
 ABCDEFGHIJKLMNOPQRS T U 
 VWXYZ 
 
 ■P&aks. small VcMeofS. Islands aruL Jhinti 
 
 (with Cap. xniiials) 
 
 abcdefghijklmnopqrstuvwxy? 
 
 PU BLIC WOR KS 
 
 Railroads, Tunnels. Srvdge'S, F&rrie,s, Wagon-roada , 
 Trails. Fords and Darns foapiteds anfyj 
 
 .A$COEr.GHIJKLMNOP9RSTUVIVJ(yZ 
 
 C OIST TQUR KUMBE RS 
 
 JiMUfy contour's .I23A-S61B90 
 Light contours- Ks^e^aBo 
 
 MAR GINAL LETTE RING 
 
 AB CDEFG-HIJKLMNOPQRSTU 
 VWXYZ 
 
 fWUh, Cap, initials J 
 
 abcdefghijklmnopqrstuvwxyz 
 
 1234.567890 
 
TOPOGRAPHY, MAP EEADING, AND RECONNAISSANCE. 
 
 67 
 
 1- R. F. = -Jg-=8:33 to 1"= 633:6 to 1 mile. 
 
 10 98765*8810 
 
 10 feet. 
 
 2- R. F. = f^Q- = 10' to f — 528" to 1 mile. 
 
 10 987 6 543210 10 feet. 
 
 3- R. F. =^ 5^=41'.66 to f= 1261? to H mile. 
 
 so 
 
 25 
 
 50 feet. 
 
 4'- R. F.= -ggg- = 50' fo l"=105:6 to 1 miJe. 
 
 50 
 
 25 
 
 50 feet' 
 
 5- R. F.= :^^=352' to !"=• 15' to 1 mile. 
 
 100 
 
 50 
 
 ICO 
 
 200 yds. 
 
 6- R. F. = -?5M=440' to 1"= 12" to 1 mile. 
 
 20 yda. 
 
 5280" 
 
 100 50 
 
 H H H H-hU 
 
 100 
 
 7- R. F.'= :fo5Qo=833:3 to 1=6:34 to 1 mile. 
 
 100 
 
 HHHHHl7= 
 
 ICO 800 . 300 400 SOOyds. 
 
 8^ R.F.= inbi==-880'to t" 
 
 10560 
 
 6" to 1 mile. 
 
 1MU„0, 
 
 ■ H M M H H t — 
 
 100 200 300 400 SO Oyds. 
 
 — I I II I 
 
68 TOPOGRAPHY, MAP BEADING, AND BEOONNAISSANCB. 
 
 9'R- F"f==-2o5oo = ''^^^''' ^° l"=3'.'l7to l:mile. 
 
 100 MO 100 yds. 
 
 100 500 1000 yds. 
 
 imit- — I , i-^.„. . h-H . .I — J. I— I I 
 
 11' R- F^52800 ^ '^^^^' ^^ ^"^ ^-^ *° ^ '"'''®- 
 
 IDOO 1000; 2000 yis. 
 
 ta n 1-1 i-i r-i , . i . . i . .i 
 
 '2'R. f^6336r*'52^°"^° 1"=^1'.'00 to 1 mile. 
 
 iqOO 1000 2000 yda. 
 
 O . M t-l 1-1 M f-T- . I , , , J 
 
 13- R. Ff-i26720 ^105^0' ^'^ 1"=0"50 to 1 mile. 
 
 10 00 1000 2000 8000 ,4000 SOO O yda.- 
 
 HHUHHI— — J I— -— j _ , l_ L 
 
 14- R. F.=g3|gQo=52800' to 1"=10 miles to 1" 
 
 l A 9 8 7 6 5 4 3 2 10 10 miles. 
 
 1^ I— I h-< K- 1 l-l .1 ==l 
 
 15- R. Ff= )384Q0Q -132000' to 1=25 miles to 1" 
 
 ■ 1 0-. ,10 ao 30 40 miles. 
 
 i-j ^llM MM I I ' I I I 
 
TOPOGRAPHY, MAP READING, AND RECONNAISSANCE. 
 
 69 
 
 s S 8 iq 
 
 s s 
 
 I 
 
 r 
 
 Miles per hou!* 
 i 
 
 I ! 
 
 
 :s. 
 
70 
 
 TOPOGBAPHY, MAP EEAWNG, AND EECONNAISSANCE. 
 
 00^1 
 
 Length of pace ! I 
 
 I i I ■ ' 1 J 
 
 ^ ^ 3 ° * ~ 
 j In Inches 
 
 0088 
 
 
 OOfZ 
 
 
 tn 
 
 
 0008 S 
 
 u 
 
 
 
 0091 °- 
 
 :^ 
 
 ■s 
 
 »- 
 
 cost a, 
 
 o 
 
 008 3 
 
 ^" 
 
 CO 
 
 
 00& 
 
 
 
 
 
 )0S 
 
 
 00» 
 
 
TOPOGRAPHY, MAP EEADING, AND EECONia'AISSANCE. Yl 
 
 LESSON xxm. 
 
 MAP KEPRODUCTION. 
 
 156. The TiektograpJk is one of the most satisfactory means of repro- 
 ducing maps, drawings, etc., in the field on account of the simphcity 
 of the equipment. Staple compound for hektographs is sold by 
 Frederick Post Co., Chicago, at 25 cents per potmd and the hekto- 
 graph can be readily made therefrom, or it can be purchased set up 
 from the company. The only apparatus needed is the hektograph 
 and hektograph ink and the paper necessary for impressions. The 
 drawing must be traced with hektograph ink on a paper with a non- 
 absorbent^surface. The drawing is then placed ink side down upon 
 the hektc^aph and gently rubbed until the ink has thoroughly taken 
 oh the hektograph material. This , may be detemuned by liftiag 
 one corner of the drawing from time to time. When the trans- 
 ference is completed, impressions are made by laying a piece of paper 
 on the hektograph and rubbing over it when it readdy takes the 
 mpression. From 30 to 75 readable copies can be thus obtained. 
 When the impressions begin to fad, the original drawing may be again 
 applied to the hektograph, which has been washed off, and a number 
 of additional copies secured. When the work is completed the sur- 
 face of the hektograph is washed off with light touch of a damp sponge, 
 aJl ink readily clears, and the hektograph is ready for further work or 
 to be stored until needed. 
 
 The compound above referred to ^ves most excellent line impres- 
 sions and is guaranteed to stand any climatic conditions. The inks 
 come in a number of colors and added value attaches to a field 
 process that will give you in one process reproduction ia several 
 colors. 
 
 Twenty-five readable impressions have been obtained from a sketch 
 made with an ordinary purple copying, or iadehble pencil. 
 
 The hektograph was considerably used by the Japanese for map 
 reproduction work during the Manchurian War. 
 
 Photographic, brown print, black print, or blue print papers being 
 available, reproduction of maps or drawings from photographic 
 plates, films, or tracings can be readUy made. Using the thinner 
 brown or black print paper, negatives can be prepared, and the speed 
 of production of prints increased by printing from these and they 
 also furnish a most excellent form m which to store such work for^ 
 future reproduction. With most of the above papers it is possible 
 to print direct from an original drawing by greasmg it and rendering 
 it tiiereby translucent. A most excellent grease for this purpose is 
 so-caUed "banana oil." _ , , ,, -, . , 
 
 The brown and black prmt papers are developed by washing only, 
 and are fixed by a weak hypo solution. Blue print paper is devel- 
 oped by washing and no fixing is reqmred. The photographic papers 
 mU be handled exactly as in ordinary photographic work, employing 
 the same developer and fixing bath. „ , ^ , .• - 
 
 157 Zincographic processes. — The method of reproduction ot 
 most general application is the zincographic process, which may be 
 employed in two ways— the photographic and autographic. 
 
 158 The autographic process has the advantage of being entirely 
 independent of hght. It 'consists in drawing on transfer paper (a 
 
72 TOPOGRAPHY, MAP READING, AND RECONNAISSANCE. 
 
 trade article) vnth autograpliic ink (a* trade article wliich, _caii be 
 obtained in either liqxiid or stick form) and transferring this drawing 
 to the surface of the zinc. This transfer is accomplished by placing 
 the transfer paper, after completion of the drawing, between dam- 
 pened sheets of^blotting paper untU it is thoroughly moistened, but 
 not wet. The transfer paper is then placed, ink side down^ upon the 
 zinc plate, the latter having been given a slight grain by unrnersion 
 for one minute in a solution of 1 gallon of water to 2 ounces ot com- 
 mercial nitric acid and 1 ounce of alum, after which it is dried. The 
 zinc plate with the transfer paper upon it is then run through the 
 Ethographic press with a very light pressure, just enough to 
 straighten the paper out upon the plate. The pressure is then 
 gradually increased until a maximum is reached at the 'fifth run 
 through. Take the plate out ai;id run it through the press in the 
 opposite direction five times, increasing pressure as before. Moisten 
 the transfer paper with a damp sponge and remove it from the plate, 
 when the ink will be found to have entirely left the paper and the 
 image will appear on the surface of the plate. Wash the plate to 
 remove the glue particles which were transferred from the glossy 
 side of the paper (upon which side the drawing mustbe made, care 
 being taken not to touch it with the hands) and which wiU adhere 
 to the zinc and which, if left on the plate, will cause trouble later by 
 taking up the etching powder. The plate will stand hard scrubbing 
 with a sponge, after which it should be wi_ped off with a damp chamois 
 skin and then dried by natural or artificial heat. 
 
 The lines now should be built up a little and this accomplished 
 by dusting the plate, which should be at about the same temperature 
 as its surroundmgs, with etching powder. This powder will adhere 
 to the ink fines and should be brushed from the clean surface of the 
 zinc with a camel's hair brush or a tuft of dry cotton. An overheated 
 plate will take etching powder everywhere. The lines of the plate 
 now appear red and the plate should be heated (gas or kerosene 
 stove) until the color changes to a briUiant black, mdicating com- 
 bination of the ink and powder. In heating the plate wiU warp 
 sfightly, and it should be placed on a flat surface to cool. Now look 
 over the plate and remove with a fine needle any spots of etchiag 
 powder which may have been burned in on the whites of the plate. 
 
 The plate is now ready for the first etching, which means the 
 immersion of ft in a tray containing a solution of 5 ounces of com- 
 mercial nitric acid, 38 per cent, in two gallons of water. It should 
 remain in this solution for 1^ to 2 minutes, the tray being continually 
 rocked to keep the acid in motion across the plate, and the surface 
 of the plate berag brushed fightly with a bristle brush to insure a 
 fresh sirrface upon which the acid may act. At the expiration of the 
 1^ to 2 minutes, the plate is removed from the etching bath, imme- 
 diately washed thoroughly, if possible imder running water, the 
 surface being gone over with a wad of absorbent cotton. It is rare 
 that one etching will give a satisfactory plate, and it will usually be 
 found desirable to now dry the plate, again powder with etching 
 powder, heat in and etch for from 1^ to 3 minutes. Upon again 
 being thoroughly washed the plate, while still . wet, is flowed with 
 the dextrine solution, and ~then placed on an inclined surface to 
 drain and dry. This takes one to the process of printing from the 
 
TOPOGEAPHY, MAP EEADING, AND KECONNAISSANGE. - 73 
 
 plate, and as this is the same for all plates from this stage on, we will 
 now pass to the photo-zincographic process. 
 
 159. Photo-zincograpliic process. — This process is the sensitizing 
 of the zinc plate and the transfer thereto of impressions by the action 
 of light through some medium immediately superposed upon the zinc 
 plate. The zinc used for this purpose is No. 19 gauge and is usually 
 purchased highly polished. If the plate is not polished this must 
 be done, and in any event it is always necessary to go over the poHshed 
 plate with a willow charcoal polisher, which should be kept under 
 water when not in use. Go over the plate with an even stroke, 
 always keeping the same direction and not applying too much 
 pressure. The same object may be accomplished by using powdered 
 pumice, applying with a tuft of cotton. The plate must be then 
 thoroughly washed and is ready to receive the sensitizing solution. 
 If instead of a new plate, a plate that has already been printed from 
 is beiag used, first remove all ink therefrom by means of turpentine, 
 go over the plate with lye to remove all grease, and scrub down the 
 slightly relieved lines of the former image with a scotch hone. The 
 plate is then polished, as before, with charcoal or pumice. 
 
 A satisfactory medium speed sensitizhig solution, which can be 
 used in any climate, is the following: 
 
 The white of one fresh egg, or 120 gr. dry albumen; bichromate ammonia (C. P. 
 Bichromate), 15 gr.; -water, 7 oz. 
 
 Beat the egg with an egg beater, or dissolve the dried albiunen in 
 the 7 ounces of water; pulverize the dichromate with mortar and 
 pestle and mix it with me water; add a few drops of 28 per cent 
 ammonia water until the solution assumes a clear yellow color. 
 This solution is not aflfected by light as long as it is in liquid form, but 
 is sensitive to light when dry. 
 
 Rinse the now polished zinc plate imder the tap, or when running 
 water is not at hand squeeze a wet sponge oyer it and allow the water 
 to run off the plate. Then, while the plate is stiU w^t pour the 
 sensitizing solution over the plate by ghding slowly with the glass 
 containing the solution . along the upper edge of the plate which is 
 held on me left hand in an incHned position. AUow the surplus 
 solution to run off the plate and then repeat the operation having 
 one of the formerly inchned edges of the plate uppermost. A third 
 pouring of the solution over the plate is sometimes recommended. 
 Watch out for dust particles and air bubbles during the sensitizing^ 
 of the plate. 
 
 The plate is now placed in the whirler and whirled over a heater to 
 give an even distribution of the solution and also to dry the plate. 
 Do not whirl too fast at first as this may throw too much solution 
 from the edges of the plate and make the comers come weak. "While 
 whirling, keep the plate also swinging in order as far as possible 
 to give even neating over the entire surface. As soon as the plate 
 begins to dry a dark circle wiU appear in tjie center and wiU almost 
 icamediately extend to the edges if the plate has been projperly 
 handled. The plate is now removed from the. whirler and, being 
 sensitive to light, is placed in a dark place to cool. 
 
 The plate, having cooled, is put into the printing frame, sensitive 
 side toward the glass, with the maduro negative, stripped film, or 
 tracing between it and the glass, and so placed as to give the reversed 
 
74 TOPOGBAPHY, MAP READING, AND BECONNAISSANOB. 
 
 impression on the zinc plate. Absolute contact between plate and 
 negative is most important and consequently the printing frame and 
 glass are heavy and considerable pressure is applied. While the 
 sensitized plate is affected by hght, it is entirely practicable to carry- 
 out the above-mentioned operations without hurry m the light of an 
 ordinary room, without damage to the plate. It is only really 
 sensitive to the rays of sunlight or intense artificial light. 
 
 The plate is now exposed to sunlight or artificial light and by this 
 action the albumen reached by the hght rays is rendered insoluble in 
 water, while that not so acted upon remains soluble. The time of 
 exposure is dependent upon so many conditions that it must be 
 arrived at by a beginner by experiment. As a guide it may be 
 stated that a zinc plate sensitized as above and exposed under a 
 
 Eerfect negative to the rays of the sim during July and August 
 etween 10 in the morning and 3 in the afternoon in the central part 
 of the United States will take a 1 ^-minute exposure if the drawing 
 to be produced consists of ordinary heavy lines. 
 
 After exposm-e the plate is'taken from the frame and laid on a level 
 surface and immediately roUed up with etchiiig ink. The surface 
 upon which the plate is laid must be absolutely a plane or distortion 
 will result. An ordinary lithograph stone makes a most excellent 
 bed for this purpose, but a heavy block of wood with care may be 
 made to serve. The exposed plate is evenly coated with the etching 
 ink, rolling it up in two directions at right angles to each other. 
 Very little ink is used and the yellow of the film should always show 
 through the ink. 
 
 Etching ink is a trade preparation resembling ordinary black 
 lithographic ink, but it has a greater proportion of resinous material 
 and less coloring matter. A very small quantity, about as much 
 as can be placed on a 5-cent piece, is worked up with the mixing 
 knife and then spread upon the ink slab (which must be plane sur- 
 face) and roUed out with the roller tmtil both slab and roUer show 
 an even coating of ink. If the ink is too hard to work up it may be 
 thinned with a drop of aniseed oU, but great care must be used to 
 thin the ink only enough to make it work, otherwise it will not have 
 sufficient body when applied to the plate. Rollers must be frequently 
 washed with turpentine to keep m proper condition. 
 
 The preparations for applying etching ink are supposed to have 
 been made prior to exposing the plate, as any delay m working up 
 theplate should be avoided. 
 
 We now have a zinc plate covered with a bichromatized albumen 
 film, portions of which, corresponding to the lines, etc., on the original 
 drawm^have been rendered insoluble in water by the action of 
 light. The whole surface has then been given a coating of ink. 
 The plate is now washed, rubbing the surface gently with a tuft of 
 cotton. If the plate has been correctly exposed, the albumen cor- 
 responding to the whites of the original drawing will be readily 
 removed, taking \dth it the ink, and there will be left the reverse 
 of the drawing m ink lines on an otherwise clean zinc plate. If the 
 plate has been underexposed the lines will begin to disappear early 
 m the washing and another plate with longer, exposure should at 
 once be tried. If overexposed the ink will smear over the plate and 
 the whites wiU not come up. A slight overexposure may sometimes 
 
TOPOGEAPHY, MAP BEADING, AND KEOONNAISSANOE. 75 
 
 be corrected by adding a few drops of stronger anunoma to tbe water 
 m the washing tray. 
 
 As the zinc plate will usually be larger than the negative used in 
 exposure the outer edges will have been exposed and rendered 
 insoluble and wiU show a black frame around the reproduction. 
 This must be removed by scrubbing with pumice stone and a cotton 
 tuft. 
 
 3?ake the plate from the tray and dry it by pattiag it with a damp 
 chamois skin; do not wipe it as the lines may smear. Look carefully 
 over the reproduction, and if there are any Hues missing paint them 
 in by means of a Httle etching ink moistened with tm-pentine and 
 apphed with a red sable brush. Also remove any unnecessary ink 
 spots on the plate. Be careful in this work not to touch the plate 
 with the arm or hand for fear of smearing. 
 
 From this point on the development of the plate is the same as 
 ia the autographic process and will not be again described, 
 
 PRINTING FROM ZINC PLATES. 
 
 160. Wlule the plates have been etched so as to give the image a 
 slight rehef the maia principle involved in the printing from zinc 
 plates is that "grease attracts grease and is repeUed by water." 
 
 The gummed plate, having dried, is wiped over with a damp rag 
 to remove surplus dry gum. After that it is gone over before each 
 rolUng up with a rag moistened in water with a Httle dextrin, in it 
 to keep the plate from drjTihg too rapidly. The proper proportion 
 is usually obtained by taking about one part of the solution for 
 gumming plates and mixing it with eight parts of water. 
 
 A smdl portion of hthographic ink is thoroughly worked up with 
 an ink knife. (This ink slab may be a piece of poMshed marble or a 
 lithographic stone.) The ink is then thoroughly distributed by the 
 roller. Never put ink directly upon the roUer. Do not put varnish 
 in the ink, but be careful to have the roller saturated with varnish. 
 
 The roller being now inked up, the plate, which must be on a perfect 
 plane surface, is wiped over with the damp rag (dextrin mixture), 
 and the roller is passed once over and back, quickly it the ink is soft 
 and liquid, slowly if the ink is very stiff. The gummed portion of 
 the plate takes moisture from the rag and will reject the greasy 
 ink upon the roller. The Hues being greasy reject the wetting and 
 take the ink. When first working up a plate it should be inked 
 and wiped several times, and it may then be necessary to make a 
 half dozen impressions before a satisfactory one is secured. If an 
 ordinary sHding contact hand press is used, the plate can be inked on 
 whatever is used as the bed of this press. If the clothes-wringer 
 press is used, a separate surface will have to be provided for inking 
 up the plate. To get an impression the paper is laid upon the plate 
 immediately after rolling up, it is passed through the press, and when 
 taken off it will show the impression. 
 
 Avoid flat and glossy paper for printing purposes. If you have to 
 use it fan the plate dry after each rolling up or the paper wiU stick. 
 
 If in roUing up some of the whites take ink, it is because, due to 
 carelessness, the entire plate was not covered in the wiping with the 
 damp rag. Go over it again with the damp rag and then give it a 
 very qidck roU and the ink wUl be removed. If the plate blocks up 
 
76 TOPOGRAPHY, MAP READING, AND RECONNAISSANCE. 
 
 and the lines of the impression become blurred, ^um the plate witii a 
 rather thick sohxtion of dextrin, and, while the dextrin is still liquid, 
 wash ofip the plate with turpentine, being careful that no pressure is 
 exerted upon the lines. The ink being removed keep the plate for 
 several mmutes under running water in order to remove the turpen- 
 tine. Then gum up with the ordinary solution of dextrin and let the 
 gum dry. The plate prepared in this way can be kept for years. 
 
 There is almost no mnit to the number of impressions that can be 
 obtained from the good plate which is taken proper care of. In the 
 choice of papers to be used, some discretion is necessary if very par- 
 ticular results are to be obtained, and the papers should be selected 
 after experiment has established their suitability. 
 
 Direct Tnethods. — It may not always be practicable to make a brown 
 print negative of the drawing to be reproduced, owing to lack of 
 paper, nor may it be possible to make and strip a photographic nega- 
 tive. The reproduction can be made directly from a tracing, or if 
 the drawing be on opaque paper this may be rendered translucent hj 
 the apphcation of banana oU," or a mixture of 3 parts castor oil 
 and 10 parts alcohol. The impression on the zinc plate may be 
 obtained by one of the following methods : 
 
 (o) 1. Prepare sensitizing solution: 
 
 Albumen 1 oz. 
 
 Water 6 oz. 
 
 Ammonium bichromate 27 grams. 
 
 Ammonia, 28 per cent 6 drops. 
 
 2. Sensitize a polished plate by "flowing" twice with the solution. 
 
 3. Dry by whLrUng over heat, being careful not to heat plate any more than ia 
 
 absolutely necessary. 
 
 4. Let plate cool. 
 
 5. When printing from tracing, give 40 seconds at 8.30 a. m., with good sun- 
 
 light, decreasing time as sun gets stronger so that by 10.30 a. m. time will 
 be from 25 to 30 seconds. 
 
 6. When printed, roll up with stiff etching ink. Put on thin coat of ink and 
 
 smooth out with smooth leather roller. Ink and roller should be free from 
 dust, or pin holes will result. 
 
 7. Place plate under water tap until all lines are clear. Going over surface 
 
 with a tuft of cotton while holding' under tap will assist in removing ink 
 from lines. 
 
 8. Immerse four seconds in solution of 2 ounces nitric acid to 3 gallons water. 
 
 9. Remove from solution and wash thoroughly to remove acid. 
 
 10. Dry plate by patting with chamois skin. Dry back of plate with rag. 
 
 11. Warm slightly over stove to complete dryinfe of lines. 
 
 12. Roll up witK the same etching ink, using composition" roller. Put on 
 
 medium coat of ink and roll into lines with smooth leather roller or bv 
 continued rolling with composition roller without the addition of any 
 more ink. •' 
 
 13. When lines show black through coating of ink place plate in a 10 per cent 
 
 solution of 28 per cent acetic acid and water. Rock tray for one or two 
 minutes, then start removing ink from ground of plate by dragging medium- 
 sized tuft of cotton over surface. Plate should be clear in five to seven 
 minutes. 
 
 14. When clean remove and rinse, then dry by whirling over heat let cool 
 
 then powder up, biu:n in, etch, and print as usual. ' ' 
 
 (6) 1. Expose a sensitized zinc plate under the drawing. 
 
 2. Develop in the usual manner. This gives white lines on a black eround 
 
 3. Etch the plate loijig enough in the nitric acid bath to give a slight depth 
 
 to the lines. ^ 
 
 4. Wash and dry. 
 
 5. Bub ordinary asphaltum paint into these etched lines and dry. 
 
TOPOGRAPHY, MAP EBADING, AND EECONNAISSANCE, 77 
 
 6. Scour whole surface of plate -with, charcoal stick. The lines which con- 
 
 taia the asphaltum paint, being deeper than the remainder of the plate, 
 are not affected by the charcoal stick. Thus there is obtained a polished 
 plate bearing the impression in depressed lines filled with asphaltum 
 paint. 
 
 7. Etch the plate again. This takes down the whites and leaves the asphal- 
 
 tumlines in relief. 
 
 8. Gum up and dry, ready for use. 
 
 Excellent results may be obtained by this method, which, however, takes 
 a little longer than when using a negative, on account of the time required 
 to scour the plate. 
 
 Field expedients. — It is entirely possible that work which. wiU give 
 satisfactory rough results in the field can be done with a much re- 
 duced apparatus. For instance, it has been found that impressions 
 can be gotten without the press by simply laying the paper on the 
 inked zmc plate and passing a rubber roller over it so as to press 
 upon all parts. There is a tendency for the paper to shift and give 
 a blurred impression, but this can be sufiiciently avoided by a little 
 care, thus disposing with the press.- 
 
 It has also Deen found that a satisfactory exposure can be made 
 by placing the negative upon the sensitive side of the zinc plate, 
 fastening it in any convenient manner, then bowing the sensitive 
 side of the plate to the front, which will stretch the negative and 
 give fair . contact. During exposure the plate must be so moved 
 as to give all portions an even illumination. The above are, of 
 course, makeshifts and will not give such results as are usually de- 
 sired, but they may save the day in an emergency. 
 
 Negatives. — ^If the original be a tracing the best negative is a, 
 brown print. This negative is made in the usual manner on thin 
 paper and is placed brown face down upon the zinc plate. 
 
 Photographic negatives : 1. Make the usual photographic negative 
 on a "stripping" plate, strip o£E and place, reversed, upon a clear 
 plate. To print on the zinc plate, place film side down, thus giving 
 reversed impression. 2. Expose a "process" photographic dry plate 
 in the camera backward — ^that is, with the emulsion side away irom 
 the lens, correcting the focus for the thickness of the plate. Develop 
 as usual. To print on the zinc plate, place emulsion side down. 
 3. The most satisfactory method of producing photographic nega^ 
 tives for lithographic work is the "wet-plate" process. This, how- 
 ever, is not adapted to field purposes. 4. The cameragraph: This 
 apparatus wiU make at a single operation a positive negative brown 
 prmt by which to transfer any drawing to a zinc plate. It consists 
 of a copying board on which to place the map or drawing to be repro- 
 duced, a camera^fitted with a reversing prism, and special attachments 
 for carrying a roll of brown print paper, a developing and a fixing bath. 
 To produce a negative, sufficient paper therefor is fed down from the 
 roll into the focal plane and the exposure made. By means of a 
 mechanical attachment this exposed piece is cut off the roU and is 
 then fed through the developing and fixing tanks, removed, and 
 dried. FuU directions for operation come with the cameragraph. 
 
78 TOPOGRAPHY, MAP READING, AND RECONNAISSANCE. 
 
 PART n. 
 
 DETERMINATION OF AZIMtlTHS. 
 
 161. The compass is the standard instrument for the determination 
 of azimutlis ia topographical reconnaissances. It consists of case, 
 needle, dial, pivot, and stop. 
 
 The dial may be fixed to the case or it may be movable, that is, 
 moving with the needle to which it is attached. The stop raises the 
 needle from the pivot and clamps it against the glass cover. A good 
 compass must have a needle sufficiently magnetized to settle accu- 
 ratefy and a pivot which is true. If the needle becomes too weak, 
 it may be remagnetized by rubbing gently from pivot to poiat on a 
 permanent or electromagnet, each end of the needle to be rubbed 
 on the pole of the magnet which attracts it. In returning the 
 needle for another stroke, carry it a foot or more froni the magnet. 
 The pivot may be polished with Putz pomade or a similar substance 
 on a soft stick. If possible, turn in a defective compass and get one 
 in its place. 
 
 A needle loses part of its magnetism if kept for a long time out 
 of the plane of the magnetic meridian. In storing a compass, there- 
 fore, care should be taken to see that the needle is ia the magnetic 
 meridian with the N. end of the needlfe pointing north. 
 
 A symmetrical needle tends to point downward toward the nearer 
 magnetic pole of the earth. This displacement from the horizontal 
 is called dip, and is measured ia degree of arc. Immediately over 
 the magnetic poles the needle stands vertically or has a dip of 90°. 
 Near the Equator, where North and South Poles of the earth exert 
 an equal influence, the needle will be horizontal, or the dip 0. 
 
 For reading azimuths the needle must be kept ia a horizontal 
 plane, which is done by a smaU movable counterweight (to overcome 
 the dip). For considerable changes in latitude, as in passing from 
 the United States to the Philippines, the counterweight wUl require 
 adjustment to keep the needle norizontal, and in passiag from the 
 Northern to the Southern Hemisphere, the counterweight must be 
 changed to the other end of the needle. 
 
 There are two adopted forms of compass for topographical recon- 
 naissance, one ia which the dial is fixed to the case and one in which 
 the dial moves with the needle to which it is fixed. 
 
 THE BOX COMPASS. 
 
 162. The dial or face on which the graduations are marked is 
 rigidly attached to the case. The type of box compass best adapted 
 to running courses by azimuth is constructed as follows : The gradua- 
 tions read counterclockwise continuously from to 360 ; the instru- 
 ment reads when pointiag south and 180° when pointing north; 
 the E. and W. points, if marked, are reversed. 
 
 To determine the azimuth of a line point the north and south line 
 of the case along the line (the north point away from the observer) 
 and read the N. end of the needle. The dial is graduated to single 
 degrees, but when the needle is stationary the reading can be esti- 
 mated to half degrees. 
 
 Many box compasses are not graduated in the manner above 
 dscribed. To use such compasses for azimuth reading they should 
 
TOPOGRAPHY, MAP READING, AND REOONNAISSANOE. 79 
 
 be altered to conform to the conditions cited. This is ordinarily- 
 done by pastiag paper over the stamped numbers on the* dial and 
 renumbering in ink or pencil. 
 
 THE PRISMATIC COMPASS. 
 
 163. The dial containing the graduations is attached to the needle 
 and moves with it. _ It is read by means of a small prism, adjustable 
 for focus. This prism is mounted on a hinge joint and can be turned 
 down for carrying. The line of sight of the instrument is determined 
 by front and rear sights, which lold down when not in use, at the 
 same time stopping the needle. The needle may be compensated 
 for dip by a bit of sealing wax on the under side of the dial card. 
 The graduations on the dial should be numbered so as to read azi- 
 muths, as above described, beginning at the south point. If the 
 Graduations are not so numbered, they should be altered as follows: 
 'he zero should be at the north end of the needle (which is on the 
 under side of the dial) and the graduations should run clockwise 
 continuously to 360°. It is to be noted that with such numbering 
 the instrument will not read azimuths if used as a box compass. 
 The index is a point on the case, and as the dial is movable the 
 graduations are numbered clockwise, instead of counterclockwise, 
 as in the box compass. Readings should be made through the prism. 
 To determine the azimuth of a line with this instrument, adjust the 
 prism until the graduations on the dial are distinct, raise the front 
 sight; look through the sht in the prism plate and bring the front 
 sight in line with the forward station; wben the needle comes to 
 rest, read the azimuth through the prism. 
 
 COMPASS ERRORS. 
 
 164. The magnetic and the true meridian generally do not coin- 
 cide. The angle between them at any point is called the magnetic 
 declination at that point. If the needle points east of the true 
 meridian, it is called an east declination; if west, a west declina- 
 tion. Magnetic declination varies in amount and direction at dif- 
 ferent points on the earth. The figure facing par. 166 (p. 80) is a chart, 
 called an isogonic chart, which for the epoch gives, by curved 
 lines connecting points of equal dechnations, the approximate dech- 
 nation of points on the earth. At no point is the declination constant. 
 It is subject to the following variations: The daily variation consists, 
 of a swing from the extreme easterly position at about 8 a. m. to the 
 extreme westerly position about 1.30 p. m.; the mean position occur- 
 ring about 10 a. m. and 5 p. m. The daily variation is from 5' to 15' 
 of arc The secular variation is a long slow swing, covering many 
 years. In the United States all east declinations are now gradually 
 decreasing and all west declinations gradually increasing at the rate 
 of about 3' per year. The annual variation is very small (less than 1' 
 per year) and need not be considered in surveying work. The lunar 
 declination is still smaller. All of the foregomg variations are 
 periodic in character. Irregular variations due to so-called magnetic 
 storms are uncertain in character and can not be predicted. Such 
 variations are sometimes large. local attractions may greatly 
 disturb the needle, and often come from unknown sources. Ihe 
 
80 TOPOGRAPHY, MAP BEADING, AND RECONNAISSANCE. 
 
 observer should have them constantly in mind and endeavor to keep 
 all magnetic iafluences^ such as magnetic bodies, electric wire^ etc., 
 at a distance from the mstrument when the needle is being read. 
 
 The geometric axis of a needle may not coiacide with its magentic 
 axis, hence the readings of two compasses at the same station may 
 differ slightly. 
 
 165. A simple way to detect— not measure— such disturbances 
 is to take frequent back azhnuths. If the position of the needle is 
 normal at both stations, the azimuth and back azimuth will ditter 
 by 180°. If there is local attraction on the course, it will usually be 
 stronger or cause a greater deflection at one station than at the other, 
 and the azimuth and back azimuth will not differ by 180°. 
 
 Another way is, when taking the bearing to a station, to select a 
 well-defined point beyond and on the same course. On arriving at 
 the new station take a bearing from there to the selected point ahead. 
 If it is the same as the first bearing to that point, there probably is 
 no local disturbance. If the two bearings to the same point differ, 
 there probably is local disturbance. 
 
 Corrections for abrupt deflections of the needle due to local attrac- 
 tions must not be distributed uniformly over the traverse. A course 
 in which local attraction is detected or suspected should be noted, and 
 if, on closing, an azimuth correction is necessary, it should be appUed 
 to the suspected courses. 
 
 USE OF COMPASSES. 
 
 166. A good needle requires time to settle, even when the case is 
 firmly supported, and the user should cultivate the knack of catching 
 it at the middle of its swing, which is the desired reading. If the 
 compass can be supported, it is always better to do so. Then the 
 sight can be carefully taken and the position of the eye changed to 
 read the needle. Wait till the swing gets down to 4° or 5°, which it 
 wfll usually do in a few seconds. Then catch the highest and the 
 lowest readings on the same swing and take their mean for the true 
 reading. If the first swings are very large, catch the needle with the 
 stop near the middle of the swing and release it quickly. This wiU 
 suddenly check the swings and shorten the time in which the reading 
 can be taken. 
 
 In using the box compass without a support, hold it sufficiently 
 below the eye so that the swing of the needle can be seen. Point 
 the line of sight in the required direction, catch the needle with the 
 stop in the middle of the swing, and hold it stopped until the reading 
 is taken. Stop readings are less accurate than sight readings, due 
 to the difficulty in stopping the needle at the middle of the swing 
 and to the tendency to displace the needle slightly in lifting it off 
 thepivot. When the stop is used, press it firmly and quickly. 
 
 With the prismatic compass the stop is not used except to check 
 the swings. UtOize a support if practicable. The pnsm having 
 been adjusted for focus, level the case so as to bring the scale into 
 focus, and when the swing becomes small read the extremes and take 
 the mean. 
 
TOPOGRAPHY, MAP BEADING. AND EECONNAISSANCE. 81 
 
 58740°— 18: 6 
 
82 
 
 TOPOGRAPHY, MAP EEADIKG, AND EECONNAISSANCB. 
 
 TO DETERMINE THE DECLINATION OF THE COMPASS. 
 
 167. First method; from the sun. — Prick a small hole in a piece 
 of tin or opaque paper and fix securely over tlie south edge of-a table 
 or other surface perfectly lerel, so that the sunhght coming through 
 the hole wiU fall on a convenient place on the surface, figure 15, 
 The hole may be 2 feet above the table for long days and 18 inches 
 for short ones. HaK an hour before to half an hour after noon, mark 
 the position of the spot of sunlight on the horizontal surface at equal 
 time intervals of about 10 minutes. Draw a curve, as hd, figur6 15, 
 through the points marked, and from point c in the horizontal surface 
 and in a vertical fine with the hole, a, sweep a circular arc, ef, iater- 
 secting hd in two points. The form of the curve id will vary with the 
 dechnation of the sun. 
 
 168. Second method; from Polaris. — The true North Pole is about 
 1 12' distant from Polaris on a line joining that star with one in the 
 handle of the dipper, and another in Cassiopeia's chair, figure 16. 
 One of these stars will always be above the horizon, wherever Polaris 
 is visible. The polar distance of Polaris is decreasing at the rate of 
 19" per year. It also varies during the year by as much as 1'. 
 Both variations may be neglected in this work. 
 
 Imagine Polaris to be the center of a clock dial, figure 16, with the 
 line joining 12 and 6 o'clock vertical and with the position of one of 
 the fines described (from Polaris to star in handle of dipper, or from 
 Polaris to star in chair) as the hour hand of the clock. The distance 
 in angular distance of Polaris from the true north may be taken from 
 the following table : 
 
 Table I. 
 
 169. Table showing the angular distances of Polaris in different posi- 
 tions with respect to the pole. Epoch 1911 ; polar distance 70'. Lati- 
 tude 0° to 18° north. 'This table may beused until 1930. 
 
 Clock reading of — 
 
 Angular 
 distances. 
 
 Clock reading or— 
 
 Angular 
 distances. 
 
 "Cass. 
 
 .ZTJrsae 
 . Maj. 
 
 "Cass. - 
 
 Z tTrsae 
 Maj. 
 
 Xn:30 
 I 
 
 1:30 
 II 
 
 m 
 mi 
 
 mi:30 
 
 v 
 
 V:30 
 
 , VI:30 
 
 vn 
 
 Vn:30 
 VIII 
 IX 
 X 
 
 X:30 
 XI 
 XI:30 
 
 o 
 
 E18 
 35 
 '49 
 61 
 70 
 61' 
 49 
 35 
 18 
 
 VI:30 
 
 vn 
 
 VII:30 
 VIII 
 IX 
 X 
 
 X:30 
 XI 
 XI:30 
 
 XII30: 
 I 
 1:30 
 
 n 
 
 m 
 
 1111 
 
 1111:30 
 V 
 V:30 • 
 
 W18 
 35 
 49 
 61 
 70 
 61 
 49 
 35 
 18 
 
 _ For higher latitudes multiply the tabular readings by the follow- 
 ing: 
 
 Latitude 19° to 30° i i 
 
 Latitude 31° to 37° 12 
 
 Latitude 88° to 42° 1 o 
 
 -Xatitude 43° to 46° {1 
 
 Latitude 47° to 50° i ' c 
 
 Latitude 51° to 53° IB 
 
 Latitude 56° to 57° 17 
 
 Latitude 58° to 59° " ' " 1 8 
 
 Latitude 60° to 61° "' jg 
 
- TOPOGBAPHT, MAP EEADIKG, AND BECONlsrAISSAN-CE. 83 
 
 170. It is well to keep track of the position of Polaris by notiag it 
 frequently and taking the correspondong clock time. Then, if on a 
 cloudy night a glimpse of Polaris is had, the observation may be 
 taken, even though the other stars can not h& seen. 
 
 171. For practical details of the observation, the following may 
 serve as a guide: Select a clear space of level ground not too near 
 buUdings or any other object which might cause local disturbance 
 of the needle. Drive a picket, leaving its top smooth and level, 
 about 18 inches above the ground. Six feet north of the picket 
 suspend a plumb line from a point high enough so that Polaris, 
 seen from the top of the picket, will be near ftie top of the line, 
 figure 17. The cord should be hard and smooth, about one-tenth 
 inch in diameter. The weight at the bottom of the line should 
 hang in, a vessel of water or in a hole dug in the ground to lessen its 
 vibration. Drive a second picket in range with the first one and 
 the plumb line a short distaiice north of the latter. Make a peep 
 sight by punchiiig a hole one-tenth inch in diameter in a piece of 
 paper and hold it on the top of the first picket; adjust it so mat the 
 star is behind the plumb line when looking through the peep. Note 
 the position of one of the stars on the imaginary clock face at the 
 moment the observation is taken. Mark the position of the peep 
 on the top of the first picket and lay a straightedge or stretch a 
 cord from that point touching the plumb line to the second picket. 
 Place the north-and-south edge of the compass box against the cord 
 or straightedge and read the needle. 
 
 The compass azimuth read is the magnetic azimuth to Polaris at 
 the instant of observation. To find the magnetic azimuth of the 
 tme meridian, correct the compass reading by the angular distance 
 of Polaris as given in Table I, adding the correction if marked W ; 
 subtracttQg, if marked E. This method will give results true to 
 within one-fourth of a degree. 
 
 From an examination of the table it will be seen that when either 
 S, Cass, or Z Ursae Major are at XII or VI, no angular distance is 
 given. At these times, Polaris is on the true meridian and the mag- 
 netic azimuth to Polaris is the true azimuth. For a rough check on 
 the magnetic declination, an observation on Polaris, taken when 
 either the dipper or Cassiopeia is above the pole (near 12 o'clock on 
 the imaginary clock dial) will give the magnetic azimuth of the true 
 meridian direct to within less than the least reading of the ordinary 
 compass. 
 
 THE SEXTANT. 
 
 172. This instrument is shown and its parts indicated in figures 
 18 and 19. The former is a very compact form, called the pocket 
 sextant. The larger form, figure 19, has telescopes of different 
 powers and also a telescope tube without lenses, which is used for 
 reconnaissance work at short ranges. The pocket sextant has a 
 telescope for use in astronomical and long-range terrestrial work. 
 For ordinary reconnaissance and surveying, the pocket sextant is 
 used without the telescope, the sight being taken through a small 
 hole in a shutter which closes the telescope opening. 
 
 The adjustments are as follows: For the index glass, place the 
 vernier at about 30° of the limb and examine the arc and its image in 
 the index glass. If the arc and image appear continuous, the glass 
 is in adjustment. If the image appears above the arc, the mirror 
 
84 
 
 TOPOGRAPHY, MAP READING, AND RECONNAISSANCE.' . 
 
 leans forward; if below, it leans backward. Adjust with screws if 
 provided, or with slips of paper inserted between the nairror and its 
 frame. 
 
 For the horizon glass. — Set at zero and observe a well-defined dis- 
 tant point, using the telescope. If the direct and reflected images 
 coincide, the horizon glass is in adjustment. If not, adjust it until 
 
 Fig.B 15 
 
 they do, or if that can not be conveniently done, move the arm a 
 short distance from zero until comcidence occurs. Read the vernier 
 and apply that readmg with its proper sign to aU angles measuJSd 
 Such a readmg apphed as a correction is called the index error M 
 
TOPOGBAPHY, MAP BEADING, AND RECONNAISSANCE. 
 
 85 
 
 the index error is off the arc, that is, between zero and the end, it 
 is additive. If on the arc, subtractive. 
 
 In the pocket form the horizontal glass only_ is adjustable. To 
 adjust the pocket sextant, select a distant object with a clearly 
 defined straight outKne. Set the vernier carefully at the zero of the 
 arc and look at the object through the peephole and the lower por- 
 tion of the horizon glass. Turn the sextant about the line of sight 
 
 A-Ac/jt/shn^ kej/ 
 
 SCO/Pl 
 
 fi/ha 
 
 Fiq. \QPccAef Sex font 
 
 A- Index glass, 
 B- Horizon «• 
 0- Arc. 
 
 O- Vernier clamp. 
 E-' 'fi 'tang, screw. 
 G- Reading glass, 
 V- Vernier. 
 T- Telescope. 
 
 !9 
 
 as an axis until the straight line appears to be perpendicular to the 
 straight bottom edge of the horizon glass. If the mstrument is not 
 perfectly adjusted for this position, the straight hne of the observed 
 object will appear broken, in which case unscrew the smaller milled 
 head A of the top plate, and using its small end as a key, turn 
 the single adjusting screw in the cyhndrical surface while looking 
 at the object through the peep. The part of the ima,ge seen m 
 the mirror will appear to move, and by turmng the key in the 
 
86 TOPOGEAPHT, MAP READING, AND EECONNAISSANCE. 
 
 proper direction the two parts may be brought together. -N^ext 
 turn the sextant about 60° about the line of sight, and. H tne 
 straight line again appears broken, use the key to sUghtly loosen one 
 of the two adjusting screws in the top plate while looking througn tne 
 instrument. If this brings the two parts nearer in line, the proper 
 screw has been selected; if not, try the other one. Then turn the 
 two adjusting screws in the top plate by correspondmg amounts 
 and in opposite directions and continue turning them alternately 
 tiU the straight hne becomes continuous. The two screws^ are 
 opposed to each other, and care must be taken to use no consider- 
 able force and to always unscrew one before screwing up the other. 
 When the adjustment is complete, the Hne should remain continu- 
 ous and straight while the sextant is slowly revolved about the line 
 of sight. If the index arm is then moved back and forth by tiimuig 
 the large mQled head, the reflection of any object may be made to 
 pass exactly over that object as seen through the clear glass. 
 
 For adjusting at night, screw the telescope in place. Pull its 
 inner tube wdl out. Remove the sunglass from the eyepiece. > 
 Focus the telescope on a bright star by pushing in the tube tm the 
 image of the star is clear. Then, by turning the large milled head, 
 make the star's reflected image pass through the field of view. If it 
 does not pass exactly over the stationary image of the star, adjust 
 the horizon glass with the two screws in the top plate till one image 
 will pass exactly over the other. Next set the "vemier accurately to 
 the zero of the arc, and with the single adjusting screw in the cylindri- 
 cal surface make the two images appear as one. The instrument ia 
 then completely adjusted. The dtayhght method is most conven- 
 ient, but it is well to test the adjustment by the star method before 
 attempting to do any astronomical work. 
 
 In thtf cylindrical surface just below the zero degree end of the 
 arc are two projecting levers which move colored glasses to be used 
 in looking at the sun. At other times these glasses should be de- 
 
 Eressed through the opening in the bottom plate by first sliding the 
 rass stud ia the plate and then pushing the two levers. The tele- 
 scope also has a colored sun glass secured on the eye end which 
 must be removed when observing any other object. 
 
 Adjustment of the line of sight. — ^Two parallel wires are placed ia 
 the focus of the objective of the telescope, the middle point between 
 which marks the center of the field of view. The line joining this 
 point and the optical center of the objective is the axis of the telescope 
 or the line of sight. This hne should be parallel to the frame of the 
 instrument. To test the adjustment, turn the' telescope in its collar 
 until the wires are parallel to the frame. Select two objects which 
 arc at a considerable distance apart, as the sun and moon when 
 distant 100° or more from each other. Point the telescope to the 
 moon and brmg the unage of the sun tangent to it on one of the 
 wires. Move the instrument until the images appear on the other 
 wire. If they are stiU tangent, the telescope is adjusted ; if otherwise 
 the adjustment is made by two screws in the coUar loosenine one 
 and tightenmg the other. In some instruments the adjustment 
 of the parallelism bemg supposed to be carefully attended to by the 
 makers, the screws are wanting. With a properly adjusted ii^tr'- 
 ment, two images seen in contact on the wires will overlap in the center 
 
TOPDGBAPHY, MAP EEAMNG, AND EBCONNAISSANCE. 87 
 
 of the field. If the two images are tangent on the lower wire and 
 appear to separate on the wire farthest from the frame, the obiect 
 end ot the telescope droops toward the frame. 
 
 ERRORS OF THE SEXTANT. 
 
 173. To whatever division of the arc the index may point when the 
 mirrors are parallel, this division is the piont of beginning of all 
 angle measurements. In other words, it is the temporary zero, and the 
 difference in arc between this temporary zero and the actual zero of the 
 arc is the index error. Due to unequal expansion and contraction, this 
 mdex error will not remain the same, it should, therefore, be deter- 
 mined anew each time the instrument is to be used. To measure it, 
 bring the muror to p arallelism by producing a perfect coincidence of the 
 direct and reflected images of a distant point or star ; read the vernier, 
 diving the result the proper sign— minus if coincidence occurs when 
 the index is on the same side of zero as the greater part of the arc; 
 plus if on the same side of the small portion of the arc, called "off the 
 arc." Another error which must be looked out for, is that due to 
 " eccentricity." This error is caused either by an original defect in 
 the instrument or by a bending of the frame by varying temperatures 
 or by accidental blows. 
 
 To determine this error, measure with a transit the angle between 
 two distant points having the same elevation. Make several readings 
 of the same aiigle with the sextant and take the mean. The difference 
 between the transit determination and the mean of the sextant 
 determination, will be the effect of the eccentricity for that particular 
 reading of the sextant. The operation should be repeated for the 
 whole arc at short angular distances and the results tabulated. From 
 . time to time this tabulation should be checked to see that no change 
 has occurred. 
 
 USE OF THE SEXTANT. 
 
 174. When angles between terrestrial objects are to be taken 
 with the sextant, the index is set at zero, and holding the instrument 
 in the right hand so that the plane of the frame coincides with the 
 plane through the objects to be observed, with the telescope on the 
 upper side 3 the angle is approximately horizontal and on the left 
 sid^ if the angle is vertical, sight the left-hand object. There wiU be 
 a slight lack of coincidence in the two images due to parallax, even if 
 the instrument has been adjusted for index error by sighting at a star. 
 Move the index arm until there is coincidence and read the vernier. 
 Use this reading as index error. Now keeping the left-hand object 
 in the field by sighting through the transparent part of the horizon 
 glass, move the index arm with the left hand until the other object 
 appears in the mirror portion of the hbrizon glass opposite the Crst 
 point. Bring the second point exactly opposite by the tangent 
 screw. Test the coincidence of the images by twisting the instrument 
 so as to make the reflected image move back and forth across the 
 direct image. Eead the vernier and apply index and eccentricity 
 correction. In rapid work the telescope is not used; sight is taken 
 through the telescope ring. Make it a rule to commence takmg angles 
 from the object farthest to the left, then from the next farthest; and 
 so on, always working from left to right. Avoid very large or very 
 
88 
 
 TOPOGBAPHY, MAP BEADING, AND BECONNAISSANOB. 
 
 small angles. Though the angles measured with the sextant are 
 seldom horizontal angles, it is usual to plot them as such m tUing m 
 a topographical or hydrographic sUrvey . The errors due to o bhquity 
 will^be nondiscemible in work plotted, with the ordmary protractor. 
 
 THE ENGINEER'S TRANSFT. 
 
 175. This instrument is shown, and the names of its parts indicated 
 in figure 20. i x j j ^ 
 
 To set up the transit.— Place the tripod with the legs extended far 
 enough to give a stable base and so as to make the top surface of the 
 
 A— Tripod. 
 
 B- <■ head. 
 
 C [ P1^%='^'"P- 
 Vernier ■" 
 
 ^ (Plate tang.iscrew 
 D or^ 1 
 
 'Vernier " •• 
 Et- Linnb clamp. 
 F- " tang, screw. 
 G— Main leveling 
 
 screws. 
 H-H-Verniers 
 l-1-Plate levels. 
 K- Vert. limb. 
 L- '■ "Vernier. 
 M— " "tang, screw 
 N- !' "- clamp. B' 
 O— Attached level. 
 P— Telescope. 
 Q— Eye. piece 
 R-R-Reticle screws. A 
 Y- Support. 
 
 Fig. 20. 
 
 tripod head horizontal or nearly so. On level ground the legs will 
 be equally extended. On inclined ground, two legs on the lower 
 side should be on the same level and relatively close together. The 
 third leg is moved straight uphill at right ajigles to the line of the, 
 lower two; the amount this leg is thrown uphill is that sufficient to 
 bring the tripod head roughfy level. If the instrument has not 
 already been screwed to the tripod, remove the tripod cap and screw 
 on the instrument in its place. Hang the plumb line on the hook 
 (depending through the tripod head). Level the instrument as 
 follows: Unclamp the vernier plate and turn the transit so that one 
 
aX)POGRAPHY, MAP BEADING, AND KECONlirAISSANOE. 89 
 
 of the plate levels is parallel to one pair of leveling screws. The 
 other plate level will be parallel to the other pair. Bimg the bubbles 
 of the levels to the center in succession by means of the leveling 
 screws. Always turn one of a pair down as the opposite one is turned 
 up and avoid more pressure of the screws against the plate than is 
 necessary for a firm bearing. If a screw turns hard at any time it is 
 either sprung or has been set up too tight. In turning a pair of 
 leveling screws always move the thumbs toward each other or away 
 from each other. The bubble will follow the motion of the left 
 thumb. If the screws are too tight, unscrew either but not both. 
 
 With the level bubbles in the centers of their tubes, the plate will 
 be level if the bubbles are in adjustment. Turn the transit slowly 
 in azimuth and watch the bubbles. If they remain in the centers, 
 the plate is level and the levels are also correct. If either bubble 
 leaves the ctenter, the amount of its motion indicates the amount 
 by which it is out of adjustment. If the amount is small, it may be 
 neglected; if large, the adjustment should be made as hereafter 
 described. For short lines the level error may be neglected if the 
 entire bubble remains in sight during the entire revolution. Adjust 
 the leveling screws in this case so that the travel of the bubble will 
 be equal on both sides of the center. ' 
 
 Parallax. — ^Having leveled the instriunent, point the telescope at 
 the sky. Focus the eyepiece until the cross hairs appear sharply 
 distinct. This should ehminate parallax. To test the adjustment, 
 point the telescope at some terrestrial object and bring it to a proper 
 locus by means of the focusing screw of the object glass. Bring the 
 intersection of the cross hairs on some well-defined point of the image. 
 Now move the head laterally, watching the intersection. If there is 
 no relative motion of the cross hairs and the image, parallax has been 
 eliminated. The eyepiece once adjusted for parallax need not again 
 be focused for the same observer unless it has been disturbed. The 
 test, however, should be occasionally repeated. The object glass 
 must be focused for each separate sight. The instrument is now ready 
 for use or adjustment. 
 
 The adjustments of the transit are : (1) To make the plane of the 
 plate bubbles truly perpendicular to the vertical axis or the instru- 
 ment; (2) to make the line of sight truly perpendicular to the hori- 
 zontal axis; (3) to make the horizontal axis of the telescope truly 
 perpendiciilar to the vertical axis of the instrument. These three 
 adjustments are made to depend on the principle of reversion, the 
 effect of an error being doubled by a reversal of the instrument. 
 The adjustments should always be made in the order given. 
 
 (1) Adjustment of the bubble tubes.— One level tube is adjusted 
 at a time. Clamp the lower limb. Bring the bubble in the center 
 of its tube with the leveling screws. Revolve the vernier plate 180°. 
 If the bubble axis is not truly perpendicular to the axis of revolution, 
 the error will be indicated by the bubble leaving the center of the 
 tube. The movement of the bubble measures double the error. Cor- 
 rect the error by bringing the bubble (by means of the small capstan 
 screws on the tube) halfway back to the center. If it is brought 
 exactly halfway back, the error is eradicated. Verify the adjustment 
 by recentering' the bubble, as before, with the leveling screws and 
 revolving 180°. If the bubble again leaves the center, there is some 
 error remaining. Correct this residual error as before and verify. 
 
90 TOPOGBAPHY, MAP BEADING, AND KEOONNAISSANCB. 
 
 Two attempts should be sufficient to correct all the error. Adjust 
 the other bubble tube in the same manner. WMle adjusting one 
 bubble see that the other is centered. 
 
 (2) Adjustment of the line of sight. — Clamp the lower limb. First 
 make the vertical hair perpendicular to the horizontal axis. To do 
 this sight the vertical cross hair on some well-defined point, clamp 
 both ^ateg, rotate telescope about horizontal axis. If point does 
 not appear to travel along vertical hair, loosen screws (K.) holding 
 cross hair ring, and by lightly tapping on one screw rotate ring untu 
 above condition is fulfilled. Tighten screws and proceed with 
 second part of adjustment as follows: See that lower limb is clanrped, 
 unclamj) the upper or vernier plate. Direct the intersection of the 
 cross hairs at a snarply defined point A, 200 or 300 feet away; clamp 
 the vernier plate, then plunge the telescope (revolve on its horizontal 
 axis), and have an assistant set a point B (a marking pin or a pencil 
 mark on a vertical wall) in line with the intersection of the cross 
 hairs and at approximately the same distance away (points A and B 
 should be at about the same elevation), but in the opposite direction. 
 Unclamp the vernier plate and revolve the telescope in azimuth 
 (about the vertical axis) until the intersection of the cross hairs is 
 again accurately on point A; 'clamp the vernier plate; and again 
 plunge the telescope and set a point in line with the intersection of 
 the cross hairs and beside the point B. The distance between the 
 points B and C is four times the error. Mark a point D one-fourth 
 of the distance between B and C, measured frojn C. Move the cross- 
 hair ring by loosening the reticle screw on one side of the telescope 
 and tightening the one on the opposite side until D is at the inter- 
 section of the cross hairs. To verify, repeat the whole operation. 
 Two attempts should be sufficient for accurate adjustment. 
 
 (3) Adjustment of the horizontal axis. — ^The plate bubbles being 
 truly perpendicular to the vertical axis of revolution and the line of 
 sight being truly perpendicular to the axis of the trunnions, set up 
 and level the transit. Clamp the lower limb and release the vernier 
 plate. Now with the cross hairs bisect some sharply defined point 
 A, at a very high angle (gable of a house near by) ; clamp the veraier 
 plate. Depress the telescope and mark down, at a convenient point 
 under the point A and at about the level of the telescope, the point B 
 in line of sight. Now unclamp the vernier plate and revolve the 
 instrument in azimuth (about vertical axis), pltmge the telescope 
 (revolve about horizontal axis), and sight at A. The telescope is 
 now inverted (bubble up). Depress the telescope and set a pomt C 
 in the line of sight and beside the point B. The distance between 
 B and C measures twice the error. Correct for one-half the error 
 by the adjusting screw underneath one end of the horizontal axis. 
 
 Adjustment of the telescope level, — If there is a level attached to 
 the telescopfe, it may be adjusted by the "peg" method after the other 
 adjustments are made, as follows: Set up midway between two stakes 
 which have their tops at about the same elevation, level the transit' 
 and with the bubble of the attached level at the center read a rod 
 on each stake. The difference in the readings is the true difference 
 in level of the tops of the stakes. Move the instrument toward one 
 of the stakes, and set it up so that the eyepiece is about over the 
 center of the stake. Place the rod on the stake near the eyepiece 
 and set the target in the middle of the field as seen through the 
 
TOPOGRAPHY, MAP HEADING, AND EECOH-NAISSANCE. 91 
 
 object glass Set up the rod on the far stake tvith a target set at the 
 reading ]ust taken through the object glass, plus or minus the differ- 
 ence ot level between stakes— plus if lower, minus if higher. Bisect 
 the target with the cross wires. The line of sight must now be hori- 
 zontal, and, keepmg the vertical motion clamped so as to retain the 
 pomtmg, adjust the bubble of the attached level to the center by 
 naeans of the small screws at the movable end of its tube. Both line 
 of sight and axis of bubble are now horizontal, and therefore parallel. 
 
 Note that the position of the intersectio^ of the cross wires in the 
 field is a matter of convenience mainly. It is best to have it near 
 the middle of the field, and it can be placed there by inspection with 
 aU needful precision before making the adjustment of^the line of 
 sight. 
 
 Vertical circle adjustment. — While the line of sight and attached 
 bubble are still horizontal, the screws holding the vernier for the 
 vertical arc should be loosened and the vernier moved until the read- 
 ing is zero. If the vernier is not adjustable, the reading of the vernier 
 when the attached level and line of sight are horizontal may be taken 
 as the index error and applied to afl readings (or the line of sight 
 may be adjusted to the vernier when reading zero; this will involve 
 a retest of all previous adjustnients). 
 
 An instrument may at times appear to be out of adjustment because 
 some part is loose. — ^The object glass ma,j be partly unscrewed or 
 an adjusting screw may be only partly tightened. Level bubbles 
 or cross wires occasionally become loosened; therefore, before com- 
 mencing the adjustment of an instrument, look out for such defects. 
 When it is thought that an adjustment has been completed, always 
 test the instrument before using. All adjusting screws should be 
 screwed tight enough to hold, and yet not so tight as to injure the 
 threads or put a severe strain on any other part. Especial care 
 should be taken not to strain the cross-wire screws. 
 
 To eliminate effects of errors in adjustment, the instrument should 
 be used as follows: To avoid errors in plate bubbles, level up, turn 
 180° in azimuth, and bring bubbles halfway back by means of levehng 
 screws. This makes vertical axis truly vertical, and the bubbles 
 should remain in the same parts of their respective tubes as the 
 instrument is turned about the vertical axis. Errors in the line of 
 sight and horizontal axis are avoided by using the instrument with 
 its telescope direct, and then in its reversed position and taking the 
 mean of the results, whether the work is running lines or meastuing 
 angles. Errors of eccentricity are eliminated by taking the mean 
 of the reading of two opposite verniers. Errors of graduation are 
 nearly eliminated hj reading the angle in different parts of the 
 circle, or by measuring the angle by repetition. Where only one 
 vernier is read in determining an angle, always read the same one. 
 
 USE OF THE TRANSIT. 
 
 176. To measure a horizontal angle, set up over the vertex of the 
 angle to be measured and direct the telescope along one of the sides 
 of the angle. Clamp Hmb and plate — if the latter is set at zero it is 
 more convenient— and with the tangent screw of the hmb bring the 
 intersection of the cross hairs on a definite point of the line. Read 
 each of the two verniers and record, caUing one vernier A and one B, 
 
92 TOPOGEAPHY, MAP READING, AND RECONNAISSANCE. 
 
 Unclamp the plate— not the limb— and direct the telescope along 
 the other line. Clamp and bring the cross hairs to a defimte point 
 with the vernier tangent screw. Read and record as before. Take 
 the differences of the two readings A and B, respectirely. If these 
 differences are the same, it is the Talue of the angle. If not, take 
 the mean of the differences as the value. Por greater accuracy, the 
 method of repetition is used. After the first measurement is made, 
 unclamp the limb— not the plate — and resight on the first point hj 
 means of the hmb tangent screw and proceed as before. The read- 
 ing of the vernier is now twice the angle. Continue the repetitions 
 until the desired number are made. The last reading divided by the 
 number of measurements is the value of the angle. To guard against 
 errors, it is well to read and record after each measurement. 
 
 To measure a vertical angle. — ^Point the instrument, clamp the 
 horizontal motions, and make the readings on the vertical Hmb. For 
 greater accuracy when there is a complete vertical circle, revolve 
 the instrument through 180°, plunge the telescope, and take new 
 readings. If the results differ, use the mean. . 
 
 To run out a straight lime. — Set up accurately over the initial point. 
 Point the telescope in the required direction and estabhgh a second 
 point. These two determine the line which is to be run out. Set 
 up over the forward or second point; lay the telescope on the initial 
 point; clamp hmb and plate, plunge telescope, and get a point for- 
 ward. If the adjustments are good, this third point will be in line 
 with the first and second and the fine may be prolonged by repeating 
 the steps taken at the second point. 
 
 If the adjustments are not good, set a third point as before. Then 
 unclamp the hmb and turn 180° in azimuth and lay on the initial 
 point. Clamp and plunge again and set another third point beside 
 the first onoi Take the middle point between the two for the true 
 third point. This method ehminates errors of adjustnient, except 
 those of the plate levels. These are so easily observed and corrected 
 that they should never exist when close work is required. 
 
 TRAVERSING. 
 
 177. The transit must be set at each station with a 0-180° line of 
 the azimuth circle parallel to its position at preceding stations. This 
 is called carrying an azimuth. The direction chosen for the 0-180° 
 Une is usually the true N. and S., or as near it as data at hand wiU 
 permit. 
 
 Having observed the second station from the first, proceed to the 
 second, set up, and get one of the verniers at its reading from the 
 first to the second station, plus 180^°, or at the back azimuth. Point 
 at the first station and clamp the limb. The fine 0-180° is now in a 
 position parallel to thait at the first station. Unclamp the plate, 
 direct the telescope to the third station, and proceed as before. 
 
 Also see paragraph 135. 
 
 A VERNIER. 
 
 178. A vernier is an auxihary scale by means of which the principal 
 scale can be read more closely than can be shown by actual subdi- 
 visions on the principal scale. 
 
 Consider AB, figure 21, as part of a scale of equal parts. Construct 
 the auxihary scale or vernier CD, the total length of which is equal 
 
TOPOGEAPHY, MAP BEADING, AND EECONNAISSANCE. 93 
 
 to 9 of the smallest divisions of the principal scale, but divided into 
 10 equal parts instead of 9, which makes each division of the vernier 
 ^ the length of the division of the scale. 
 
 When the zero division of the vernier, indicated by an arrow, is 
 coincident with a division, as 31, of the scal6, the reading is 31 and 
 it is obvious that the first division of the vernier is to the left of 32 
 in the scale by ^ of the distance between 31 and 32. Similarly, the 
 second, third, etc., division of the vernier is 2, 3, etc., tenths to the 
 left of the 33, 34, etc., division of the scale. To make any division 
 of the vernier, as 2d, 3d, 5th, or 8th, coincide with the division of 
 the scale next ahead of it the vernier must be moved to the right 2, 
 3, 5, or 8 tenths of the length of one division of the scale, and the 
 arrow will then be opposite a point on the scale 2, 3, 5, or 8 tenths of 
 the distance from 31 to 32, or at 31.2, 31.3, 31.5, or 31.8. The quan- 
 tity obtained by dividing the value of one division of the scale by the 
 number of divisions of the vernier is called the least coniit of the 
 vernier. Only one intermediate vernier division can coincide with 
 a scale division at the same time and the number of the coincident 
 vernier division, counting from the arrowhead, is the number of 
 times the least count must be added to the last scale division passed 
 by the arrow to get the true reading. 
 
 To read any vernier note the value of the last scale division passed 
 by the zero of the vernier and to it add the least count multiplied by 
 the number of the coincident vernier division. 
 
 Mistakes will be avoided and the reading facilitated by estimating 
 in advance the fractional part of the division of the principal scale. 
 
 A vernier constructed as described is always read ahead of the 
 zero, or in the direction in which the scale graduations increase, and 
 is called a direct vernier. Verniers may also be constructed by 
 dividing the length of a certain number of divisions of the scale, as 
 11, into equal parts one less in number, as 10. The principles of 
 operation and method of reading are the same, except that the 
 coincident line is to be found behind the zero of the vernier, or in 
 the direction in which scale graduations decrease. This form is 
 called retrograde. It is but little used. 
 
 If the scale is graduated in both directions, as is often the case, 
 the vernier is doubled, the zero in the middle and each side forming, 
 a direct vernier for the graduations increasing in the same direction. 
 This form is called double direct, figure 22. The most compact form 
 is that shown in figure 25, called the folded vernier, in which the 
 graduations are numbered from the middle to one end and continue 
 from the other end to the middle. This is read as a direct vernier 
 in either direction. If the coincident line is ahead of the middle or 
 in the direction of iTicreasing gradvMion, take its number from the 
 middle as zero. If it is ieMna the middle, or in the direction of de- 
 creasing graduation, take its number from the nearest end, counting 
 the end line as numbered on the vernier. 
 
 Verniers are also constructed on cylindrical surf aces and on corneal 
 surfaces. The principles and method of reading are the same for aU. 
 
 179. The plane table is shown and its parts mdicated in figure 26. 
 The a'diustments of the instrument are entirely analagous to those 
 of the transit. The plate levels are carried either on the alidade or 
 on the declinator. In reversing for level tube adjustment, care must 
 be taken to have the alidade or the decHnator, as the case may be. 
 
94 
 
 TOPOGRAPHY, MAP BEADING, AND EECONNAISSANOE. 
 
 cover the same part of the board in both positions by marking two 
 corners on the paper by faint pencil lines. 
 
 To set np the plane table over a known station. — In the foUowinff 
 discussion, it is assumed that a number of stations, the locations or 
 which have been secured by transit triangulation, have been plottedi 
 on the plane table sheet as will usually be Qie case in plane table work.; 
 The^eticaUy before any work can be done on the drawing from any 
 station, the instrument must be so set up that the plotted position of 
 the point is vertically over the corresponding station on the ground; 
 that the board must be truly horizontal; and that the meridian (true 
 
 Fig. 24. 
 
 or magnetic) of the point on the paper must he in the plane of the 
 mendian of the correspondm^ station. In the figure, there is shown 
 a devise for plumbing any pomt on the paper over the corresponding 
 pomt on the ground. Such a refinement is necessaiy only in very 
 close work on a very large scale. For maps on a scale of 6 inches to 
 the mile or smaller it is sufficient to place the point over the station bv 
 the eye. •' 
 
 The board is leveled by the leveling screws or as in some instru- 
 ments by mampulation of the board by the hands, the board beina? 
 mounted on the tnpod by a ball-and-socket joint. Wilson sava tbnf 
 the mchnation of the board from the true horizontal plane or the 
 
TOPOGRAPHY, MAP READING, AND RECONNAISSANCE. 95 
 
 amount 'which it is out of lerel affects the location in azimuth far 
 less than ■would, be at first estimated. For an inclination of 15° the 
 azimuth is affected only 1°. For an inchnation of about 3°, the error 
 in azinxuth Would amount to about less than 2^ minutes (about 4 
 feet in a^mile). An undue amount of time should not, therefore, 
 be spent in leveling_ 'when orientation is to be secured from nearby- 
 stations and short side shots only are to be taken. 
 
 The third requirement for a proper set up is met by orienting the 
 board, after it has been set orer the station and leveled. By orient- 
 ing is meajit the adjustment of the board in azimuth so that the line 
 from the station point to any other point shall be parallel to the 
 corresponding line between the two stations in nature. To orient 
 the board, therefore, place the edge of the ruler of the alidade on the 
 station point and any other distant point. Unclamp and swing the 
 board in azimuth until the line of sight of the ahdade intersects the 
 other station. Check by a sight on another station or so. 
 
 If the mragnetie meridian has not been plotted on the paper, place 
 the declinator on the board, after orientation, and allow the needle to 
 come to a rest. Draw a line the fuU length of the declinator side and 
 mark the north end. The magnetic meridian will ass^t in orienting 
 the board when plotted stations can not be seen and will always be 
 valuable as a check. 
 
 locations by intersection. — ^The plane table finds its greatest use 
 in quickly securing a secondary control for topographical work. In 
 open, billy country, this control is most rapidly made by locating a 
 number of points by intersection and resection. Points are located 
 by intersection by setting up the plane table and orienting as de- 
 • scribed above over a known sta,tion and drawing rays to natural or 
 artificial signals, being careful to number each ray and note in a note- 
 book the object sighted by each numbered ray. ,A second known 
 point is then visited, the plane table set up and oriented, and a series 
 of rays to the same signals drawn. The intersection of corresponding 
 rays locates the sign^. j. ,, , 
 
 AH stations which are important m the propagation ot the plane 
 table triangulation, should be checked by a third ray drawn from a 
 third known station. Where it is difficult to get 9, third mtersec- 
 tion locations by two rays wiU serve for tertiary pomts of control. 
 But'if such a station is occupied by the instrument for propagation 
 of the triangulation, the location and orientation should be checked 
 bv resection methods. It is sometimes desirable to locate a plane 
 table station from a line only one end of which can be occupied by 
 the table. Let A and B represent the points on the ground at the 
 ends of the hne: is the signal which is to be located, and ab repre- 
 sents the hne plotted on the plane table sheet. Set up at A the 
 endof the line which is accessible, and orient the table by sightmg 
 B with the ahdade along ah. Then, centering the ahdade on a, 
 draw an indefinite hne toward C. This hne should be drawn the full 
 length of the ahdade. The table is then taken to C and oriented by 
 melns of the line just drawn. Since the position of c on the mdefi- 
 nite line is now known it is necessary to estimate its position on the 
 Sap Sd to use this pomt in setting the table over C If the ahdade 
 !^ now centered on 6 and sighted toward B, a resection hne may be 
 L^^ aTd thS 5ne\5u cut the first indefinite hne thus locating the 
 t^^c desired! Tirposition of c found by this method shoufd be 
 
96 TOPOGEAPHT, MAP READING, AND EECONNAISSANOE. 
 
 checked if possible by resection lines from other points whose posi- 
 tions are known to be correct. 
 
 THE THBEE-POINT PROBLEM. 
 
 180. The plane table may be set tip at any place where three 
 triangulation points (plotted on the sheet) can be seen and me 
 position of th& plane table station can be determined and plotted 
 on the sheet by observations from this point. The foUowmg is one 
 of the graphical solutions: . . , j. 
 
 If three signals A, B, and C hare their plotted positions at a,b, 
 and c, and if the table be set up at any point and oriented correctly, 
 the resection lines drawn from o, h, and c will pass through rf, the 
 plotted position of P. Since there is no means of accurately onent- 
 mg the table, the position of f being unknown at the start the table 
 
 A C 
 
 A . A 
 
 Pig. 25. 
 
 A 
 
 B 
 
 The +able is on sta+ion" P'on +be qround . The 
 problem^ is +o -find The poinT'*P"on the sheeT. 
 
 must be oriented approximately by the compass. If the place 
 table is not oriented exactly, the three resection lines wiU not ordi- 
 narily pass through a common point but will form a triangle known 
 as the triangle of error (fig. 25). From this triangle of error the 
 true position of p may be estimated, and by a second trial a new 
 triangle of error may be obtained which is smaller than the former. 
 By successive trials the triangle may be made so small that it is 
 almost a point. In practice very few trials are necessary, the tri- 
 angle often being reduced to a point in the second trial. 
 
 If the table is on the circumference of the circle through the three 
 points, its position is indeterminate. When point ji is inside the 
 triangle ABC it is in a favorable position for an accurate location. 
 If the table is outside this triangle there are certain positions of the 
 signals which are not favorable, especially when the angles subtended 
 by the sides of the triangle formed by the signals are small and the 
 
TOPOGRAPHY, MAP EEADIlsrG, AHD KECONNAISSAN-CE. 97 
 
 stat&y"go?d. '^'"^ ^' ^"* ^ *^^ ^"^^'^ -g-'^i^ -- ^ 
 
 ■ of ^erfor^^?S£ *^^ ^"'^fi^ ^^^. *^«^ 2^ ^^ lie inside the triangle 
 StlSecti^ oFfL ^^•.- ^^ V ^""^^^ ^^ P^««d through a, b and the 
 SeTue Win,? nf "^ -'^ n ^^' *"<?°^, « ^'^'i ^ it ^W p^s through 
 section o?thP^««l/^" ^^^'JJ ^ circle through I, c and the intlr- 
 ?h??r«i.Lr. r ^f^^^^T ^^ ?°,™ ^ ^'I'i '^ may be sketched and in 
 this manner a close estunate of the position of } made for the second 
 
 Srtinr. nf ?r ' }^^ T^'H ^^^ ^°* actually constructed. A smaU 
 K^? ^'■'' ?^ ^^''^ ^'"■^l^ ^^ sketched, so as to indicate the posi- 
 
 tion ot p, preparatory to a second trial. The correct distance of p 
 irom any resection hne is proportional to the distance of the plane 
 table from the signal from which that line was drawn. 
 
 iracmg-paper solution of the three-point problem. By the use 
 01 tracmg paper the three-point problem is solved approximately 
 with great rapidity. Setting up the table on the unknbwn point P, 
 lasten on it a piece of tracing paper of sufficient size to include the 
 positions ot aU four points. ' A fine point is marked upon the tracing 
 paper to represent the position of p. The alidade is then centered 
 about the point p and pointed successively at the three known points 
 n^ O, and the hues pa, pi, and pc are drawn on the tracing paper, 
 ine ahdade being then removed and the tracing paper released, 
 -■■^e latter is then so shifted over the plane table sheet that the line 
 pa shall pass through the located point a, the line pi through h, and 
 the hne pc through c. Then with all three lines passing through the 
 known points, the poiat p is exactly over its correct position on the 
 plane table paper and may be pricked through to the latter. The 
 pomt p is now located and orientation is secured by placing the alidade 
 over p and any other known point and moving the table in azimuth 
 untd. the other point is intersected by the line of sight. Check by a 
 sight on two other points. It will usually be found that there is a 
 slight error in the location of p. In case the location of the point is 
 desired more accurately, draw rays from each of the three known 
 stations, and proceed as in the solution given next above. 
 
 BANGING IN AND LYING IN. 
 
 181. It will often be desirable to locate the table at a point from 
 which only two known points are visible. This may be done by 
 ranging in or lining in or by use of the magnetic meridian and needle. 
 This method depends upon the fact that 3 the table can be oriented 
 in any way the location of the point where the table is set up can be 
 found by resection rays drawn from any two stations in view or by 
 resection from one point, if station is on a plotted line. Therefore u 
 any line has been drawn on the place table sheet which represents a 
 physical line on the ground, as a straight fence, a railroaa tangent, 
 road tangent, etc., the plane table can be set up at any point along the 
 fence or on the tangent and oriented by laying the alidade along the 
 line on the paper and turning the table until the paper hne is parallel 
 to the physical Hne represented. As soon as the table is satisfac- 
 torily oriented, pivot the alidade over any known signal and draw a 
 resection ray from that station. The intersection of this resection 
 Lne and the line of the fence or tangent is the location of the instru- 
 ment. Similarly it will often be possible to set up the instrument on 
 
 58740°— 18 7 
 
98 TOPOGBAPHY, MAP BEADING, AND KECONNAISSANCE. 
 
 the line throtigh, two stations and orient by sighting along the line, 
 locating by a resection line from a third station. Or, set up and 
 orient by the needle and draw two resection rays from two known 
 points. The iatersection of these will be the location of the table. 
 These methods are satisfactory only for the location of station from 
 which local topography is to be secured. They are not good for 
 extension of tnaagumtion. 
 
 PLANE TABLE TRAVERSES. 
 
 182. Once the plane table has been set up and properly oriented 
 over any plotted point, any other point in the vicinity may be located 
 by drawing a ray through the station in the direction of the point and 
 laying off to scale on the ray the distance to the point as ascertained 
 by stadia or other method of measurement. 
 
 A second jjoint so located on the sheet may, of course, serve as an 
 instrument station, and if the plane table be moved to the second 
 point it may be set up with the plotted position of the point over the 
 actual point on the ground and the instrument oriented by placing 
 the alidade along the ray and moving the table in azimuth until the 
 first point is "backsighted" by the line of sight. With the instrument 
 oriented, a third station may be located from the second in the same 
 manner and by thus proceedingf rom point to point a plane table traverse 
 may be run and plotted more quickly than can be done by transit 
 and stadia and computation by latitudes and longitudes. As with 
 all other traverses of any considerable length, plane table traverse 
 should not be made a part of the map until they have been adjusted 
 to locations made by more accurate methods. The graphical adjust- 
 ment described in paragraph 148 is very convenient for the adjust- 
 ment of plane table traverses. In very large scale werk, back flags 
 must be used in order to secure correct orientation and the orienta- 
 tion of the table should be checked whenever possible on triangula- 
 tion stations. In work on smaller scales better results will be secured if 
 orientation is made by magnetic needle instead of backsights. Wher- 
 ever local attraction is suspected, backsights should be taken in this 
 work to check the needle orientation. 
 
 DETERMINATION OP DISTANCES. 
 
 183. Distances passed over may be determined by the stride of 
 man or horse, by the time taken by a rated horse, by the revolutions 
 of a wheel, by chain or tape, and by stadia. Distances which are 
 not passed over may be determined by estimation, stadia, or by 
 intersection. 
 
 REDUCTION TO THE HORIZONTAL. 
 
 184. Distances measured along a slope may require a correction 
 before plotting them on a map, as all map distances are, or are 
 supposed to be, measured in a horizontal plane. Such corrections, 
 when made, are called redaction to the horizontal. The following 
 table gives horizontal distances corresponding to sloping distances 
 for gradients up to 30°. The correction for slopes of 6° and less is 
 too small to be plotted on the customary scales and is usually 
 neglected. In flat or ordinary rolling country, the correction wiU 
 rarely be necessary. 
 
A-iTable. 
 6^ Alidade. 
 C- Tele scope. 
 D-Vert. circ)©, 
 
 , E- Vernier. 
 F- Level, 
 
 AA. 
 
 Fig. m. 'Z7 ~"'-~-~-l^ 
 
 Plane. Table 
 
 Position at04 
 
 Traversing o^^<^ 
 Inlersecfin^ ^J/ 
 Flane-Tpihle 
 
100 TOPOGKAPHY, MAP BEADING, AND BECONNAISSANOE. 
 
 Table II. 
 
 185. Horizontal distances for gradients of 0° to 30° corresponding 
 to distances on the slope: 
 
 
 
 Horizontal distances for sloping distances of 
 
 - 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1 
 
 09998 
 09994 
 09986 
 09976 
 09962 
 09945 
 09925 
 09903 
 09877 
 09848 
 09781 
 09703 
 09613 
 09510 
 09397 
 09272 
 09135 
 09063 
 08988 
 08910 
 08829 
 08746 
 08660 
 
 19997 
 19988 
 19972 
 19951 
 19924 
 19890 
 19851 
 19805 
 19754 
 19696 
 19563 
 19406 
 19225 
 19021 
 18794 
 18544 
 18271 
 18126 
 17976 
 17820 
 17659 
 17492 
 17320 
 
 29995 
 29982 
 29959 
 29927 
 29886 
 29836 
 29776 
 29708 
 29631 
 29544 
 29344 
 29108 
 28838 
 28532 
 28191 
 27815 
 27406 
 27189 
 26964 
 26730 
 26488 
 26238 
 25981 
 
 39994 
 39976 
 39945 
 39902 
 39848 
 39781 
 39702 
 39611 
 39607 
 39392 
 39126 
 38812 
 38450 
 38042 
 37688 
 37087 
 36542 
 36252 
 35952 
 35640 
 35318 
 34985 
 34641 
 
 49992 
 49969 
 49931 
 49878 
 49810 
 49726 
 49627 
 49513 
 49384 
 49240 
 48907 
 48515 
 48063 
 47553 
 46985 
 46369 
 45677 
 45315 
 44940 
 44550 
 44147 
 43731 
 43301 
 
 59991 
 59963 
 69918 
 69854 
 69772 
 59671 
 59653 
 69416 
 69261 
 59088 
 68689 
 68218 
 67676 
 57063 
 56381 
 55631 
 54813 
 54378 
 53928 
 63460 
 52977 
 52477 
 61961 
 
 69989 
 59967 
 69904 
 69829 
 69733 
 69616 
 69478 
 69319 
 69138 
 68936 
 68470 
 67921 
 67288 
 66574 
 65778 
 64903 
 63948 
 63441 
 62916 
 62370 
 61806 
 61223 
 60622 
 
 79988 
 
 79951 
 
 79890 
 
 79805 
 
 79695 
 
 79562 
 
 79404 
 
 79221 
 
 79015 
 
 78785 
 
 78252 
 
 77624 
 
 76901 
 
 76084 
 
 75175 
 
 74175 
 
 73084 
 
 72506, 
 
 71903 
 
 71280 
 
 70636 
 
 69969 
 
 69282 
 
 89988 
 
 2 
 
 89945 
 
 3 
 
 89877 
 
 4 
 
 89781 
 
 5 
 
 89657 
 
 6 :. 
 
 89507 
 
 7 
 
 89329 
 
 8 
 
 89124 
 
 9 
 
 88892 
 
 10 
 
 88633 
 
 12 
 
 88033 
 
 U 
 
 87326 
 
 16 
 
 86513 
 
 18 
 
 85595 
 
 20 
 
 84572 
 
 22 
 
 83446 
 
 24 
 
 82219 
 
 25. 
 
 81568 
 
 26 
 
 80891 
 
 27 
 
 80190 
 
 28 
 
 79466 
 
 29 
 
 78716 
 
 30 
 
 77942 
 
 
 
 The horizontal distance corresponding to o»y 'sloping distance and ant/ angle or gradient may te found 
 by multiplying the sloping distance by the cosine of the angle. 
 
 MEASURING DISTANCES WFTH STEEL TAPE. 
 
 186. The steel tape furnishes a convenient, rapid, and economical 
 means of measuring any distance for any desired degree of accuracy 
 up to about 1 in 300,000. For topographical surveying a length of 100 
 feet is most convenient. For base-hne measurement the length 
 should be from 300 to 500 feet and its cross section from two to tnree 
 one-thousandths of a square inch. 
 
 The length of a topographical base will depend on the stretch of 
 level or evenly sloping ground available. Such bases have varied in 
 length from ^ mile to 3 or 4 miles. It is more important to be able 
 to extend the base by well-conditioned triangles than to measure a 
 long base. The ideal site 'is on level, even ground, from which good 
 views of the surrounding country can be obtained. 
 
 MEASUREMENT OF BASE. 
 
 187. A topographical base should always be measured with a steel 
 tape. This should be carefuUy compared with a standardized steel 
 tape before and after using. A standardized steel tape is one where 
 exact length for a given temperature pull as well as its coefficient 
 of expansion has been determmed by the National Bureau of Stand- 
 ards at Washington, D. C. Any steel tape may be standardized 
 by sending it to that bureau. The manufacturers for a sHght extra 
 cost will furnish a certificate giving the temperature and tension at 
 ,which a tape agrees with their standard (facsimile of that in the 
 national bureau) or they will furnish the tape with the certificate 
 of test made by the national bureau itseK. The standardized tape 
 itself should not be used in the field lest it be broken. 
 
 For an accuracy of 1 in 5,000 the tape may be used in all kinds 
 of weather, held and stretched by hand, the horizontal position and 
 
TOPOGRAPHY, MAP BEADING, AND BECONNAISSANOE. 101 
 
 amount of pull estimated by the chainman. The temperature may 
 be estimated or read from a thermometer carried for the purpose. 
 On uneven ground the end marks are given by plumb line. 
 
 FOR AN ACCURACY OF 1 IN 10,000 TO 1 IN 50,000. 
 
 188. The line of the base should &st be cleared of trees, bushes, 
 or high grass; smaU mounds, etc., should be removed. 
 
 Each terminal point of the base should, if time allows, be marked 
 by burying a bottle 2 feet underground- accurately centered over 
 this should be built a small masonry pillar, surface flush with the 
 ground. A fine mark should be made m a metal plug sunk into the 
 surface of the pillar. 
 
 Haying marked the terminal points, set a transit over one of them 
 and direct it on the other, over which a pole should have been fixed ; 
 with the transit ahgn stakes at intervals of about 300 yards. By 
 means of thisahgnment hammer into the ground square-headed stakes, 
 accurately in the hne of the base and at tape lengths from each other, 
 starting from one end of the base. The tops of the stakes should be 
 just flush with the ground; on these stakes nail strips of wood or zinc. 
 
 In measuring the base, stretch the tape from terminal point to first 
 stake and from stake to stake; the tension should be taken on a spring 
 balance and should be that given for the standardized tape. The 
 end of the tape should be marked by a fine pencil Hne or knife cut on 
 a wood or zinc strip nailed to top of each stake. The temperature 
 of the tape should be taken at frequent intervals by letting the bulb 
 of the thermometer come in contact with the underside of the tape. 
 
 At the other terminal the small space between the last graduation 
 and the end of the base should be measured with a pair of dividers. 
 
 Now measure the inclination to horizontal from peg to peg with 
 level or transit. Minor undulations crossed by the tape are to be 
 disregarded. 
 
 The base should be measured at least oace la each direction ; on a 
 duU day, if possible, or in early morning or late afternoon. 
 
 Base calcTilations. — The measured length of the base is subject to 
 five corrections : 
 
 I. For • standard; II. For temperature; III. For inchnation; 
 IV. For sag; V. For height above sea. 
 
 The following case may be taken as an example (taken from 
 Close) : 
 
 One tape was kept as a reference tape (the standardized one) and 
 not used for measuring. This, when standardized, had been found 
 to be 0.05 foot short at 62° F. 
 
 The tape used (supposed to be 100 feet long) was 0.02 foot long on 
 refereace tape at 71° F. „ t^ 
 
 The temperature of tape used in the measurement was 82 Jb . 
 
 Height of base above sea, 4,520 feet. . , , i 
 
 For 2,300 feet, base had a slope of 1° 15'; remamder level. 
 
 I. Standard: „ 
 
 Reference tape was 0.05 short at 62 J) . 
 
 Steel expands 0.00000625 of its length for 1°F. „„„.„, , 
 
 Therefore, at 71° it was 9X100X62.5/10,000,000 longer, or 0.0056 foot longer, 
 
 or 05 less 0. 0056 or 0.0444 short of standard. 
 Reference tape was therefore . . 0.0444 short at 71° F. 
 
 Tape used 0.02 long on reference tape at 71° F. 
 
 Tape used WM therefore 0.0244 short at 71° ..... . (a). 
 
102 TOPOGEAPHY, MAP READING, AND RECONNAISSANCE. 
 
 TT TsDQ.'DGrflit'lirG * 
 
 Temperature of tape used during measurement was 32°; when compared was 
 71*. 
 
 Increase of length, 11X100X62.5/10,000,000, orO.0069 (6). . 
 
 Combining (a) and (6) we find that the tape used was 0.0175 short, dunng 
 measurement; correction for standard and temperature is therefore 
 0.0175X5301/100, or 0.928 foot. 
 III. Inclination: 
 
 For 2,300 feet the base has a slope of 1° 15'. 
 Correction, 2,300 (1-cos 1° 150=0.55. 
 rV. Sag: , , , 
 
 In the measurement of a base it is desirable that the whole length of the tape should 
 as far as possible be supported by the ground, so that no correction is necessary for the 
 sag of the tape. Where the ground is uneven, it is customary to support the tape_ at 
 intervals by stakes, but it may happen in the measurement of a base that a ravine 
 has to be- crossed. In such a case the sag, i. e., the difference between the length 
 when suspended and when laid on a plane surface, must be determined and corrected 
 for. 
 
 If s=the correction for sag, 
 
 l=the length of the tape suspended between two supports (in feet), 
 w=the weight of the tape in pounds, 
 t=the tension applied m pounds. 
 s=l wy24t2. 
 V. Height above sea: 
 
 If the base is measured at a mean height (h) above sea level, it will require a correc- 
 tion of h/EXlength of base, where R, the radius of the earth, may be taken as 20,900,000 
 feet. 
 
 In this case h=4,520. Then correction=4,520X5,301/20,900,000=1.146. 
 Collecting these corrections: ^^t- 
 
 I and II, standard and temperature : —0. 928 
 
 III, inclination —0. 550 
 
 rV,sag ,. , -0.000 
 
 V, height above sea '. —1.146 
 
 Total correction —2. 624 
 
 Measured length 5,301.240 
 
 Corrected length of base 5, 298. 62 
 
 Corrections III and IV must always be negative; I and II may be 
 either positive or negative. 
 
 189. For measurements requiring an accuracy greater tban 50,000 
 the precaution taken is more elaborate in the following particulars. 
 The tape is usually supported at intervals of 20 feet by adJTistable 
 standards, aU standards being aligned and set to uniform grade for each 
 tape length. The correction for sag may then be dispensed with. 
 
 For securing steady tension, several devices exist. The simplest is 
 a pointed lever (e. g., a survey sighting rod) . A movable collar with a 
 hook or ring and secured by a set screw slides up and down the rod. 
 The end of the tape is secured to this collar at any desired height 
 above the ground. The toe of rod is stuck m the groimd and the 
 required tension secured by hauliug back on the rod. 
 
 For greater accuracy still, the tape ends are secured to movable 
 boxes which are weighted with stones. The ends of the graduation 
 pass over wooden cages spiked into the ground. On the top of the 
 cages are zinc plates for scratching the ends of tape lengths. 
 
 For detailed methods of the measurements of lines cafling for high 
 accuracy see treatises on surveying. 
 
 There are but few localities in the United States that can not con- 
 veniently be connected with known positions and distances and there- 
 fore, before base-line measurements are undertaken, the records 
 of the Coast and Geodetic Survey, the Lake Survey, the Geological 
 
TOPOGRAPHY, MAP BEADING, AND EECOKNAISSANCE. 103 
 
 Survey, the Corps_ of Engineers, United States Aimy, and other 
 Lrovermnent organizations should be examined in order to ascertain 
 what positions m the area surveyed have been determined and are 
 available for use in the work at hand. 
 
 MEASTJHEMENT OF DISTANCES BY STADIA. 
 
 190. The stadia is primarily intended to secure rapidity rather 
 than accuracy; nevertheless with proper care to eliminate the chief 
 sources of error, a high degree of accuracy may be obtained. Where 
 properly handled, it will produce results as good as, and frequently 
 better than, those with the tape, especially in rough country where 
 variations in the slope of the ground affect taping seriously. 
 
 The degree of precision is dependent upon several conditions, 
 chief among which are: 
 
 1, Length of sight; 2. Magnifying power of the telescope; 3. Fine- 
 ness of the cross hairs; and 4. Precautions taken to avoid errors due 
 to refraction. 
 
 _ Experiments show that the average error increases rapidly for 
 sights over 800 feet. For short sights errors are less with a 25-power 
 than with a 15-power telescope. On the Mexican boundary survey, 
 the ratio of error between tnangulation and taping was 1 in 1,436, 
 and between triangulation and stadia 1 in 1,166. In all there were 
 measured 182.5 miles by stadia which were triangulated and in 
 which the total difference in length was 60 meters, or 1 in 5,873. 
 Experiments show that refraction is a variable quantity, dependent 
 upon temperature of air and ground, that is much greater near the 
 ground than 3 feet above it; also at noon than before or after; that the 
 effects vary for different distances. Twelve miles of stadia measure- 
 ments with sights averaging 600 feet, in the morning and evening 
 hours, showed an accuracy of 1 in 2,685. The same distances meas- 
 ured at midday showed an accuracy of 1 in 655. 
 
 191. To obtain accuracy in stadia work it is best to make short 
 sights, avoid readings near the ground (bottom of the rod), and to 
 avoid worMng during the midday hours, except on duU, overcast 
 days. For accurate work the most approved practice is to use rods 
 of standard division and to determine the rod interval factor to be 
 applied to the observed distance as a correction. The speed of 
 stadia traverses is far greater than that of taping where the surface 
 of the ground is rough, since sights of one to two thousand feet length 
 can be taken. Under similar circumstances the chain has to be laid 
 down and stretched every hundred feet, or less. Over smooth 
 country, traverses may be made by stadia still more rapidly than 
 by taping if the rodmen be mounted and the instrument man rides 
 or drives. On the Mexican survey above cited as many as 16 miles 
 a day were often covered, including the determination of height and 
 the sketching of topography. 
 
 THEORY OF STADIA MEASUREMENTS. 
 
 192. The relation between the size and distance of an object and 
 the size of its image in the telescope is given by the expression 
 
 YVY = F/X 
 
 where Y denotes the height of the object, Y^ that of its image (the 
 distance between stadia wires in this case), F the focal length of the 
 
104 TOPOGEAPHY, MAP BEADING, AND BECONNAISSANCB. 
 
 object glass, and X the distance of the object (rod) from the first 
 principal focal poiat. This point Hes ia front of the object glass at a 
 distance equal to the focal length (distance between the object glass 
 and the stadia wires). To reduce the measured distance X to the 
 true distance from the center of the instrument, add to X a constant 
 equal to the distance of the first principal focal point from the center 
 of the instrument. This constant is called c+f. Its value varies 
 for different instruments and is usually marked on the inside of the 
 transit box by the makers. It may be directly measured, however, 
 with all the accuracy necessary. F is the distance from the object 
 glass to the cross hairs when the object glass is focused for a distant 
 object, and C is the distance from the object glass to the center of 
 the instrument when focused for an average sight. The average 
 value of c + f for a transit is about 1 foot. 
 
 GRADUATING THE STADU ROD. 
 
 193. Eods should be of hght, straight-grained, well-seasoned wood, 
 12 to 14 feet long, 5 inches wide, and | inches thick, dressed smooth 
 aU around and covered with at least two coats of white paint. To 
 graduate the rod it ip necessary to know what space on the rod cor- 
 responds to a hundred feet in distance. Measure off c+f from the 
 plumb bob and set a point. From this point measure off any con- 
 venient distance on level ground, as 500 feet. Hold the blank rod 
 in a vertical position at the end of this base, or if possible fasten it 
 rigidly in that position. Have a fixed mark or target on the upper 
 part of the rod on which the upper wire is set. Have an assistant 
 record the position of the lower wire as he is directed by the observer. 
 Some sort of open target is good for this purpose, but any scheme is 
 suitable that wiU enable the observer to fix the position of the extreme 
 wires at the same moment with exactness. The work should be 
 done when there is no wind and when the atmosphere is very steady: 
 a cahn day is best. Eepeat the observation until the ntimber of 
 results, or their accordance, shows that the mean will give a good 
 result. Divide the mean space so obtained into five equal parts^ 
 thus obtaining the intercept on the rod for 100 feet. By dividing 
 this distance into 20 equal parts diagrams similar to Fig. 24 can be 
 constructed and painted on the rod. 
 
 A method of graduation in common use is to divide all rods exactly 
 in feet and tenths. This method has the advantages that the rod 
 may be used as a level rod, that the same accurate tranplate may be 
 Used to graduate all rods, etc. By the use of standard templates 
 for patterning the rods large numbers of rods may be painted or 
 repainted each, season with much facility and with, the TniniTrmTn 
 risk of mistakes. 
 
 194. The, stadia is used in connection with a transit having a 
 vertical hmb, or a plane table with^ telescopic alidade having a verti- 
 cal limb. Distances are read by setting either the upper or lower 
 stadia hairs on a foot mark by means of the vertical arc clamp and 
 tangent screw and counting the feet, tenths, and hundredths between, 
 the stadia hairs. The vertical angles are taken by sighting the 
 middle cross hair on a point on the rod whose distance above the 
 foot of the rod is equal to the distance from the horizontal axis of 
 the telescope" to the station beneath the transit. The distance is 
 
TOPOGRAPHY, MAP EEADING, AND EECONNAISSANCE . 105 
 
 fes'^oi^t *^Tf ?>fi^^M mstrument (HI). It is convenient to mark 
 8et 4? ^ ^^ ^ ™^^^^ ^^^<i ^1^^^ is adjusted for each 
 
 recSd iTfcli^S ^""-h"^ asfollows: First read the distance and 
 E 1 tl.r^-!f^/^%'^l'^'^^^ ^^'^ °^ *^« "^^^er band and the vertical 
 S« L^n^L + "^^^ ?^ the rod Signal the rodman away and while 
 he IS movmg to next point read the azhnuth and the vertical angle. 
 
 EEDUCTION OF STADIA FIELD NOTES. 
 
 195. This is done by means of tables, diagrams, or stadia sUde 
 rule rather than by direct use of formula. Table III gives for 
 vertical angles up to 20°, the difference m elevation for ah inclined 
 readmg ol 100 feet. If the incHned reading is 612, the instrumental 
 constant .(F + c) 1 foot, and the vertical angle 4° 42', the difference 
 in elevation is 6.13x8.17 = 50.1 feet. The horizontal distance is 
 obtained by use of the table of horizontal corrections (Table II), 
 which gives the distances to be subtracted from the inclined reading 
 increased by the instrumental constant, c+f. 
 
 Stadia reduction tables are pubhshed (such as the Noble and 
 Casgram) in which the corrections are made by addition only, instead 
 of multiplication as in Table III. These tables are more convenient 
 than multipUcation tables. 
 
 The most rapid method of reducing stadia readings^ is by means 
 of stadia slide rules or computers, such as the Kern, the Colby, the 
 Webb, the Matthes, the Cox, etc. For accurate traverses the read- 
 ings should be reduced by table and checked by sUde rule. For less 
 accurate work the shde rule alone is sufficient. 
 
 PLOTTING. 
 
 196. The plotting of stadia rfotes which are "cold" is fraught 
 with. error. They should when practicable be plotted in the field 
 in the sight of the facts. This may be done by means of a circular 
 protractor and scale, a small plane table or sketching board being 
 used as a plotting boa.rd. Some surveyors prefer to use the small 
 plane table as a plane table in conjunction with the transit; the 
 
 Elane table being used to obtain and plot all directions, the transit 
 eing used as an auxfiiary instrument to read distances and vertical 
 angles. 
 
 CAUTION. 
 
 It is sometimes desirable to take long sights where the intercept 
 of the stadia wires exceeds the length of the rod. In such cases it 
 should be remembered that the middle wire is seldom exactly mid- 
 way between the upper and lower wires. The sum of the readings 
 between the upper and middle wire and between the nuddle and 
 lower wire wiU give closer results than double the reading between 
 the middle and one extreme wire. For accurate azimuth reading, 
 bisect with the vertical wire the edge of the rod instead of the face. 
 
106 
 
 TOPOGEAPHT, MAP BEADING, AND EECONKAISSANCB. 
 
 1 
 
 .3 ai 
 
 •S Eh 
 W 
 
 o 
 
 <¥ 
 
 m 
 
 Hi 
 
 m 
 
 Eh 
 > 
 
 00000>-Hi-4iHFHl-Hi-4f-4rHi-^T-H(Hr-4^i-Hi-HrHi— <rH>-Hi— lf-HrH?4C^(NC4 
 COCOMCOCQCOeOMCOCOeOCOeOMeOCOCOMeQCQCOCOPOCOMCOCOCOCOMCQ 
 
 gSSSgSS^SSgS§S2?3SSS3S§giSS!SSSSE;SSS8 
 
 oiOsoOi-ii-ic5c5coM'*TiiiococDr-i--ooooososoorti~icfl«eoco'*'* 
 tH-i--5NMC^c>icgc^c4NNNeqc<iCToSc>ic4cgMCgco«5« 
 
 ■^oa■^omoeD1-^c01H^-.e^l^-«QDc<Doo^Oi"*Ol^oo»oocO'-^01-^^-P^ 
 coMTf(m»oStDt»-c-oooooi0500«HrH(Nca«eQ"<i<»o»ocooi>-i>ooooo> 
 
 l>.t^obCOm©OtH,HNMCOim'^xail>,lO<0 1.-'l--.OOC)00>CTiOO.H.-H(NMCO 
 
 OtDi-iepCMt^eOOOTtlOblOOtO'H'CNt^COQO'^OS'^O'O'-IO.-ID-C'IOOCO 
 i-li-HC<INCOCO"*Tf<lOlOtDI>t>0000010SOOi-li-(C!|e«3W»JH'*WlOCOCCll~ 
 
 ioweo«t-ccoo^o>u3QcOTHr-NOQeoOiT*;ou3iH5D«r-coog^mino 
 ^-ooooo»oor^lHp5«coTf^■«J^lf^locD^ot^^-oowo>oo'-lT-<&^Me*^■^■^ 
 
 ^HtHCMCSMCCI*'^ to CO OI>r~C0000>OO>-li-lMMC0MTf<W3meD«OI>.t^ 
 
 OlOi-H-NCO-^OHOi-lOCqcOCOO>»OOcONt-COai-^OcOiHt-«00-^0 
 
 ^'^ioioa3eor-t-ooosOTOO'-Hi-HCicoco"*Tt<iou3«oc^r-oDODOi030.-t 
 
 CDC^00Q00>0»OOi-<CMC5c0C0'*I'">*<i0OCDt-l-0000daOO.-t^C^NC0^ 
 
 1-ll-li-li-Hi-Hi— iT-t^H 
 tONI>-eOO>»(30CONCOC001iOi-<eC«OQ^O>lOi-Ht--C<10Q-^t'0»0(-lt-COOO 
 
 o)OOi-Hf-<cqcop3-^'^tn»oeot*t^ooGoa>oio>-(THCicsM'ti*Ti4iou3«5<o 
 
 uSu3ia>rjtaio>ou3^^^u5uSu6^cococdcocoocD«o^cocdcdocooco 
 
 Oi lO'O COMOO-#Oy5^r-COOO-*OeDNOOC001U5iHSD«00-Tf(OSvOi-it^M 
 T}'w3CO«Df-l>OD010»OOi-li-HNeOCO'*'*lOlO«5£~-I>OOOOCbOiOr-li-lCq 
 
 ti00a0010>OOi-lWC<»C0C0-*«3«3eDtDt--t^000109OOi--l«WWC0'#'SJ 
 
 g I I !■;';;;;;;;;;;;;;;;;;;;;;;•• • 
 
TOPOGBAPHV, MAP BEADING^ AJSTD BECOIiriirAISSAlSrCE. 
 
 107 
 
 o 
 
 EH 
 
 o 
 
 o 
 o 
 
 <! 
 
 I 
 
 o 
 
 a 
 
 o 
 
 m 
 
 
 lOT-t^OWt-eooO-^OSiO 
 
 p.ga 
 
 JiO <-l CO C4 £^ CO flO Ti< ffi lO 
 
 J> »0 c3 O 00 J^ CO O CO «o 
 
 i-« « so ioo ■* *o so eo t^ 
 
 t^ ■* rH OO >0 « O <D -W O 
 
 tt t-«3 "^ 00 ^ O 00 1> »0 
 
 »-ij-iC«)NeO-*-*»au5cO 
 
 eoa>ocoo<»N(©a>(N 
 
 cseo-osodooior-ioo^ 
 
 « ro rf c^ »d 00 ci -^ t; S 
 
 oa 13a CO CO f> CO CO U3 lO '^ 
 ■4iflo uS r^ o iH £<3 u5 1> ^ 
 
 ■ua 10 -O aO -(B ^ 03 -d< o» 
 rH flO -* O t^ BD d iH « ■'^ 
 
 th ci e*5 -^ »o «D t-^ 06 ^ g 
 
 Iff d -^ -00 CO C- OT <D 1-1 M3 
 
 « «» « O OD no eo 1-1 OD o 
 o rt ic4 «o «6 ■* «5 eo «3 1-^ 
 
 10 O »0 O »0 0>-* Oi ■^ Oi 
 
 o f-5 i-i cq«c3<6« -* ■* 
 
 0C U300r-wv<coaac^-4<t^ 
 o d d-rH th th T-5 «5 ci N 
 
 »^ CT ■^ 1©^ !>-«> O «-< « 
 
 dddddddrHT-fi-t 
 
 Oi-irHiHC<)cr4cqC4ccen 
 oooddddod^ 
 
 00^0000000 
 
 oddoddoo^^ 
 
 ® ;:;::::: is 
 
 SodoQOOQSS® 
 
 .a 
 
108 TOPOGRAPHY, MAP EEADIITG, AND EECONNAISSANCE. 
 
 DETERMINATION OP DIFFERENCES IN ELEVATION. 
 
 197. In topograpliical work elevations are referred to a common 
 level surface called the datum. The datum is taken low enough so 
 that no point of the area to be mapped will be below it.. This makes 
 aU elevations positive. For topographic surveys the datum in general 
 use throughout the world is mean sea level. This should always be 
 used where practicable. 
 
 The difference in elevation of the two points may be determined: 
 (a) By means of the angle which the hne connecting the two points 
 makes with the horizontal, and the horizontal or inclined distance 
 between the two points; (b) by means of an aneroid barometer car- 
 ried from one point to the other; (c) by means of the differential spirit 
 levehng between the two points. 
 
 The vertical angle may be read with some form of clinometer, or 
 with a transit having a vertical circle. 
 
 BAROMETRIC LEVELING. 
 
 198. The weight of the atmosphere at sea level is 14.703 pounds per 
 square inch, equal to the weight of a colimm of mercury 29.92 inches 
 high, or a column of fresh water 34,7 feet high. 
 
 The aneroid barometer records the pressure of the atmosphere in 
 inches, the same as a mercurial barometer, the reading being taken 
 from a pointer moving on a circular scale. It must be carefully 
 handled as it issensitive to shocks. A screw head will be seen through 
 a hole in the back of the outer case by which the needle may be. 
 brought to any desired reading, and the instrument corrected when- 
 ever it can be compared wiUi a standard. With the aneroid, correc- 
 tions for instrumental temperature can not be made, and for this 
 reason small pocket instruments are preferable, as carried in the 
 pocket they are not exposed to so great changes in this respect. 
 
 The pressure of the atmosphere varies with the altitude above sea 
 level, and it also varies with the moisture, temperature, and latitude, 
 which do not depend upon the altitude. 
 
 In measuring altitudes with the barometer these other causes of 
 variation must be eliminated so far as possible. It is best done by 
 simultaneous observation at both stations. If the stations' are not far 
 apart all disturbing conditions will be substantially the same at each 
 and therefore eliminated, except temperature, which, with consider- 
 able difference of altitude, wiU always oe less at the upper than at the 
 lower station. 
 
 If simultaneous observations can not be made, the stations should be 
 occupied with as little interval of time between as possible, and better 
 results will be obtaiaed if the time of observation can be so chosen as 
 to take advantage of cahn, bright, dry weather. 
 
 When the hygrometric conditions are very imiform an aneroid read 
 at intervals on a day's march over a rough conutry will give a fairly 
 good idea of the profile. 
 
 199. Table of elevations above sea level from barometer readings 
 (United States Coast and Geodetic Survey), for mean hygrometric' 
 conditions and mean temperature of 50° F. : 
 
TOPOGRAPHY, MAP EEADING, AKD KECONNAISSANCE. 109 
 
 Barome- 
 
 eter 
 reading. 
 
 IncTtes. 
 
 18.0 
 
 18.1 , 
 
 18.2 
 
 18.3...... 
 
 18.4 
 
 18.5 
 
 18.6 
 
 18.7 
 
 18.8 
 
 18.9 
 
 19.0 
 
 19.1 
 
 19.2 
 
 19.3 
 
 19.4 
 
 19.6 
 
 19.6 
 
 19.7 
 
 19.8 
 
 19.9 
 
 20.0 
 
 20.1 
 
 20.2 
 
 20.3 
 
 20.4 
 
 20.5 
 
 20.6 
 
 20.7 
 
 20.8 
 
 20.9 
 
 21.0 
 
 21.1 
 
 21.2 
 
 21.3 
 
 21.4 
 
 21.5 
 
 21.6 
 
 21.7 
 
 21.8 
 
 21.9 
 
 22.0 
 
 22.1 
 
 Altitude 
 
 above 
 sea level. 
 
 Feet. 
 13,918 
 13,707 
 13,617 
 13,46S 
 13,319 
 13,172 
 13,025 
 12,879 
 12,733 
 12,689 
 12,445 
 12,302 
 12, 100 
 12,018 
 11,877 
 11,737 
 11,598 
 11,459 
 11,321 
 11, 184 
 11,047 
 10,911 
 10,776 
 10,642 
 10,508 
 10,375 
 10,242 
 10,110 
 9,979 
 9,848 
 '9,718 
 9,589 
 9,460 
 9,332 
 9,204 
 9,077 
 8,951 
 ' 8,825 
 8,700 
 8,575 
 8,451 
 8,327 
 
 Differen- 
 tial tor 
 0.01 inch. 
 
 Feet. ' 
 -15.1 
 -15.0 
 -14.9 
 -14.9 
 -14.7 
 -14.7 
 -14.6 
 -14.6 
 -14,4 
 -14.4 
 -14.3 
 
 ' -14.2 
 -14.2 
 -U.l 
 -14.0 
 -13.9 
 -13.9 
 -13.8 
 -13.7 
 -13.7 
 -13.6 
 -13.5 
 -13.4 
 -13.4 
 -13.3 
 -13.3 
 -13.2 
 -13.1 
 -13.1 
 -13.0 
 -12.9 
 -12.9 
 -1?.8 
 -12.8 
 -12.7 
 -12.6 
 -12.6 
 -12.5 
 —12.5 
 -12.4 
 -12.4 
 -12.3 
 
 Barom- 
 eter 
 reading. 
 
 Inches. 
 
 22.2 , 
 
 22.3 
 
 22.4 
 
 22.5 
 
 22.6 
 
 22.7 
 
 22.8 
 
 22.9 
 
 23.0 
 
 23.1 
 
 23.2 
 
 23.3 
 
 23.4.;.... 
 
 23.5 
 
 23.6 
 
 23.7 
 
 23.8 
 
 23.9 
 
 24.0 
 
 24.1 
 
 24.2 
 
 24.3 
 
 24.4 
 
 24.5 
 
 24.6 
 
 24.7 
 
 24.8 
 
 24.9 
 
 25.0 
 
 25.1 
 
 25.2 
 
 25.3 
 
 25.4 
 
 25.5.%.... 
 
 25.6 
 
 25.7 
 
 26.8 
 
 25.9 
 
 26.0 
 
 26.1 
 
 26.2 
 
 26.3 
 
 Altitude 
 
 above 
 sea level. 
 
 Feet. 
 8,204 
 8,082 
 7,960 
 7,838 
 7,717 
 7,697 
 7,477 
 7,358 
 7,239 
 7,121 
 7,004 
 6,887 
 6,770 
 6,654 
 6,538 
 6,423 
 6,308 
 6,194 
 6,080 
 5,967 
 5,854 
 5,741 
 5,629 
 5,618 
 5,407 
 5,296 
 5,186 
 5,077 
 4,968 
 4,859 
 4,751 
 4,643 
 4,535 
 4,428 
 4,321 
 4,215 
 4,109 
 4,004 
 3,899 
 3,794 
 3,690 
 3,586 
 
 Differen- 
 tial for 
 0.01 Inch. 
 
 Feet. 
 -12.2 
 -12.2 
 -12.2 
 -12 1 
 -12.0 
 . -12. 
 -11.9 
 -11.9 
 -11.8 
 -11.7 
 -11.7 
 -11.7 
 -11.6 
 -11.6 
 -11.5 
 -11.5 
 -11.4 
 -11.4 
 -11.3 
 -11.3 
 -11.3 
 -11 2 
 -11.1 
 -11.1 
 -11.1 
 -11.0 
 -10.9 
 -10.9 
 -10.9 
 -10.8 
 -10.8 
 -10.8 
 -10.7 
 -10.7 
 -10.6 
 -10.6 
 -10.5 
 -10.5 
 -10.5 
 -10.4 
 -10.4 
 -10.3 
 
 Barom- 
 eter 
 reading. 
 
 Incites. 
 
 16.4 , 
 
 16.5 
 
 25.7. 
 26.8. 
 26.9. 
 27.0. 
 27.1. 
 27.2. 
 27.3. 
 27.4. 
 27.5. 
 27.6. 
 27.7., 
 27.8., 
 27.9., 
 28.0., 
 28.1., 
 28.2., 
 28.3., 
 28.4., 
 28.5.. 
 28.6.. 
 28.7.. 
 
 28.9. 
 29.0. 
 29.1. 
 29.2. 
 29.3. 
 29.4. 
 29.5. 
 29.6. 
 29.7. 
 29.8. 
 29.9. 
 30.0. 
 30.1. 
 30.2. 
 30.3. 
 30.4. 
 30.5. 
 
 Altitude 
 
 above 
 sea level. 
 
 Feet. 
 
 3,483 
 
 3,380 
 
 3,277 
 
 3,175 
 
 3,073 
 
 2,972 
 
 2,871 
 
 2,770 
 
 2,670 
 
 2,570 
 
 2,470 
 
 2,371 
 
 2,272 
 
 2,173 
 
 2,075 
 
 1,977 
 
 1,880 
 
 1,783 
 
 1,686 
 
 1,589 
 
 1,493 
 
 1,397 
 
 1,302 
 
 1,207 
 
 1,112 
 
 . 1,018 
 
 924 
 
 830 
 
 736 
 
 643 
 
 650 
 
 468 
 
 360 
 
 274 
 
 182 
 
 91 
 
 00 
 
 - 91 
 
 -181 
 
 -271 
 
 -361 
 
 -451 
 
 Differen* 
 tial lor 
 0.01 inch. 
 
 Feet. 
 -10.3 
 -10.3 
 -10.2 
 -10.2 
 -10.1 
 -10.1 
 -10.1 
 -10.0 
 -10.0 
 -10.0 
 
 - 9.9 
 
 - 9.9 
 
 - 9.9 
 
 - 9.9 
 
 - 9.8 
 
 - 9.8 
 
 - 9.7 
 
 - 9.7 
 
 - 9.7 
 
 - 9.7 
 
 - 9.6 
 
 - 9.6 
 
 - 9.5 
 
 - 9.5 
 
 - 9.5 
 
 - 9.4 
 
 - 9.4 
 
 - 9.4 
 
 - 9.3 
 
 - 9.3 
 
 - 9.2 
 
 - 9.2 
 
 - 9.2 
 
 - 9.2 
 
 - 9.1 
 
 - 9.1 
 
 - 9.1 
 
 - 9.0 
 
 - 9.0 
 
 - 9.0 
 
 - 9.0 
 
 - 8.9 
 
 COEFFICIENTS FOR TEMPERATURE COKREC(riON. 
 
 200. Argument (i+^') = STim of temperatures at the two stations: 
 
 t+t'. 
 
 Coeflcient 
 C. 
 
 t+t'. 
 
 Coefficient 
 C. 
 
 t+t'. 
 
 Coefficient 
 C. 
 
 
 
 -0.1024 
 -0.0915 
 -0.0806 
 -0.0698 
 -0.0592 
 -0.0486 
 -0.0380 
 
 70 
 
 -0.0273 
 -0.0166 
 -0.0O58 
 +0.0049 
 +0.0156 
 +0.0262 
 
 130 
 
 +0.0368 
 
 10 
 
 80 
 
 140 . 
 
 4-0 0472 
 
 20 
 
 90 
 
 150 
 
 +0. 0575 
 
 30 
 
 100 
 
 160 
 
 +0.0677 
 
 40 
 
 110 
 
 170 
 
 +0. 0779 
 
 50 
 
 120 
 
 180 
 
 +0. 0879 
 
 60... 
 
 
 
 
 
 
 Examples : 
 
 station. 
 
 Sacramento. 
 Summit 
 
 Temper- 
 ature. 
 
 °F. 
 59.9 
 42.1 
 
110 TOPOGRAPHY, MAP BEADING, AlfD BEC01irSfAISSA2T0E. 
 
 From table of elevations: Feet. 
 
 Sacramento = —12. 7 
 
 Summit =6,901.0 
 
 Differential =6,913.7 
 
 i+f =102" 
 
 ... C - ^- = 0.0070 
 
 .-. Temperature correction, 6,613.7x6.007 = +48. 4 
 
 H =6,962.1 
 
 c. .. i Barome- 
 
 Station. j ^p 
 
 Temper-' 
 ature. 
 
 Inches. 
 
 l/ower i 28.075 
 
 Upper 22.476 
 
 "J. 
 57.3 
 38. S 
 
 From table of elevations: Teet. 
 
 Lower = 7,867.0 
 
 Tipper , = 1,807.0 
 
 Differential = 6,060.0 
 
 l+t' , = 95''.08 
 
 .-. C - = +0.0004 
 
 .-. Temperature correction, 6,060 X0.0004 = +2.4 
 
 H. = 6,062.4 
 
 GENERAL ETJI.es FOR CSING ANEROBD BAROMETERS. 
 
 201. The best type of aneroid 'barometer for wse in reconnaissance 
 is one ■with a dial ahout 2^ inches in diameter, graduated to 3,000 
 feet on the scale, with a least reading of 10 feet. In using the bar- 
 ometer — 
 
 (1) Keep it at a temperature as nearly constant as is practicable. 
 This is best done by keeping it in an inner pocket, where it will have 
 nearly the temperature of the body. Remove 'it from the pocket 
 only for the pnTpose of reading and return it as soon as possible. 
 
 (2) Always hold the barometer with its dial horizontal when reading 
 it and tap it gently two or three times with the finger or pencU 
 before reading. 
 
 (3) In clear settled weather it will be found that the pressure vari- 
 ation due to change of temperature follows a regular law. Beginning 
 at about 9 a. m. the elevation scale will show a rise of about 10 feet 
 per hour for about four hours. It will then remain stationary until 
 about 4 p. m., and will then fall regularly until about 7 p. m., when 
 the same reading as at 9 a. m. will be reached. A knowledge of tiaa 
 change will enable proper corrections to be made. 
 
 (4) In unsettled weather, before or after a storm, note, if possible, 
 the movement of- the needle for an hour before starting work to ascer- 
 tain its direction and rate of change, and thus be enabled to make 
 proper corrections. 
 
 LEVELING WITH THE HAND LEVEL. 
 
 202. Differences in elevation can be determined with considerable 
 accuracy by means of the hand level or by means of the clinometer, 
 using level sights. For reconnaissance work without an assistant, 
 
TOPOGRAPHY, MAP READING, AND EECONNAISSANCE, 111 
 
 fc^^ ^^"""^^ "^P ^^^^' ^ote where tlie level line of sight strikei 
 tne ground, advance to that point and repeat the operation. Each 
 advance corresi)onds to a difference of elevation equal to the height 
 of the observer s eye. If an assistant is avaHable, leveling can be 
 done doym ^ade as weU as up, in which cage much longer sights 
 are possible by the use of an improvised level rod. 
 
 iJie hand level m connection with a standard level rod, carried 
 DT an assistant, has a wide application in construction work. It 
 admits of great rapidity in cross-section leveling and gives results 
 sumciently accurate for construction purposes. 
 
 The locator's hand level^ combining the virtues of the hand level 
 and the chnometer, is an instrument of peculiar value in all recon- 
 naissance work. 
 
 THE ENGINEER'S LEVEL. 
 
 203.. This instrument is shown and its parts indicated in Fig. 29. 
 The instrument is focused and set up as described for the transit, 
 except that, as there is but one level, the telescope must be turned in 
 the direction of one pair of leveling screws and leveled, then turned in 
 the direction of the other pair and leveled again. The second leveling 
 usually disturbs the first and the latter should then be releveled. 
 
 The level consists essentially of two geometric straight lines — the 
 line of sight and the vertical axis. The adjustment of the instrument 
 consists in making these two lines truly perpendicular to each other. 
 
 This is effected by the use of the level tube. The line of sight (the 
 geometric line through the center of the object glass and the inter- 
 section of the cross hairs) is made parallel to the axis of the bubble 
 tubes by making each parallel to the axis of the Y's. The verti- 
 cahty of the vertical axis, in the adjusted instrument, is secured by 
 the operation of leveHng. 
 
 First, make the line of sight parallel to the axis of the Y's, as 
 follows : Set up the instrument and level carefully; note a small object 
 about 300 feet away that one end of the horizontal cross hair touches; 
 turn the instrument in azimuth a few degrees and note whether the 
 other end of the cross wire cuts the point; if it does the horizontal 
 wire is horizontaL Now unlock the i dips. Bisect with the inter- 
 section of the cross hairs some sharply defined point at a convenient 
 distance; revolve (not reverse) the telescope m the Y's until the 
 bubble comes on top. E there is not' coincidence between the line 
 of sight and the axis of the Y's, the intersection of the cross hairs 
 will, in the revolved position, not be ia the original mark and the 
 amount that it has moved measures double the error. Correct the 
 horizontal hair, moving it (by tightening the top and loosening the 
 bottom capstan screws which hold the reticle or by tightening the 
 bottom and loosening the top, as the case may be) halfway back 
 toward the original mark. Verify the correction by repeating the 
 entire operation. Two attempts should usually be sufficient to 
 eliminate all error. Then adjust the vertical wire in the same way. 
 
 Second : Make the bubble axis parallel to the axis of the Y's, as 
 follows : Set up the instrument and unlock the Y cMps. Bring the 
 telescope to a position over one diagonal set of levelmg screws and 
 clamp m azimuth. Using the leveling screws, now center the bubble. 
 Then lift the telescope carefully out of the Y's and replace it in them. 
 
112 TOPOGRAPHY, MAP EEA.DING, AND EECONNAISSANOE, 
 
 reverse end for end. If the bubble axis is not parallel to the axis 
 of the Y's, it wiU be shown by the bubble leaving the center and the 
 amount of movement measures twice the error. Bring the bubble 
 halfway back by means of the adjusting screws at one end of the level 
 tube. 
 
 Now verify the adjustment by repeating the entire operation. 
 Two attempts should usually be sufficient to elimiaate all error. 
 
 The line of sight and the axis of the bubble tube now being each 
 parallel to the axis of the Y's are parallel to each other, and the level 
 is in adjustment. This is the only essential adjustment of the level, 
 and if the bubble be centered carefully ia every new direction that 
 a sight is taken there will be no error ia the work. 
 
 It is desirable, however, as a matter of convenience, that the ver- 
 tical axis be made truly vertical, so that if the instrument is leveled 
 in one direction the bubble will remain centered whUe the instru- 
 ment is moved in azimuth. 
 
 To make the vertical axis truly vertical, proceed as follows : 
 
 The instrument must first be adjusted as above indicated. Set up 
 the instrument and bring the telescope over a diagonal set of leveling 
 screws. Center the bubble by the leveling screws. Turn the instru- 
 ment 180° in azimuth. If the vertical axis is not truly vertical, the; 
 bubble will leave the center, and the amount of movement wiU 
 measure twice the error. Correct by moving the bubble halfway 
 back by means of the large nut under one Y. verify the adjustment, 
 and, if necessary, repeat. 
 
 "Peg method." The method of adjustment of the level given as- 
 sumes that the two rings on the telescope tube which rest in the Y's 
 are circular and exactly equal by construction. This is looked to by 
 good instrument makers. 
 
 The line of sight and axis of the bubble may be made perpendicular 
 independently of these two rings and the axis of the Y's by the method 
 known as the "peg method." This method is described in fuU under 
 the adjustment of the transit, paragraph 166. 
 
 level rods are of two kinds, target and self-reading or speaking. 
 The target rod is finely graduated and has a metal target sliding on 
 it, which is graduated as a vernier. The levelman signals to the 
 rodman, who moves the target up or down until it is in the correct 
 position, when the reading is taken by the rodman, or else the rod 
 is carried to the levelman to be read. The ordinary form is the New 
 York rod. The rod proper is in two parts, which shde on each other. 
 For readings up to 6.5 feet the target is moved on the rod and read 
 from the graduation on the front part by a vernier on the target. 
 For greater readings the target is damped at 6.5 feet and the back 
 part of the rod slid up on the front part, the reading being taken 
 irom a scale on the side of the back part of the vernier on the side of 
 the front part. The rod is graiduated to hundredths of feet and the 
 verniers read to thousandths. 
 
204. 
 
 TOPOGKAPHY, MAP EEAMNG, AND KEOONNAISSA.NCB. 
 FIELD NOTES FOR PROFILE LEVELING. 
 
 113 
 
 station. 
 
 B. S. 
 
 H.I. 
 
 r. S. 
 
 Eleva- 
 tion. 
 
 B. M. and 
 T. P. ele- 
 vation. 
 
 Eemarlrs. 
 
 B.M.16 
 
 
 7.825 
 
 115.089 
 
 
 
 107.264 
 
 X out on east end of 
 
 
 
 6.32 
 
 108.77 
 
 
 south abutment of 
 
 2 
 
 
 
 5.01 
 
 110.08 
 
 
 Main. Street Bridge 
 
 
 
 4.78 
 
 110.31 
 
 
 over Jones Creek. 
 
 
 
 
 3.22 
 
 111.87 
 
 
 
 T. P. 
 
 7.326 
 
 119.978 
 
 2.437 
 
 
 112.652 
 
 
 4 
 
 
 
 8.28 
 
 111.70 
 
 
 
 4 70 
 
 
 
 7.32 
 
 112. 66 
 
 
 
 5 
 
 
 
 S.26 
 
 114. 72 
 
 
 
 6 
 
 
 
 3.15 
 
 116.83 
 
 
 
 6 96 
 
 
 
 2.14 
 
 117.84 
 
 
 
 „ X, '' 
 
 
 
 2.05 
 
 117.93 
 
 
 
 T. P. 
 
 9.326 
 
 127.867 
 
 1.437 
 
 
 118.541 
 
 X cut on northwest 
 
 9 
 
 
 
 7.25 
 
 120.62 
 
 
 corner top step St. 
 Lukes Church. 
 
 
 
 
 
 
 
 
 
 
 15. 151 
 
 
 3.874 
 
 
 
 
 
 3.874 
 
 
 
 
 
 
 11.277 
 
 
 107,264 
 
 
 
 
 
 
 118.541 
 
 FIELD NOTES FOE DIFFERENTIAL LEVELING. 
 
 Station. 
 
 B. M. 21 
 T. P. 1 
 T. P. 2 
 T. P. 3 
 B.M. 
 
 B. S. 
 
 8.752 
 9.365 
 10.213 
 6.428 
 
 34.758 
 8,782 
 
 25.976 
 374. 256 
 
 n. L 
 
 383.008 
 389.936 
 398.003 
 401.215 
 
 F. S. 
 
 2.437 
 2.146 
 3.216 
 0.983 
 
 8,782 
 
 Eleva- 
 tion. 
 
 374.266 
 380. 571 
 387. 790 
 394. 787 
 400.232 
 
 Remarks. 
 
 X cut on northeast corner of 
 coping of bridge corner Main 
 and Second Streets. 
 
 Station No. 3. 
 
 PROFILE LEVELING. 
 
 205. To determine a profile: The line to be profiled is first sta- 
 tioned, every 100 foot point or such other distance as is desired being 
 distinctly marked, usually with a stake. The level is set up and a 
 rod reading called a back sight (B. S.) taken on a point, called a 
 benck mark (B.-M.) whose elevation is known. When the B. S. is 
 added to the elevation of the B. M., it gives the height of the instra- 
 ment (H. I.). Rod readings called fore sights (F. S.), may then be 
 read on as many station pomts as can be conveniently seen from the 
 instrument. The elevation of the point on which the rod rests, 
 when A. F. S. is taken is found by subtracting the F. S. from the H. I. 
 
 F. S.'s are taken to all station stakes and also to all changes of 
 slope whether they fall at a station or not; and intermediate sta- 
 tions at such pomts are located by tape measurements. When 
 aU the required F. S.'s have been taken that are desired from first 
 set up and it becomes necessary to move tlie instrument to a new 
 
 58740°— 18 8 
 
114 TOPOGEAPHY, MAE ^EEADIliTG, AND EECONKAISSANCE. 
 
 position to proceed along the line to be profiled, a turning point 
 (T. P.) is selected and its elevation is determined by a careful F. S. 
 This elevation is to be used to determine the height of instrument at 
 the new position. This is done by setting up the instrument at the 
 
 «4 
 
 
 new position and takmg a back sight at the T. P. This B S added 
 to the predetemuned elevation of the T, P., is the H. I, for the new 
 position. As soon as this is found F. S.'s may be taken on the line 
 Then a new T. P, is selected from which to determine the H I of a 
 
TOBOGBAPH?, MA^ EEADHTO, AND EEOOlTWAISSAlirOE. 115 
 
 £Vhf t*fwt«*^^ ^^''^' ^^'. ^^^^i^SS O'l B- ^-'s and T. P.'s 
 The B M*« «?«IV^ T^f ^r^""^! P}^°« tJ'a^ *Ji«se for the proflla. 
 Ihe J3 Ms are aU carefully described in the notes. As a rule the 
 t-^oi^^JhLl ^ described, as they are of temporary use only, 
 except when they are taken on easily identiaed poiits. 
 bA «t^^« T^^l Y^^^^^i-eat accura,cy is not needed, ihe T. P.'s may 
 Wk!5. oT «r /takes dnren mto the ground at convenient pointi. 
 When accurate Work is attempted a better T. P. must be used: 
 such as the head of a hatchet the blade of which is firmly driven 
 into the ground, or a We stone well embedded; still better, a steel 
 pin « mches long, camed by the rodman for the purpose. Such a 
 pm should be driven mto the ground for use and its top should be 
 kept rounded hj frequent dressing so that there shall l)e but one 
 pomt ot contact With the rod; as soon as the new H. I. has been 
 determmed from the T. P.j the steel pia is pulled from the ground 
 by a cord through an eye m the pin, and used at the next T. P. 
 
 The H. I. must always be obtained from a B. M. or proper T. P.; 
 it should never be determined from a stake on the line which is not 
 perfectly firm in the fflroimd and provided with an upper surface 
 givmg only one point ot contact. 
 
 BIFPEEENTIAL tEVEJLING. 
 
 206. In differential levelmg, properly speaking, fte only result 
 sought is lihe determination of the difference of elevation of two 
 poiats, such as between an existing B. M. and one newly established, 
 etc. No stationing of such a line is necessary and all sights taken 
 are to T. P.'s or B. M.'s. Differential and profile levefing merge 
 into each other and there are few cases in practice where the line 
 run is not a combination of the two. 
 
 LEVEL NOTEBi 
 
 207. The forms given in paragraph 207 are recommended. The 
 column arrangement gives the order of Work. All level lines should 
 be checked as shown on the forms. The difference between the 
 sum of the B. S.'s and the sum of the F. S.'s, between any two eleva- 
 tions, added or subtracted, as the case may be, to the first elevation 
 will give the other elevation. This check applies only to the sights 
 ■which are included in the chain of the levels. No F. S. on other 
 than T. P.'s or B. M.'s are part of the chain of levels. This check is 
 arithmetic only and therefore exact. It is in no Way a check on the 
 leveling, but only on the accuracy of the notes. 
 
 A level line is a curved line everywhere perpendicular to the plumb 
 line, while the line of sight of the ijEistrtunent is a horizontal hno 
 tangent to the level line through the instrument. In order, there- 
 fore, to run out a line of levels (curve) by means of a straight fine 
 of sight, it is necessary that the F. S. and B. S. be equal in length. 
 The varying densities of layers of air cause a ray of light to be bent 
 in a verticS direction. The effect of this bending (refraction) can 
 be eliminated by limiting the sights to 300 to 400 feet and making 
 the F. S. and B. S. equal. The boiling of the air may make neces- 
 sary even shorter sights than these. 
 
 208. The accuracy of level lines inay be checked as follows: Double 
 rodded lines may be run. In such lines, for every set up of the instru- 
 
116 TOPOGEAPHY, MAP BEADING, AKI) BECONNAISSANOE. j 
 
 ment two (2) T. P.'s are read. They should be close together but 
 should vary in elevation by at least a foot. This gives two inde- 
 pendent determinations for each H. I. When these independent 
 determinations differ by small amoimts, the work is accepted and 
 notes of both determinations are taken. If they show sudden marked 
 differences, the work is investigated on the spot till the error or 
 mistakes are discovered and corrected. By this method the leveler 
 knows at all times the measure of the precision of his work before 
 he leaves the field. 
 
 Another method is to run what are called level loops. The level 
 line is carried forward a mile or so and then closed back to the initial 
 point. If the circuit closes within the allowable limit of error, the 
 work is accepted, the error distributed properly; the work then 
 
 f)roceeds from the outer end of the loops as a B. M. and a second 
 oop is run, etc. The allowable error iu the precise leveling of the 
 United States Coast and Geodetic Survey is 4 millimeters multiplied by 
 the square root of the distance run in kilometers. For such work sen- 
 reading" level rods are used and in the construction of the instruinent 
 and its manipulation. The observations are made in such a manner 
 as to elimiaate the recognized common errors in leveling, which are 
 setting o* level on soft ground; unequal expansion tod contraction 
 of different parts of the instrument due to temperature changes; 
 irregular refraction of air near the ground; unequal length of bacK 
 and fore sights; poor turning points; rod not held plumb; bubble 
 not centered when reading is taken. These errors may be eliminated 
 in levels run with the engineer's level and results of considerable ac- 
 curacy obtained. The ordinary Y level has given results well within 
 the accuracy of precise work. The allowable error of the United 
 States Geological Survey primary levels is equal to the square root of 
 the number of miles run multiphed by 0.04 of a foot (0.04 foot-mUes). 
 Work of such accuracy is done only on the main level control lines of 
 a survey; between points thus accurately determined are run level 
 lines of a lesser accuracy by the level of the transit and stadia or by 
 the aneroid, these latter classes of lines being adjusted to the more 
 accurate lines. ' 
 
 SPEEa> OF LEVELING. 
 
 209. The speed with which levels may be run varies greatly with 
 the accuracy desired, the character of the country run over, the atmos- 
 pheric conditions, the method of leveling employed, and the skiU of 
 the levelmen and rodmen. Engineering levels of considerable accu- 
 racy, such as the primary levels of the United States Geological Survey 
 are run at speeds varying from 60 to 90 miles per month; the precise 
 levels of the Coast and Geodetic Survey have been at the rate of 3 to 
 5 miles per day (precise levels are, of course, usually run over the 
 best and most favorable grades). Flying levels, in which T. P.'s 
 only are taken, may be run under the most favorable conditions at a 
 rate of over 10 miles per day. 
 
 210. The logarithm of a number is the exponent of the power to 
 which a certain other number, called the base, must be raised to pro- 
 duce the given number. The base of the system most used, called 
 common logarithms, is 10. 
 
 In any system— 
 
 The log. of a product equals the sum of the logs, of the factors. 
 
TOPOGEAPHT, MAP KEAMNG, XND RECONNAISSANCE.. 117 
 
 «f "!? f i°^: °^ "^ quotient equals the log. of tlie dividend minus the log. 
 of the divisor; or the log. of a common fraction equals the log. of S 
 numerator minus the log. of the denominator. 
 
 The log. of 1 is ; since the log. 1 = the log. ~ = the log. 1 - log. 1 = 0. 
 The log^ of a power of a number equals the log. of the number mul- 
 tipbed V the exponent of the power. The log. of a root of a number 
 equals the log. of the number divided by the index of the root 
 
 The first property above is utilized in the construction of the tables. 
 Each log. is the sum of the logs, of two factors of which its number 
 is composed, and the factors may be so chosen that the log. of one ia 
 a whole number, called the characteristic, and the log. of the other 
 is a decimal fraction, called the mantissa. Any number may be 
 resolved into two factors, one of which is the number itself with the 
 decimal point after the first significant figure, and the other the figure 
 1, alone, or followed or preceded by one or more ciphers. 
 
 Thus; 
 
 3760=3.76X1000 log.=3.57518 
 
 376=3.76X100 log.=2.57518 
 
 37.6=3.76X10 log.=1.57518 
 
 3.76=3.76X1 log.=0.57518 
 
 0.376=3.76X0.1 log.=1.57518 
 
 0.0376=3.76X0.01 log. =5.57518 
 
 0.00376=3.76X0.001 log.=3.57518 
 
 The log. of the constant factor, 3.76 in the above example, is alwaya 
 a positive decimal fraction, and is called the mantissa. The log. of 
 the variable factor in the third column above is a whole number and 
 may be positive or negative. It is called the characteristic. The 
 logs, of all numbers presenting the same combination of significant 
 fi^ires have the same mantissa regardless of the position of the deci- 
 mal point. logarithmic tables contain mantissas only, since the char-; 
 acteristics may be written by inspection and mental calculation. To 
 this rule tables of logarithmic circular functions are an exception, as 
 win be explained later. If the number is whole or mixed, the charac- 
 teristic of^its log. is positive, and one less than the number of places 
 of figures ia the integral part, or on the left of the decimal point. If 
 the number is a decimal fraction, the characteristic of its log. ia 
 negative, and one greater than the number of ciphers immediately 
 following the decimal point. See example preceding. If the char- 
 acteristic is positive, the log. is a mixed number and naay be treated 
 as such in addition, subtraction, multiplication, and division. 
 
 If the characteristic is negative, the log. is not a true mixed number 
 and special treatment is necessary. A negative characteristic may 
 be considered as composed of two numbers, one negative and the 
 other positive. The positive number, prefixed to the mantissa, forms 
 a mixed number for arithmetical operations. The positive and nega- 
 tive parts may be simultaneously increased numerically by the same 
 number without altering the value of the log. 
 
 Thus: 
 
 3.4281=3+0.4281 
 =1+1.4281 
 =5+2.4281, etc. 
 
118 TOPO&BAPHY, MAP BEAMFG, AND KECONNAISSANCE. 
 
 For example, to multiply 4.7265 by i. 
 
 1+0-7265 
 4_ 
 
 16+2.9060=14.9060, wHci is the required result. 
 To subtract 1.8432 from 3.1329=1 + 1.1329. 
 
 3+1.1329 
 1+0.8432 
 
 3+0.2897=3.2897 
 To divide 5.2368 by 7. 2.2368 = 7 + 5.2368. ' 
 
 ^±^^=1+0.7481=1.7481 
 
 In this case the number added to the minus characteristic should 
 be just enough to make it exactly divisible by the divisor. 
 
 In the logs, of circular functions a characteristic is given in the 
 tables which is larger by 10 than the true characteristic. These logs, 
 may be used by the above rule by prefixing 10 to each. Thus the 
 log. sine of 21 min. as given in the table = 7.78594. The true log. is 
 10 + 7.78594, or 3.78594. Those who are familiar with the use of 
 these logs, perform the operation on the 10 mentally. The inex- 
 perienced wiU do well to write them out in fuU. 
 
 EXPLANATION OF THE TABLE. 
 
 211. Table IV gives to five decimal places the common logs, of 
 numbers from to 999 directly, and by interpolation from to 9,999. 
 If the log. of a number larger than 10,000 is desired, factor it and 
 take the sum of the logs, oi the factors. Thus, log. 99,225= log. of 
 75,000 plus the log. of 1.323 = 4.87506+0.12156 = 4.99662. Or con- 
 vert the number into a mixed number less than 1,000 and find its 
 log. Thus, log. 992.25 = 992 + J difierence between 992 and 993 = 
 99,662, which is the mantissa for 99,225. 
 
 In the table the logs, of 2 to 9, inclusive, are found at the tops of the 
 coluinns. For numbers above 10, the first two figures are in the 
 first column, the third at the tops of the columns, and the fourth is 
 interpolated. The right-hand column contains the average differ- 
 ence in each line between logs, in successive columns. For the 
 fourth place multiply one-tenth of the difference on the same line 
 by the fourth figure, and add the product to the log. of the first three 
 figures. Thus: 
 
 To find the log. of 4,827, look for 48 in the left-hand column; fol-^ 
 low the line to the column headed 2, and take out the mantissa 
 0.68304 for the number 482. In the right-hand coliman on the same 
 line is the difference 90, one-tenth of which, 9, multiplied by the 
 fourth figure, 7, = 63, to be added to the log. of 482, making the man- 
 tissa of 4,827 = 0.68367. The characteristic is 3 or 1 less than the 
 number of places of integral figures in the number, hence the com- 
 plete log. of 4,827 is 3.68367. 
 
 When the difference exceeds 200, if close results are desired, use 
 the difference obtained by subtracting the number found for the 
 third figure from that in the column for the next higher figure. 
 
 The number corresponding to any log. may be obtainea from the 
 table by the inverse process. If the given log. is found in the table, 
 the corresponding number consists of the two figures on the left of 
 
TOPOGRAPHY, MAP EEADING, AND EECONNAISSANCE, 
 
 119 
 
 the kae, followed by the one at the top of the column. If the exact 
 log. IS not m the table, find the next one below and take out the three 
 hgures lor that. Take the difference between the giren log. and the 
 one found m the table next below it and divide this difference by 
 one-tenth the tabulated difference on the line. Write down the 
 quotient for the 'fourth figure of the required number. 
 
 Thus, to find the number corresponding to 1.49638. This is not 
 in the table and the next below is 49,554. The two figures on the left 
 of the line are 31 and the figure at top of column is 3. Hence 313 
 is the number corresponding to 49,554. The difference between 
 49,638 and 49,554 is 84, which divided by 14 or one-tenth of the 
 tabulated difference 138 on the right of the line gives a quotient 
 of 6 tobe set down as the fourth figure. Jlence the number re- 
 quired is 0.3136, since the characteristic is 1 an^ therefore the sig- 
 nificant figures are immediately after the decimal point. 
 
 Table IV. 
 
 Cozmuon^Iosarlt^uiis, 1 taOOO: 
 
 No. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 . 6 
 
 7 
 
 8 
 
 9 
 
 Diff. 
 
 10 ; 
 
 oixwo 
 
 04139 
 07918 
 11394 
 14613 
 17609 
 20412 
 23045 
 23527 
 . 27875 
 30103 
 
 32222 
 34242 
 36173 
 38021 
 39794 
 41497 
 43136 
 44716 
 46240 
 47712 
 
 49136 
 50515 
 51851 
 53148 
 54407 
 65030 
 56820 
 67978 
 59106 
 60206 
 
 61278 
 62325 
 63347 
 64345 
 65321 
 66276 
 67210 
 68124 
 69020 
 69897 
 
 70757 
 71600 
 72428 
 73239 
 74036 
 74818 
 75587 
 76342 
 77085 
 77815 
 
 OGOOO 
 00432 
 04533 
 08278 
 11727 
 14921- 
 17897 
 20682 
 23299 
 2S767 
 2SI03 
 30319 
 
 32428 
 34439 
 36361 
 38201 
 38S67 
 41064 
 43296 
 44870 
 46389 
 47856 
 
 49276 
 60650 
 61982 
 53276 
 54530 
 55750 
 56937 
 58092 
 69217 
 60314 
 
 61384 
 62428 
 63447 
 64443 
 0.5417 
 66370 
 67302 
 68214 
 69108 
 69983 
 
 70842 
 71683 
 • 72509 
 73319 
 74116 
 74896 
 75663 
 76417 
 77158 
 77887 
 
 3Q103 
 00860 
 04921 
 08636 
 12057 
 18228 
 18184 
 20951 
 23562 
 26007 
 28330 
 30535 
 
 32633 
 34635 
 36548 
 38381 
 40140 
 41830 
 43456 
 45024 
 46538 
 48000 
 
 49415 
 60785 
 62113 
 63402 
 64654 
 55870 
 67054 
 58208 
 68328 
 60422 
 
 61489 
 
 62531 
 
 63548 \ 
 
 64542 
 
 66513 
 
 66464 
 
 67394 
 
 68304 
 
 69196 
 
 70070 
 
 70927 
 71767 
 72691 
 73399 
 74193 
 74973 
 75739 
 76492 
 77232 
 77959 
 
 47712 
 01283 
 05307 
 03990 
 12385 
 15533 
 18469 
 21218 
 23804 
 28245 
 28555 
 30749 
 
 32838 
 34830 
 36735 
 38560 
 40312 
 41996 
 43616 
 45178 
 46686 
 48144 
 
 49554 
 60920 
 52244 
 53529 
 64777 
 66990 
 67170 
 6S319 
 69439 
 60530 
 
 61596 
 62634 
 63648 
 64640 
 65609 
 66558 
 67486 
 68394 
 69284 
 70166 
 
 71011 
 71850 
 72672 
 73480 
 74272 
 75050 
 75815 
 76566 
 77306 
 78031 
 
 60206 
 01703 
 056£O 
 08342 
 12710 
 16836 
 18762 
 21484 
 24064 
 26481 
 28780 
 30963 
 
 33041 
 33024 
 3a£21 
 38739 
 40483 
 42160 
 43775 
 46331 
 46834 
 48287 
 
 49693 
 61054 
 52374 
 53665 
 64900 
 66110 
 67287 
 68433 
 69549 
 60638 
 
 61700 
 62736 
 63749 
 64738 
 65705 
 •66651 
 67577 
 68484 
 69372 
 70243 
 
 71096 
 71933 
 72764 
 73659 
 74361 
 75127 
 ' 76891 
 76641 
 77378 
 78103 
 
 69897 
 02118 
 06069 
 0G6&1 
 13033 
 16136 
 19033 
 21748 
 24303 
 26717 
 29003 
 31175 
 
 33243 
 33218 
 37106 
 38916 
 40654 
 42324 
 43C33 
 45484 
 46682 
 48430 
 
 49831 
 51188 
 62504 
 63781 
 55022 
 66229 
 57403 
 58S46 
 59659 
 60745 
 
 61804 
 62838 
 63848 
 64836 
 65801 
 66746 
 67669 
 68574 
 68460 
 70329 
 
 71180 
 72015 
 72836 
 73639 
 74429 
 76204 
 75966 
 , 76715 
 77451 
 78175 
 
 77815 
 02630 
 06445 
 10037 
 13353 
 16435 
 19312 
 22010 
 24551 
 28951 
 29225 
 31386 
 
 33445 
 35410 
 37291 
 39083 
 40824 
 43488 
 44080 
 46636 
 47129 
 48572 
 
 49968 
 61321 
 52633 
 63907 
 65145 
 56348 
 57518 
 58658 
 59769 
 60852 
 
 61909 
 - 62941 
 63948 
 64933 
 65896 
 66838 
 67760 
 68663 
 69548 
 70416 
 
 71265 
 72098 
 72918 
 73719 
 74507 
 75281 
 76042 
 78789 
 77524 
 78247 
 
 84510 
 
 02938 
 
 06818 
 
 10380 
 
 13672 
 
 16731 
 
 1G5S0 
 
 22271 
 
 24797 
 
 27184- 
 
 29446 
 
 31587 
 
 33646 
 366(B 
 37474 
 39269 
 40993 
 428S1 
 4424S 
 46788 
 47275 
 48713 
 
 50105 
 614.54 
 62763 
 64033 
 65268 
 66466 
 57634 
 68771 
 69879 
 60959 
 
 62013 
 63042 
 64048 
 65030 
 66991 
 66931 
 67851 
 68762 
 89635 
 70500 
 
 71349 
 72181 
 72997 
 73798 
 74585 
 75358 
 76117 
 76863 
 77597 
 78318 
 
 80309 
 03342 
 071E8 
 10721 
 13887 
 17026 
 19865 
 22530 
 25042 
 27415 
 29666 
 31806 
 
 33845 
 3579S 
 37657 
 39445 
 41162 
 42813 
 44404 
 -46939 
 47421 
 48863 
 
 60242 
 61687 
 62891 
 64157 
 65388 
 66584 
 57749 
 68883 
 69C88 
 61066 
 
 62118 
 63144 
 64147 
 65127 
 66080 
 67024 
 67942 
 68842 
 69722 
 70586 
 
 71433 
 72263 
 73078 
 73878 
 74663 
 73434 
 76192 
 76937 
 77670 
 78390 
 
 BE424 
 03742 
 07554 
 11059 
 14301 
 17318 
 20139 
 22788 
 25285 
 27646 
 29885 
 32014 
 
 34044 
 35983 
 37839 
 39619 
 41330 
 42976 
 44660 
 46089 
 47567 
 48995 
 
 50379 
 61719 
 63020 
 64282 
 66509 
 66702 
 67863 
 68995 
 60097 
 61172 
 
 62221 
 63245 
 64246 
 65224 
 66181 
 67117 
 ' 68033 
 68930 
 69810 
 70671 
 
 71616 
 72346 
 73158 
 73957 
 74741 
 76511 
 76267 
 77011 
 77742 
 78461 
 
 415 
 
 11 
 
 379 
 
 12 
 
 319 
 
 13 
 
 323 
 
 14 
 
 300 
 
 13 
 
 2S1 
 
 16 
 
 264 
 
 17 
 
 249 
 
 18 
 
 
 19 
 
 
 20 
 
 212 
 
 21 
 
 202 
 IM 
 185 
 
 22 
 
 23 
 
 24 
 
 171 
 164 
 158 
 163 
 148 
 143 
 
 25 
 
 26 
 
 27 . -- 
 
 28 
 
 29 
 
 30 - -.'-- 
 
 31 
 
 138 
 134 
 130 
 126 
 122 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 116 
 
 37 
 
 
 38 . -. 
 
 110 
 
 39 
 
 
 40 - . .- 
 
 
 41 
 
 104 
 102 
 
 42 
 
 99 
 
 43 
 
 98 
 
 44 
 
 96 
 
 45 
 
 94 
 
 46 . -- 
 
 92 
 
 47 
 
 60 
 
 48 
 
 83 
 
 49 
 
 88 
 
 50 
 
 
 51 
 
 84 
 82 
 
 52 
 
 81 
 
 S3 
 
 SO 
 
 54 
 
 78 
 
 55 
 
 77 
 
 56 
 
 75 
 
 57 
 
 74 
 
 
 73 
 
 59 
 
 72 
 
 60 • 
 
 
12fO TOPOGEAPHY, MAP BEADING, AND EECONNAISSANOE. 
 
 Table IV — Continued. 
 Conucon logaiithms^, 1 to 999— Continued. 
 
 No. 
 
 6 
 
 Difl. 
 
 61 
 62 
 63 
 64 
 65 
 66, 
 67, 
 68, 
 69, 
 70, 
 
 71, 
 72, 
 73, 
 74, 
 75. 
 76, 
 77, 
 78, 
 79, 
 80. 
 
 81. 
 82r. 
 83. 
 84. 
 85. 
 86. 
 87. 
 88. 
 89. 
 90. 
 
 91. 
 92. 
 93. 
 94. 
 95. 
 96. 
 97. 
 98. 
 99. 
 
 78533 
 79239 
 
 80618 
 81291 
 81954 
 82607 
 83250 
 
 84609 
 
 85125 
 85733 
 86332 
 
 87606 
 
 89762 
 90309 
 
 90848 
 91381 
 91907 
 92427 
 92941 
 
 93951 
 94448 
 94939 
 95424 
 
 95904 
 96378 
 
 97312 
 97772 
 
 98677 
 99122 
 99563 
 
 78604 
 79309 
 80002 
 80685 
 81358 
 82020 
 82672 
 83314 
 83947 
 84571 
 
 85187 
 85793 
 S6391 
 
 87564 
 88138 
 88705 
 
 89817 
 90363 
 
 90902 
 91434 
 91960 
 92479 
 
 93500 
 94001 
 94497 
 94987 
 95472 
 
 95951 
 96426 
 96895 
 
 97818 
 98272 
 98721 
 99166 
 99607 
 
 78675 
 79379 
 80071 
 80753 
 81424 
 82085 
 
 83378 
 84010 
 84633- 
 
 85248 
 85853 
 86461 
 87040 
 87621 
 88195 
 88761 
 
 89872 
 90417 
 
 90955 
 91487 
 92012 
 92531 
 93044 
 93660 
 94061 
 94546 
 95036 
 96520 
 
 95999 
 96473 
 96941 
 97405 
 97863 
 98317 
 98766 
 99211 
 99651 
 
 78746 
 79448 
 80140 
 80821 
 81491 
 82161 
 
 83442 
 84073 
 84695 
 
 85309 
 85913 
 86510 
 87098 
 87679 
 88252 
 
 89376 
 89927 
 90471 
 
 91009 
 91540 
 92064 
 92582 
 93095 
 93601 
 94101 
 94696 
 95085 
 95568 
 
 96047 
 96520 
 
 97451 
 97909 
 
 99255 
 
 78816 
 79618 
 80208 
 
 81557 
 82216 
 
 83505 
 84136 
 84757 
 
 85973 
 
 87167 
 87737 
 
 89431 
 89982 
 90525 
 
 91062 
 91592 
 92116 
 92634 
 93146 
 93651 
 94151 
 94645 
 95133 
 95616 
 
 96094 
 96567 
 97034 
 97497 
 97954 
 98407 
 
 99299 
 99738 
 
 78887 
 79588 
 80277 
 80956 
 81624 
 82282 
 
 83669 
 84198 
 84818 
 
 85430 
 
 86628 
 87215 
 87794 
 88366 
 
 90036 
 90679 
 
 91115 
 91645 
 92168 
 92686 
 93196 
 93701 
 94200 
 94694 
 95182 
 95664 
 
 96142 
 96614 
 97081 
 97643 
 98000 
 
 99343 
 
 78958 
 79657 
 80345 
 81023 
 81690 
 82347 
 82994 
 83632 
 84260 
 
 85491 
 
 86687 
 87273 
 87852 
 88422 
 
 89542 
 90091 
 
 91169 
 91698 
 92220 
 92737 
 93247 
 93751 
 942.50 
 04743 
 95230 
 95712 
 
 96661 
 97127 
 97589 
 98045 
 98497 
 
 99387 
 99825 
 
 79028 
 79726 
 80413 
 81090 
 81766 
 82412 
 83058 
 
 84323 
 84941 
 
 85551 
 86153 
 86746 
 87332 
 87909 
 88479 
 89042 
 89597 
 90145 
 90687 
 
 91222 
 91750 
 92272 
 92788 
 93298 
 93802 
 94300 
 94792 
 96279 
 96760 
 
 96236 
 96708 
 97174 
 9763S 
 
 98642 
 
 99431 
 
 79098 
 79796 
 80482 
 81157 
 
 82477 
 83123 
 83758 
 84386 
 85003 
 
 85612 
 86213 
 
 87390 
 87966 
 
 90200 
 90741 
 
 91275 
 91803 
 92324 
 92839 
 93348 
 93862 
 
 94841 
 95327 
 
 96754 
 97220 
 97680 
 98136 
 98587 
 98033 
 99475 
 99913 
 
 79169 
 79865 
 80550 
 81224 
 81888 
 82542 
 83187 
 83821 
 84447 
 85064 
 
 85672 
 86272 
 
 87448 
 
 88592 
 89153 
 89707 
 90254 
 90794 
 
 91328 
 91855 
 92376 
 92890 
 
 96376 
 95856 
 
 96331 
 96801 
 97266 
 97726 
 98181 
 98632 
 99078 
 99519 
 99956 
 
 71 
 70 
 69 
 6S 
 67 
 66 
 65 
 64 
 63 
 62 
 
 61 
 60 
 59 
 SS 
 57 
 56 
 56 
 SS 
 54 
 54 
 
 53 
 53 
 52 
 61 
 51 
 50 
 49 
 49 
 4S 
 48 
 
 48 
 47 
 47 
 
 45 
 45 
 44 
 
 44 
 
rOPOGEAPHY, MAP BEADING, AKD RECONNAISSANCE. 121 
 
 Table Y.— Common logarithms of cirmlar functions. 
 
 Arc. 
 
 00 
 01 
 
 02 
 03 
 CI 
 05 
 06 
 07 
 08 
 09 
 10 
 
 11 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 30 
 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 
 41 
 42 
 43 
 44 
 45 
 
 I? 
 48 
 49 
 50 
 
 51 
 52 
 53 
 54 
 65 
 56 
 57 
 58 
 69 
 60 
 
 Sine. 
 
 Inl.neg. 
 
 6.46373 
 
 .76476 
 
 6.94085 
 
 7.06579 
 
 .16270 
 
 .24188 
 
 .30882 
 
 .41797 
 .46373 
 
 7. 50512 
 .54291 
 .57767 
 
 .63982 
 .06784 
 .69417 
 .71900 
 .74248 
 .76475 
 
 7.78594 
 .80615 
 .82545 
 .84393 
 .86166 
 .87869 
 .89508 
 
 .92612 
 .94084 
 
 7.95508 
 .96887 
 
 7.99620 
 8.00779 
 .02002 
 .03192 
 .04350 
 .05478 
 .06578 
 
 8.07650 
 
 .09718 
 .10717 
 .11693 
 .12647 
 .13581 
 . 14495 
 .15391 
 
 8. 17128 
 . 17971 
 .18798 
 .19610 
 .20407 
 .21189 
 .21958 
 .22713 
 .23456 
 
 8.24185 
 
 Cosine. 
 
 DM. 
 
 17609 
 12494 
 9691 
 7918 
 6694 
 5800 
 5115 
 4576 
 
 4139 
 3779 
 3476 
 3218 
 2996 
 2803 
 2633 
 2482 
 2348 
 2227 
 
 2119 
 2020 
 1930 
 1848 
 1773 
 1703 
 1639 
 1580 
 1524 
 1472 
 
 1424 
 1379 
 1336 
 1297 
 1259 
 1223 
 1190 
 1158 
 1128 
 1100 
 
 1072 
 1046 
 1022 
 998 
 976 
 954 
 
 877 
 
 860 
 843 
 827 
 812 
 797 
 782 
 769 
 755 
 743 
 729 
 
 Difl. 
 
 Cosine. 
 
 10.00000 
 10.00000 
 10. 00000 
 10.00000 
 lO.OOOOO 
 10. 00000 
 9. 99999 
 
 .99999 
 
 .99999 
 
 .99999 
 .99999 
 .99999 
 
 .99909 
 .99999 
 
 .99999 
 
 .99999 
 .99999 
 
 .99998 
 
 .99998 
 
 .99998 
 
 .99997 
 .99997 
 .99997 
 .99997 
 
 1.99997 
 .99997 
 .99997 
 .99996 
 
 .99996 
 
 .99996 
 .99995 
 
 1.99995 
 .99995 
 .99995 
 
 .99994 
 .99994 
 
 Sine. 
 
 Difl. 
 
 Difl. 
 
 Tang. 
 
 Inf.neg. 
 
 6.46373 
 
 .76476 
 
 6.94085 
 
 7.06579 
 
 .16270 
 
 .24188 
 
 .41797 
 .46373 
 
 7.50512 
 .64291 
 .67767 
 
 .63982 
 . 60785 
 . 69418 
 .71900 
 .74248 
 .76476 
 
 7.78595 
 .80616 
 .82646 
 .84394 
 .86167 
 .87871 
 . 89510 
 
 .92613 
 
 7.95610 
 
 7.99522 
 8.00781 
 .02004 
 .03194 
 .04353 
 .05481 
 .06681 
 
 8.07653 
 .08700 
 .09722 
 .10720 
 . 11696 
 . 12651 
 .13686 
 .14600 
 . 15395 
 .16273 
 
 8. 17133 
 .17976 
 .18804 
 . 19616 
 .20413 
 .21195 
 .21964 
 .22719 
 .23462 
 
 8.24192 
 
 Contag. 
 
 Difl. 
 
 30103 
 
 17609 
 
 12494 
 
 9691 
 
 7918 
 
 5115 
 4576 
 
 4139 
 3779 
 3476 
 3219 
 2996 
 2803 
 2633 
 2482 
 2348 
 2228 
 
 2119 
 2020 
 1931 
 1848 
 1773 
 1704 
 1639 
 1579 
 1524 
 1473 
 
 1424 
 1379 
 1336 
 1297 
 1259 
 1223 
 1190 
 1158 
 1128 
 1100 
 
 1072 
 1047 
 1022 
 999 
 976 
 955 
 934 
 915 
 896 
 878 
 
 860 
 843 
 828 
 812 
 797 
 782 
 769 
 755 
 743 
 730 
 
 Difl. 
 
 Cotang. 
 
 Inf.pos. 
 
 13.53627 
 .23524 
 
 13.05915 
 
 12.93421 
 .83730 
 .75812 
 .69117 
 .63318 
 .58203 
 .53627 
 
 12.'49488 
 .45709 
 .42233 
 .39014 
 
 .33216 
 .30682 
 .28100 
 .25762 
 .23524 
 
 12.21405 
 .19384 
 .17454 
 .15606 
 .13833 
 . 12129 
 . 10490 
 .08911 
 .07387 
 .05914 
 
 12.04490 
 .03111 
 .01775 
 12.00478 
 11.99219 
 .97996 
 .96805 
 .95647 
 .94519 
 .93419 
 
 11.92347 
 .91300 
 .90278 
 
 .88304 
 .87349 
 .86416 
 .86500 
 .84606 
 .83727 
 
 11.82867 
 .82024 
 .81196 
 .80384 
 .79687 
 .78806 
 .78036 
 .77280 
 .76638 
 
 11..76808 
 
 TaJlg. 
 
 90 00 
 59 
 58 
 57 
 56 
 55 
 64 
 53 
 52 
 51 
 60 
 
 49 
 
 48 
 47 
 46 
 45 
 44 
 43 
 42 
 41 
 40 
 
 37 
 36 
 36 
 34 
 33 
 32 
 31 
 30 
 
 29 
 28 
 27 
 26 
 25 
 24 
 23 
 22 
 21 
 20 
 
 19 
 18 
 17 
 16 
 15 
 14 
 13 
 12 
 11 
 10 
 
 Arc. 
 
122 TOPOSBAPHT, MAP BEADING, ANI> KECONNAISSANCE. 
 
 Table Y .—Common logarithms of drcular functions — Continued. 
 
 Arc. 
 
 Sine. 
 
 DifE. 
 
 Cosine. 
 
 Difl. 
 
 Tang. 
 
 Difl. 
 
 Cotang. 
 
 
 1 00 
 
 8. 24185 
 
 729 
 
 9.99993 
 
 
 8.24192 
 
 730 
 
 11.75808 
 
 60 
 
 01 
 
 .24903 
 
 718 
 
 .99993 
 
 
 .24910 
 
 Vl8 
 
 .75090 
 
 59 
 
 02 
 
 .25609 
 
 706 
 
 .99993 
 
 
 .25616 
 
 706 
 
 .74383 
 
 
 03 
 
 .26304 
 
 696 
 
 .99993 
 
 
 .26311 
 
 695 
 
 .73688 
 
 67 
 
 04 
 
 .28988 
 
 684 
 
 .99992 
 
 
 .26996 
 
 686 
 
 .73004 
 
 
 05 
 
 . 27661 
 
 673 
 
 .99992 
 
 
 .27669 
 
 673 
 
 .72331 
 
 56 
 
 06 
 
 .28324 
 
 663 
 
 .99992 
 
 
 .28332 
 
 663 
 
 .71668 
 
 
 07 
 
 .28977 
 
 653 
 
 .99992 
 
 
 .28988 
 
 654 
 
 .71014 
 
 53 
 
 08 
 
 .29621 
 
 644 
 
 .99991 
 
 
 .29629 
 
 643 
 
 .70371 
 
 52 
 
 09 
 
 .30255 
 
 634 
 
 .99991 
 
 
 .30263 
 
 634 
 
 .69737 
 
 51 
 
 10 
 
 .30879 
 
 624 
 
 .99991 
 
 
 .30888 
 
 625 
 
 ,.69112 
 
 60 
 
 11 
 
 8.31495 
 
 616 
 
 9.99991 
 
 
 8.31505 
 
 617 
 
 11.68495 
 
 49 
 
 12 
 
 .32103 
 
 608 
 
 .99990 
 
 
 .32112 
 
 607 
 
 .67888 
 
 48 
 
 13 
 
 
 599 
 
 .99990 
 
 
 
 .32711 
 
 599 
 
 .67289 
 
 
 14 
 
 690 
 
 .99990 
 
 
 .33302 
 
 691 
 
 .66697 
 
 46 
 
 15 
 
 .33875 
 
 683 
 
 .99990 
 
 
 .33886 
 
 584 
 
 .66114 
 
 45 
 
 16 
 
 .34460 
 
 675 
 
 .99989 
 
 
 .34461 
 
 576 
 
 .65539 
 
 
 17 
 
 .36018 
 
 568 
 
 .99989 
 
 
 .35029 
 
 568 
 
 .64971 
 
 43 
 
 IS 
 
 .35578 
 
 560 
 
 .99989 
 
 , ./ 
 
 ..35689 
 
 560 
 
 .64410 
 
 
 19 
 
 .3ftl31 
 
 653 
 
 .99988 
 
 
 .36143 
 
 554 
 
 .63857 
 
 41 
 
 20 
 
 .36678 
 
 547 
 
 .99988 
 
 
 .36689 
 
 546 
 
 .63310 
 
 
 21 
 
 8.37217 
 
 539 
 
 9.99938 
 
 
 8.37229 
 
 540 
 
 11.62771 
 
 39 
 
 22 
 
 .37750 
 
 633 
 
 .99988 
 
 
 .37763 
 
 633 
 
 .62238 
 
 
 23 
 
 .38276 
 
 626 
 
 .99987 
 
 
 .38289 
 
 527 
 
 .61711 
 
 37 
 
 24 
 
 .38796 
 
 620 
 
 .99987 
 
 
 .38809 
 
 520 
 
 .61191 
 
 36 
 
 25 
 
 .39310 
 
 514 
 
 .99987 
 
 
 .39323 
 
 514 
 
 .60677 
 
 35 
 
 26 
 
 .39818 
 
 508 
 
 .99986 
 
 
 .39831 
 
 608 
 
 .60168 
 
 34 
 
 27 
 
 .40320 
 
 602 
 
 .99986 
 
 
 .40334 
 
 503 
 
 .69666 
 
 33 
 
 28 
 
 . 40816 
 
 496 
 
 .99986 
 
 
 .40830 
 
 496 
 
 .69170 
 
 32 
 
 29 
 
 .41307 
 
 491 
 
 .99985 
 
 
 .41321 
 
 491 
 
 .58679 
 
 31 
 
 30 
 
 .41792 
 
 485 
 
 .99985 
 
 
 .41807 
 
 486 
 
 .58193 
 
 30 
 
 31 
 
 8.42272 
 
 480 
 
 9.S9986 
 
 
 8.42287 
 
 480 
 
 11.57713 
 
 29 
 
 32 
 
 .42746 
 
 474 
 
 .99984 
 
 
 .42762 
 
 475 
 
 .57238 
 
 
 33 
 
 ,43216 
 
 470 
 
 .99984 
 
 
 .43231 
 
 469 
 
 .56768 
 
 27 
 
 34 
 
 .43680 
 
 464 
 
 .99984 
 
 
 .43696 
 
 465 
 
 .56304 
 
 26 
 
 35 
 
 .44139 
 
 - 469 
 
 90983 
 
 
 .44166 
 
 460 
 
 .55844 
 
 25 
 
 36 
 
 .44594 
 
 455 
 
 .99983 
 
 
 .44611 
 
 455 
 
 .55389 
 
 24 
 
 37 
 
 .45044 
 
 450 
 
 .99983 
 
 
 .45061 
 
 460 
 
 .54939 
 
 23 
 
 38 
 
 .46489 
 
 446 
 
 .99982 
 
 
 .45507 
 
 446 
 
 .64493 
 
 22 
 
 39 
 
 .45930 
 
 441 
 
 .99982 
 
 
 .45948 
 
 441 
 
 .54052 
 
 21 
 
 40 
 
 .46366 
 
 436 
 
 .99982 
 
 
 .46386 
 
 437 
 
 .53615 
 
 20 
 
 41 
 
 8.46798 
 
 433 
 
 9.99981 
 
 
 8.46817 
 
 432 
 
 11.63183 
 
 19 
 
 42 
 
 .47226 
 
 428 
 
 .99981 
 
 
 .47245 
 
 428 
 
 .52766 
 
 18 
 
 43 
 
 .47650 
 
 424 
 
 .99980 
 
 
 .47669 
 
 424 
 
 .52331 
 
 17 
 
 44 
 
 .48069 
 
 419 
 
 .99980 
 
 
 .48089 
 
 420 
 
 .61911 
 
 16 
 
 45 
 
 .48486 
 
 416 
 
 .99980 
 
 
 .48506 
 
 416 
 
 .51495 
 
 IS 
 
 46 
 
 .48898 
 
 411 
 
 .99979 
 
 ^, . 
 
 .48917 
 
 413 
 
 .51083 
 
 14 
 
 47 
 
 .49304 
 
 408 
 
 .99979 
 
 
 .49326 
 
 408 
 
 .50675 
 
 13 
 
 48 
 
 .49708 
 
 404 
 
 .99979 
 
 
 .49729 
 
 404 
 
 .50271 
 
 12 
 
 49 
 
 .50108 
 
 400 
 
 .99978 
 
 
 .60130 
 
 401 
 
 .49870 
 
 11 
 
 50 
 
 .50504 
 
 396 
 
 .99978 
 
 
 .60527 
 
 397 
 
 .49473 
 
 10 
 
 51 
 
 8.50897 
 
 393 
 
 9.99977 
 
 
 8.50920 
 
 393 
 
 11.49080 
 
 9 
 
 52 
 
 .51287 
 
 390 
 
 .99977 
 
 
 .61310 
 
 390 
 
 .48690 
 
 8 
 
 53 
 
 .51673 
 
 386 
 
 .99976 
 
 
 .61696 
 
 386 
 
 .48304 
 
 7 
 
 54 
 
 .52056 
 
 382 
 
 .99976 
 
 
 .52079 
 
 383 
 
 .47921 
 
 6 
 
 65 
 
 .S2434 
 
 379 
 
 ,99976 
 
 
 .52459 
 
 380 
 
 .47641 
 
 5 
 
 56 
 
 .62810 
 
 376 
 
 .99975 
 
 
 .62836 
 
 376 
 
 .47166 
 
 4 
 
 67 
 
 .63183 
 
 -373 
 
 .99975 
 
 
 .53208 
 
 373 
 
 .46792 
 
 3 
 
 68 
 
 .53662 
 
 369 
 
 .99974 
 
 
 .53578 
 
 370 
 
 .46422 
 
 2 
 
 59 
 
 .53919 
 
 367 
 
 .99974 
 
 
 .53945 
 
 367 
 
 .46065 
 
 1 
 
 60 
 
 8.54282 
 
 363 
 
 9.99973 
 
 
 8.54308 
 
 363 
 
 11.45692 
 
 88 
 
 
 Cosine. 
 
 Dill. 
 
 .Sine. 
 
 Difl. 
 
 Cotang. 
 
 Difl. 
 
 Tang. 
 
 Arc. 
 
.TOPOGRAPHY, MAP BEADIKG, AKO RECONNAISSANCE. 123 
 
 Table Y. -Common logarithms of circular funciions-Caaliimed. 
 
 
 
 
 
 
 
 
 
 
 Arc. 
 
 Sine. 
 
 Difl. 
 
 Cosine. 
 
 Difl. 
 
 Tang. 
 
 Difl. 
 
 Cotang. 
 
 
 2 00 
 01 
 
 8.54282 
 
 303 
 300 
 357 
 355 
 351 
 349 
 346 
 343 
 341 
 337 
 330 
 
 9.99973 
 
 
 8.54308. 
 
 363 
 
 11.45092 
 
 o f 
 
 60 
 
 02 
 
 
 99973 
 
 
 .54669 
 
 361 
 
 .45331 
 
 59 
 
 03 
 
 553 f)4 
 
 . 99973 
 
 
 .65027 
 
 368 
 
 .44973 
 
 58 
 
 04 
 
 
 .99972 
 
 
 .55382 
 
 365 
 
 .44618 
 
 67 
 
 05 
 06 
 07 
 08 
 09 
 10 
 
 .56054 
 .56400 
 .56743 
 . 57084 
 .57421 
 .57767 
 
 .99972 
 .99971 
 .99971 
 .99970 
 .99970 
 .99909 
 .99969 
 
 
 .55734 
 .56083 
 .56429 
 .56773 
 . 57114 
 . 57452 
 .57783 
 
 362 
 349 
 340 
 344 
 341 
 338 
 336 
 
 .44266 
 .43917 
 .43571 
 .43227 
 .42886 
 .42648 
 .42212 
 
 56 
 66 
 54 
 53 
 62 
 61 
 50 
 
 U 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 
 i 8. 58089 
 .58419 
 .58747 
 .59072 
 .59395 
 .59715 
 
 332 
 330 
 328 
 325 
 323 
 320 
 
 9.99908 
 .99908 
 .99907 
 .99907 
 .99966 
 .99960 
 
 
 8.58121 
 .58461 
 .58779 
 . 59105 
 .59428 
 .59749 
 
 333 
 
 330 
 328 
 320 
 323 
 321 
 
 11.41879 
 . 41649 
 .41220 
 .40895 
 .40572 
 . 40261 
 
 49 
 48 
 47 
 46 
 45 
 44 
 
 .60033 
 
 318 
 
 .99905 
 
 
 . 60008 
 
 319 
 
 .39932 
 
 43 
 
 
 310 
 
 .99965 
 
 
 . 60384 
 
 310 
 
 . 39310 
 
 42 
 
 
 
 313 
 
 .99904 
 
 
 .00698 
 
 314 
 
 . 39302 
 
 41 
 
 
 .60973 
 
 311 
 
 .99964 
 
 
 .61009 
 
 311 
 
 .38991 
 
 40 
 
 21 
 
 8. 61282 
 
 309 
 
 9.99903 
 
 
 8. 01319 
 
 310 
 
 11.38081 
 
 39 
 
 
 . 61589 
 
 307 
 
 .99963 
 
 
 . 01023 
 
 307 
 
 .38374 
 
 38 
 
 23 
 
 . 61894 
 
 305 
 
 .99962 
 
 
 .01931 
 
 305 
 
 .38009 
 
 37 
 
 24 
 
 . 62196 
 
 302 
 
 .99902 
 
 
 .62234 
 
 ' 393 
 
 . 37766 
 
 36 
 
 25 
 
 .62496 
 
 300 
 
 .99901 
 
 
 .62535 
 
 301 
 
 .37465 
 
 35 
 
 26 
 
 . 62796 
 
 299 
 
 .99901 
 
 
 .62834 
 
 299 
 
 . 37166 
 
 34 
 
 27 
 
 .63091 
 
 296 
 
 .99960 
 
 
 . 63131 
 
 297 
 
 .36869 
 
 33 
 
 28 
 
 .63385 
 
 294 
 
 .99900 
 
 
 .63426 
 
 295 
 
 .30674 
 
 32 
 
 29 
 
 .63678 
 
 293 
 
 .99969 
 
 
 .63718 
 
 292 
 
 .36282 
 
 31 
 
 30 
 
 .63968 
 
 290 
 
 .99969 
 
 
 .64009 
 
 291 
 
 .36991 
 
 30 
 
 31 
 
 8.64256 
 
 288 
 
 9. 99958 
 
 
 8.64298 
 
 289 
 
 11. 36702 
 
 29 
 
 32 
 
 .64543 
 
 287 
 
 .99957 
 
 
 .64585 
 
 287 
 
 .36415 
 
 28 
 
 33 
 
 .64827 
 
 284 
 
 .99957 
 
 
 .64870 
 
 285 
 
 .35130 
 
 27 
 
 34 
 
 .65110 
 
 283 
 
 .99950 
 
 
 .66154 
 
 284 
 
 .34840 
 
 26 
 
 35 
 
 .05391 
 
 281 
 
 .99950 
 
 
 .65435 
 
 281 
 
 .34505 
 
 25 
 
 36 
 
 .65670 
 
 279 
 
 .99955 
 
 
 . 65716 
 
 280 
 
 .34285 
 
 24 
 
 37 
 
 .66947 
 
 277 
 
 .99955 
 
 
 .05993 
 
 278 
 
 - .34007 
 
 23 
 
 38 
 
 .66223 
 
 270 
 
 .99954 
 
 
 .66269 
 
 270 
 
 .33731 
 
 22 
 
 39 
 
 .66497 
 
 274 
 
 .99953 
 
 
 .66543 
 
 274 
 
 .33457 
 
 21 
 
 40 
 
 .66769 
 
 272 
 
 .99953 
 
 
 .66810 
 
 273 
 
 .33184 
 
 20 
 
 41 
 
 8. 67039 
 
 270 
 
 9,99952 
 
 
 8.67087 
 
 271 
 
 11.32913 
 
 19 
 
 42 
 
 .67308 
 
 269 
 
 .99952 
 
 
 .67360 
 
 209 
 
 .32044 
 
 18 
 
 43 
 
 .67575 
 
 267 
 
 .99951 
 
 
 .67624 
 
 268 
 
 .32376 
 
 ,17 
 
 44 
 
 .67840 
 
 265 
 
 .99951 
 
 
 .67890 
 
 266 
 
 .32110 
 
 16 
 
 45 
 
 .68104 
 
 264 
 
 .99950 
 
 
 .68164 
 
 254 
 
 - .31846 
 
 15 
 
 46 
 
 .08300 
 
 262 
 
 .99949 
 
 
 .68417 
 
 263 
 
 .31583 
 
 .14 
 
 47 
 
 .68627 
 
 261 
 
 .99949 
 
 
 .68678 
 
 201 
 
 .31322 
 
 13 
 
 48 
 
 .68886 
 
 
 .99948 
 
 
 . 68938 
 
 200 
 
 .31062 
 
 12 
 
 49 
 
 . C9144 
 
 258 
 
 .99947 
 
 
 .69190 
 
 268 
 
 .30804 
 
 11 
 
 50 
 
 .69400 
 
 256 
 
 .99947 
 
 
 .69453 
 
 267 
 
 .30547 
 
 10 
 
 51 
 
 & 69054 
 
 254 
 
 9.99940 
 
 
 8.09708 
 
 266 
 
 11.30292 
 
 9 
 
 62 
 
 .09907 
 
 263 
 
 .99940 
 
 
 .09962 
 
 254 
 
 .30038 
 
 8 
 
 53 
 
 . 70159 
 
 262 
 
 .99945 
 
 
 .70214 
 
 262 
 
 .29780 
 
 7 
 
 54 
 
 .70409 
 
 260 
 
 .99944 
 
 
 .70466 
 
 251 
 
 .29535 
 
 
 
 55 
 
 .70658 
 
 249 
 
 .99944 
 
 
 ■ .-70714 
 
 ■ 249 
 
 .29280 
 
 5 
 
 56 
 
 .70905 
 
 - 247 
 
 .99943 
 
 
 .70962 
 
 248 
 
 .29038 
 
 4 
 
 57 
 
 .71161 
 
 246 
 
 .99942 
 
 
 .71208 
 
 240 
 
 .28792 
 
 3 
 
 58 
 
 .71396 
 
 244 
 
 .99942 
 
 
 .71453 
 
 245 
 
 .28547 
 
 2 
 
 59 
 
 .71638 
 
 243 
 
 .99941 
 
 
 .71697 
 
 .28303 
 
 1 
 
 60 
 
 8.71880 
 
 242 
 
 9.99940 
 
 
 8.71940 
 
 243 
 
 11.28000 
 
 87 
 
 
 Cosine. ■ 
 
 Difl. 
 
 Sine. 
 
 Difl. 
 
 Cotang. 
 
 Difl., 
 
 Tang. 
 
 Arc. 
 
124 TOPOGRAPHY, MAP BEADING, AND BECONNAISSANOB. 
 
 Table V.— ComTJion logarithms of circular functions — Continued. 
 
 Arc. 
 
 Sine. 
 
 Dill. 
 
 Cosine. 
 
 DifE. 
 
 Tang. 
 
 Difl. 
 
 Co tang'. 
 
 
 / 
 
 3 00 
 
 8.71880 
 
 242 
 
 _ 9. 99940 
 
 
 8. 71940 
 
 243 
 
 11.28060 
 
 o / 
 
 60 
 
 01 
 
 . 72120 
 
 240 
 
 .99940 
 
 
 .72181 
 
 241 
 
 .27819 
 
 59 
 
 02 
 
 .72359 
 
 239 
 
 . 99939 
 
 
 .72420 
 
 239 
 
 .27579 
 
 58 
 
 03 
 
 .72597 
 
 238 
 
 .99938 
 
 
 . 72659 
 
 239 
 
 .27341 
 
 57 
 
 04 
 
 .72834 
 
 237 
 
 .99938 
 
 
 .72896 
 
 237 
 
 .27104 
 
 56 
 
 05 
 
 .73069 
 
 235 
 
 .99937 
 
 
 .73132 
 
 ^ 236 
 
 .26869 
 
 55 
 
 06 
 
 .73303 
 
 234 
 
 .99936 
 
 
 .73366 
 
 234 
 
 .26634 
 
 54 
 
 07 
 
 .73635 
 
 232 
 
 .99930 
 
 
 .73600 
 
 234 
 
 .26400 
 
 63 
 
 08 
 
 .73767 
 
 232 
 
 .99935 
 
 
 .73832 
 
 232 
 
 . .26168 
 
 52 
 
 09 
 
 . 73997 
 
 230 
 
 .99934 
 
 
 .74063 
 
 231 
 
 .26937 
 
 51 
 
 10 
 
 .74226 
 
 229 
 
 .99934 
 
 
 .74292 
 
 - 229 
 
 .25708 
 
 50 
 
 11 
 
 8.74454 
 
 228 
 
 9.99933 
 
 
 8. 74621 
 
 229 
 
 11.26479 
 
 49 
 
 12 
 
 .74680 
 
 226 
 
 .99932 
 
 
 .74748 
 
 227 
 
 .26252 
 
 48 
 
 13 
 
 .74905 
 
 225 
 
 .99931 
 
 
 .74974 
 
 226 
 
 .25026 
 
 47 
 
 14 
 
 . 75130 
 
 225 
 
 .99931 
 
 
 .75199 
 
 225 
 
 .24801 
 
 46 
 
 15 
 
 .75353 
 
 223 
 
 .99930 
 
 
 .75423 
 
 224 
 
 .24577 
 
 45 
 
 16 
 
 _ .75576 
 
 222 
 
 .99929 
 
 
 .-75645 
 
 222 
 
 .24355 
 
 44 
 
 17 
 
 .75795 
 
 220 
 
 .99929 
 
 
 .76867 
 
 222 
 
 .24133 
 
 43- 
 
 18 
 
 . 76015 
 
 220 
 
 .99928 
 
 
 . .76087 
 
 220 
 
 .23913 
 
 42 
 
 19 
 
 .70234 
 
 219 
 
 .99927 
 
 
 .76306 
 
 219 
 
 .23693 
 
 41 
 
 20 
 
 .76451 
 
 , .217 
 
 .99926 
 
 
 .76525 
 
 219 
 
 .23476 
 
 40 
 
 21 
 
 8. 76667 
 
 216 
 
 9.99926 
 
 
 8.76742 
 
 217 
 
 11.23258 
 
 39 
 
 22 
 
 -.76883 
 
 216 
 
 .99926 
 
 
 .76968 
 
 216 
 
 .23042 
 
 38 
 
 23 
 
 .77097 
 
 214 
 
 .99924 
 
 
 .77173 
 
 215 
 
 .22827 
 
 37 
 
 24 
 
 .77310 
 
 213 
 
 .99923 
 
 
 .77387 
 
 214 
 
 .22613 
 
 36 
 
 25 
 
 .77522 
 
 212 
 
 .99923 
 
 
 .77599 
 
 212 
 
 .22400 
 
 35 
 
 26 
 
 .77733 
 
 211 
 
 .99922 
 
 
 ..77811 
 
 212 
 
 .22189 
 
 34 
 
 27 
 
 .77943 
 
 210 
 
 .99921 
 
 
 . 78022 
 
 211 
 
 .21978 
 
 33 
 
 ■28 
 
 .78152 
 
 209 
 
 .99920 
 
 
 .78232 
 
 210 
 
 .21768 
 
 32 
 
 29 
 
 .78360 
 
 208 
 
 .99920 
 
 
 .78441 
 
 209 
 
 .21559 
 
 .31 
 
 30 
 
 .78667 
 
 207 
 
 .99919 
 
 
 .78649 
 
 208 
 
 .21351 
 
 30 
 
 31 
 
 8. 78774 
 
 207 
 
 9.99918 
 
 
 8. 78855 
 
 206 
 
 11.21145 
 
 29 
 
 32 
 
 . 78979 
 
 205 
 
 .99917 
 
 
 .79061 
 
 206 
 
 .20939 
 
 28 
 
 33 
 
 .79183 
 
 204 
 
 .99917 
 
 
 .79266 
 
 205 
 
 -.20734 
 
 27 
 
 34 
 
 . 79386 
 
 203 
 
 .99916 
 
 
 - .79470 
 
 204 
 
 .20530 
 
 26 
 
 35 
 
 .79588 
 
 202 
 
 .99915 
 
 
 .79673 
 
 203 
 
 .20327 
 
 25 
 
 36 
 
 . 79789 
 
 201 
 
 .99914 
 
 
 . 79875 
 
 202 
 
 .20125 
 
 24 
 
 37 
 
 . 79990 
 
 201 
 
 .99913 
 
 
 .80076 
 
 201 
 
 .19924 
 
 23 
 
 38 
 
 .80189 
 
 199 
 
 .99913 
 
 
 .80276 
 
 200 
 
 .19723 
 
 22 
 
 39 
 
 .80388 
 
 199 
 
 .99912 
 
 
 .80476 
 
 200 
 
 .19524 
 
 21 
 
 40 
 
 . 80685 
 
 197 
 
 .99911 
 
 
 .80674 
 
 198 
 
 . 19326 
 
 20 
 
 41 
 
 8. 80782 
 
 197 
 
 9.99910 
 
 
 8.80872 
 
 198 
 
 11. 19128 
 
 19 
 
 42 
 
 .80978 
 
 196 
 
 .99909 
 
 
 .81008 
 
 196 
 
 .18932 
 
 18 
 
 43 
 
 .81173 
 
 195 
 
 .99909 
 
 
 .81264 
 
 196 
 
 . 18736 
 
 17 
 
 44 
 
 .81367 
 
 194 
 
 .99908 
 
 
 .81459 
 
 195 
 
 .18541 
 
 16 
 
 45 
 
 .81560 
 
 193 
 
 .99907 
 
 
 .81653 
 
 194 
 
 .18347- 
 
 15 
 
 46 
 
 .81752 
 
 192 
 
 .99906 
 
 
 .81846 
 
 198 
 
 .18154 
 
 14 
 
 47 
 
 .81944 
 
 192 
 
 .99905 
 
 
 .8*038 
 
 192 
 
 .17962 
 
 13 
 
 48 
 
 .82134 
 
 190 
 
 .99904 
 
 
 .8''230 
 
 192 
 
 .17770 
 
 12 
 
 49 
 
 .82324 
 
 190 
 
 . 99904 
 
 
 .\ 32120 
 
 190 
 
 .17*79 
 
 11 
 
 SO 
 
 .82513 
 
 189 
 
 .99903 
 
 
 .82610 
 
 190 
 
 .17390 
 
 10 
 
 51 
 
 8. 82701 
 
 188 
 
 9.99902 
 
 
 8.82799 
 
 189 
 
 11. 17201 
 
 9 
 
 52 
 
 .82888 
 
 187 
 
 .99901 
 
 
 .82987 
 
 188 
 
 . 17013 
 
 8 
 
 53 
 
 .83075 
 
 187 
 
 .99900 
 
 
 .83175 
 
 188 
 
 .16825 
 
 7 
 
 64 
 
 • .83261 
 
 186 
 
 .99899 
 
 
 .83361 
 
 ISO 
 
 .16639 
 
 6 
 
 55 
 
 .83446 
 
 is! 
 
 .99898 
 
 
 .83547 
 
 180 
 
 .16453 
 
 5 
 
 56 
 
 .83630 
 
 . 99898 
 
 
 .83732 
 
 185 
 
 . 16268 
 
 4 
 
 57 
 
 .83813 
 
 183 
 
 . 99897 
 
 
 .83916 
 
 - 184 
 
 .16084 
 
 3 
 
 58 
 
 .83096 
 
 183 
 
 .99896 
 
 
 .84100 
 
 184 
 
 .15900 
 
 2 
 
 59 
 
 .84177 
 
 181 
 
 .99895 
 
 
 .84282 
 
 182 
 
 .15717 
 
 1 
 
 CO 
 
 8.84358 
 
 181 
 
 9.99894 
 
 
 8.84464 
 
 182 
 
 11.15536 
 
 86 
 
 
 Cosine. 
 
 Difl. 
 
 Sine. 
 
 Diff. 
 
 Co tang. 
 
 Dill. 
 
 Tang. 
 
 Are. 
 
TOPOGBAPHY, MAP KBAI>ING, AND BEOONNAISSANOE. 
 Table V. — Common logarithms of circular functions — Continued. 
 
 125 
 
 Arc. 
 
 Sine. 
 
 Difl. 
 
 Cosine. 
 
 Difl. 
 
 Tang. 
 
 Difl. 
 
 Cotang. 
 
 
 4 00 
 
 8.843S8 
 
 181 
 
 9.99894 
 
 1 
 
 8.84464 
 
 182 
 
 11.15536 
 
 86 00 
 
 10 
 
 .86128 
 
 1770 
 
 .99885 
 
 9 
 
 .86243 
 
 1779 
 
 .13767 
 
 60 
 
 20 
 
 .87828 
 
 1700 
 
 .99876 
 
 9 
 
 .87963 
 
 1710 
 
 .12047 
 
 40 
 
 30 
 
 .89464 
 
 1636 
 
 .99866 
 
 10 
 
 .89598 
 
 1645 
 
 .10402 
 
 30 
 
 40 
 
 .91040 
 
 1576 
 
 .99856 
 
 10 
 
 .91185 
 
 1587 
 
 .08815 
 
 20 
 
 50 
 
 . 92561 
 
 1521 
 
 .99845 
 
 11 
 
 .92716 
 
 1631 
 
 .07284 
 
 10 
 
 5 00 
 
 8.94030 
 
 1469 
 
 9.99834 
 
 11 
 
 8.94195 
 
 ^1479 
 
 11.05805 
 
 85 00 
 
 10 
 
 .95450 
 
 1420 
 
 .3?823 
 
 11 
 
 .95627 
 
 1432 
 
 .04373 
 
 50 
 
 20 
 
 .96825 
 
 1375 
 
 .99812 
 
 11 
 
 .97013 
 
 1386 
 
 .02987 
 
 40 
 
 30 
 
 .98157 
 
 1332 
 
 .99800 
 
 12 
 
 .08358 
 
 1345 
 
 .01642 
 
 30 
 
 40 
 
 .99460 
 
 1293 
 
 .99787 
 
 13 
 
 .99662 
 
 1304 
 
 .00338 
 
 20 
 
 50 
 
 9.00704 
 
 1264 
 
 .99774 
 
 13 
 
 9.00930 
 
 1268 
 
 10.99070 
 
 10 
 
 6 00 
 
 9.01923 
 
 1219 
 
 9.99781 
 
 13 
 
 9.02162 
 
 1232 
 
 10.97838 
 
 84 00 
 
 10 
 
 .03109 
 
 1186 
 
 .99748 
 
 13 
 
 .03361 
 
 1199 
 
 .96639 
 
 60 
 
 20 
 
 .04262 
 
 1153 
 
 .99734 
 
 14 
 
 .04528 
 
 1167 
 
 .95472 
 
 40 
 
 30 
 
 .05386 
 
 1124 
 
 .99720 
 
 14 
 
 .05666 
 
 1138 
 
 . 94334 
 
 30 
 
 40 
 
 .06481 
 
 1095 
 
 .99705 
 
 15 
 
 .06775 
 
 1109 
 
 .93225 
 
 20 
 10 
 
 50 
 
 .07548 
 
 1067 
 
 .99690 
 
 15 
 
 .07858 
 
 1083 
 
 . 92142 
 
 7 00 
 
 9.08589 
 
 1041 
 
 5.99675 
 
 15 
 
 9.08914 
 
 1066 
 
 10.91088 
 
 S3 00 
 60 
 40 
 30 
 20 
 10 
 
 10 
 
 .09606 
 
 1017 
 
 .99659 
 
 16 
 
 .09947 
 
 1033 
 
 .90053 
 
 20 
 
 .10599 
 
 993 
 
 .99643 
 
 16 
 
 .10966 
 
 1009 
 
 .89044 
 
 30 
 
 . 11570 
 
 971 
 
 .99627 
 
 16 
 
 .11943 
 
 987 
 
 .88057 
 
 40 
 
 .12519 
 
 949 
 
 .99610 
 
 17 
 
 .12909 
 
 966 
 
 .87091 
 
 60 
 
 .13447 
 
 928 
 
 .99593 
 
 17 
 
 .13854 
 
 946 
 
 . 86146 
 
 8 00 
 10 
 20 
 30 
 40 
 50 
 
 9.14356 
 .15245 
 .16116 
 .16970 
 .17807 
 .18628 
 
 908 
 890 
 871 
 854 
 837 
 821 
 
 9.99575 
 .99557 
 .99539 
 .99520 
 .99501 
 .99482 
 
 18 
 18 
 18 
 19 
 19 
 19 
 
 9.14780 
 .15688 
 .16577 
 .17450 
 .18306 
 .19146 
 
 928 
 
 908 
 
 889. 
 
 873 
 
 866 
 
 840 
 
 10.85220 
 .84312 
 
 - .83423 
 .82550 
 .81694 
 .80854 
 
 82 00 
 50 
 40 
 30 
 20 
 10 
 
 9 00 
 10 
 20 
 30 
 40 
 SO 
 
 9.19433 
 .20223 
 .20999 
 .21761 
 .22509 
 .23244 
 
 805 
 790 
 776 
 762 
 748 
 735 
 
 9.99462 
 .99442 
 .99421 
 .99400 
 .99379 
 .99357 
 
 20 
 20 
 21 
 21 
 21 
 22 
 
 9.19971 
 .20782 
 .21578 
 .22361 
 .23130 
 .23887 
 
 825 
 811 
 796 
 783 
 769 
 757 
 
 10.80029 
 .79218 
 .78422 
 .77639 
 .76870 
 .76113 
 
 81 00 
 60 
 40 
 30 
 20 
 10 
 
 10 00 
 10 
 20 
 30 
 40 
 50 
 
 9.23967 
 .24677 
 .25376 
 .26063 
 .26739 
 .27405 
 
 723 
 710 
 699 
 687 
 676 
 666 
 
 9.99335 
 .99313 
 .99290 
 .99267 
 .99243 
 .99219 
 
 22 
 22 
 23 
 23 
 24 
 24 
 
 9.24632 
 .25365 
 .26086 
 .28797 
 .27496 
 .28186 
 
 746 
 733 
 721 
 711 
 699 
 690 
 
 10.75368 
 .74636 
 .73914 
 .73203 
 .72604 
 .71814 
 
 80 00 
 50 
 40 
 30 
 20 
 10 
 
 11 00 
 10 
 20 
 30 
 40 
 50 
 
 9.28060 
 .28705 
 .29340 
 .29965 
 .30582 
 .31189 
 
 655 
 645 
 635 
 625 
 617 
 607 
 
 9.99196 
 ,99170 
 . 99145 
 .99119 
 .99093 
 .99067 
 
 24 
 25 
 25 
 26 
 26 
 26 
 
 9.28865 
 .29635 
 .30195 
 .30846 
 .31488 
 .32122 
 
 679 
 870 
 660 
 661 
 642 
 634 
 
 10.71136 
 .70465 
 .69805 
 .69154 
 .68511 
 .67878 
 
 79 00 
 60 
 40 
 30 
 20 
 10 
 
 7§ 00 
 50 
 40 
 30 
 20 
 10 
 
 12 00 
 10 
 20 
 30 
 40 
 50 
 
 9.31788 
 .32378 
 .32960 
 .33534 
 .34100 
 .34658 
 
 599 
 590 
 582 
 574 
 566 
 558 
 
 9.99040 
 .99013 
 .98986 
 .98958 
 .98930 
 .98901 
 
 27 
 27 
 27 
 28 
 28 
 29 
 
 9.32747 
 .33365 
 .33974 
 .34576 
 .35170 
 .36757 
 
 626 
 618 
 609 
 601 
 696 
 687 
 
 10.67252 
 .66635 
 .66026 
 .65424 
 .64830 
 .64243 
 
 
 Cosine. 
 
 DifE. 
 
 Sine. 
 
 Diff. 
 
 Cotang. 
 
 Difl. 
 
 Tang. 
 
 Arc. 
 
126 TOPOGBAPHY, MAP REAI>ING, AND RECONNAISSANCE. 
 
 Table Y .—Common logariOims of circular functioTis — Continued. 
 
 Arc. 
 
 Sine. 
 
 Difl. 
 
 Cosine. 
 
 Difl. 
 
 Tang. 
 
 Difl. 
 
 Cotang. 
 
 
 13 00 
 
 9.35209 
 
 .551 
 
 9.98872 
 
 29 
 
 9.36336 
 
 679 
 
 10.63664 
 
 o / 
 
 77 00 
 
 10 
 
 .35752 
 
 543 
 
 .98843 
 
 29 
 
 .36909 
 
 573 
 
 .63091 
 
 60 
 
 20 
 
 .36289 
 
 537 
 
 .98813 
 
 30 
 
 .37476 
 
 567 
 
 .62524 
 
 40 
 
 30 
 
 .36818 
 
 528 
 
 .98783 
 
 30 
 
 .38036 
 
 559 
 
 .61965 
 
 30 
 
 40 
 
 .37341 
 
 523 
 
 . 98753 
 
 30 
 
 .38589 
 
 554 
 
 .61411 
 
 20 
 
 50 
 
 .37858 
 
 517 
 
 .98722 
 
 31 
 
 .39136 
 
 547 
 
 .60864 
 
 10 
 
 14 00 
 
 9.38367 
 
 509 
 
 9.98690 
 
 32 
 
 9.39677 
 
 541 
 
 10.60323 
 
 76 00 
 
 10 
 
 .38871 
 
 504 
 
 .98659 
 
 31 
 
 .40212 
 
 536 
 
 . 59788 
 
 60 
 
 20 
 
 .39368 
 
 497 
 
 .98627 
 
 32 
 
 .40742 
 
 530 
 
 .59258 
 
 40 
 
 30 
 
 .39860 
 
 492 
 
 .98594 
 
 33 
 
 .41266 
 
 524 
 
 .58734 
 
 30 
 
 40 
 
 . 40345 
 
 485 
 
 .98561 
 
 33 
 
 .41784 
 
 518 
 
 .58216 
 
 20 
 
 60 
 
 .40825 
 
 480 
 
 .98528 
 
 33 
 
 .42297 
 
 513 
 
 .57703 
 
 10 
 
 15 00 
 
 9.41300 
 
 475 
 
 9.98494 
 
 34 
 
 9.42805 
 
 508 
 
 10.57195 
 
 75 00 
 
 10 
 
 .41768 
 
 468 
 
 .98460 
 
 34 
 
 .43308 
 
 503 
 
 .56692 
 
 50 
 
 20 
 
 .42232 
 
 454 
 
 .98426 
 
 34 
 
 .43806 
 
 493 
 
 .56194 
 
 40 
 
 30 
 
 .42690 
 
 458 
 
 .98391 
 
 35 
 
 .44299 
 
 493 
 
 .55701 
 
 30 
 
 40 
 
 .43143 
 
 463 
 
 .98356 
 
 36 
 
 .44787 
 
 488 
 
 .55213 
 
 20 
 
 SO 
 
 .43591 
 
 448 
 
 .98320 
 
 36 
 
 .45271 
 
 484 
 
 .54729 
 
 10 
 
 16 00 
 
 9.44034 
 
 443 
 
 9.98284 
 
 36 
 
 9.45750 
 
 479 
 
 10.54260 
 
 74. 00 
 
 10 
 
 .44472 
 
 438 
 
 .98248 
 
 36 
 
 .46224 
 
 474 
 
 .53776 
 
 50 
 
 20 
 
 .44905 
 
 433 
 
 .98211 
 
 37 
 
 .46694 
 
 470 
 
 .53306 
 
 40 
 
 30 
 
 .45334 
 
 429 
 
 .98174 
 
 37 
 
 .47160 
 
 466 
 
 .52839 
 
 30 
 
 40 
 
 .45758 
 
 424 
 
 .98136 
 
 38 
 
 .47622 
 
 462 
 
 .52378 
 
 20 
 
 50 
 
 .46178 
 
 420 
 
 .98098 
 
 38 
 
 .48080 
 
 458 
 
 .51920 
 
 10 
 
 17 00 
 
 9.46593 
 
 415 
 
 9.98060 
 
 38 
 
 9.48534 
 
 454 
 
 10.51466 
 
 73 00 
 
 10 
 
 .47005 
 
 412 
 
 .98021 
 
 39 
 
 .48984 
 
 450 
 
 .51016 
 
 50 
 
 20 
 
 .47411 
 
 406 
 
 .97982 
 
 39 
 
 .49430 
 
 446 
 
 .50570 
 
 40 
 
 30 
 
 .47814 
 
 403 
 
 . 97942 
 
 40 
 
 .49872 
 
 442 
 
 .60128 
 
 30 
 
 40 
 
 .48213 
 
 399 
 
 .97902 
 
 40 
 
 .50311 
 
 439 
 
 .49689 
 
 20 
 
 50 
 
 .48607 
 
 394 
 
 .97861 
 
 41 
 
 .50746 
 
 435 
 
 .49254 
 
 10 
 
 18 00 
 
 9.48998 
 
 391 
 
 9.97821 
 
 40 
 
 9.61178 
 
 432 
 
 10. 48822 
 
 72 00 
 
 10 
 
 .49385 
 
 387 
 
 .97779 
 
 42 
 
 .51606 
 
 428 
 
 .48394 
 
 60 
 
 20 
 
 .49768 
 
 383 
 
 .97738 
 
 41 
 
 .52030 
 
 424 
 
 .47969 
 
 40 
 
 30 
 
 .50148 
 
 380 
 
 .97696 
 
 42 
 
 .62452 
 
 422 
 
 .47548 
 
 30 
 
 40 
 
 .50523 
 
 375 
 
 .97653 
 
 43 
 
 .62870 
 
 418 
 
 .47130 
 
 20 
 
 SO 
 
 .50896 
 
 373 
 
 .97610 
 
 43 
 
 .53285 
 
 415 
 
 .46715 
 
 10 
 
 19 00 
 
 9.51264 
 
 368 
 
 9.97567 
 
 43 
 
 9.53697 
 
 412 
 
 10.46303 
 
 71 00 
 
 10 
 
 .51629 
 
 365 
 
 .97523 
 
 44 
 
 .54106 
 
 409 
 
 .45894 
 
 60 
 
 20 
 
 .51991 
 
 362 
 
 .97479 
 
 44 
 
 .54512 
 
 406 
 
 .45488 
 
 40 
 
 30 
 
 .52349 
 
 358 
 
 .97435 
 
 44 
 
 .54915 
 
 403 
 
 .45085 
 
 30 
 
 40 
 
 .52705 
 
 356 
 
 .97390 
 
 45 
 
 .55315 
 
 400 
 
 .44685 
 
 20 
 
 SO 
 
 .53066 
 
 351 
 
 .97344 
 
 46 
 
 .55712 
 
 397 
 
 .44288 
 
 10 
 
 20 00 
 
 9.53405 
 
 349 
 
 9.97299 
 
 46 
 
 9.66107 
 
 39.5 
 
 10.43893 
 
 70 00 
 
 10 
 
 .63761 
 
 346 
 
 .97252 
 
 47 
 
 .56498 
 
 391 
 
 .43502 
 
 50 
 
 20 
 
 .54093 
 
 342 
 
 .97206 
 
 46 
 
 .56887 
 
 389 
 
 .43113 
 
 40' 
 
 30 
 
 .54432 
 
 339 
 
 .97159 
 
 47 
 
 .67274 
 
 387 
 
 .42726 
 
 30 
 
 40 
 
 .64769 
 
 337 
 
 .97111 
 
 48 
 
 .67668 
 
 384 
 
 .42342 
 
 20 
 
 50 
 
 .55102 
 
 333 
 
 .97063 
 
 48 
 
 .58039 
 
 381 
 
 .41961 
 
 10. 
 
 21 00 
 
 9.56433 
 
 331 
 
 9.97016 
 
 48 
 
 9.68418 
 
 379 
 
 lOi 41582 
 
 69 00 
 
 10 
 
 .55761 
 
 328 
 
 .96966 
 
 49 
 
 .58794 
 
 376 
 
 .41206 
 
 , 50 
 
 20 
 
 .56085 
 
 " 324 
 
 .96917 
 
 49 
 
 .59168 
 
 374 
 
 .40832 
 
 40 
 
 30 
 
 .56407 
 
 322 
 
 .96868 
 
 49 
 
 .59640 
 
 372 
 
 .40460 
 
 30 
 
 40 
 
 .66727 
 
 320 
 
 .96818 
 
 60 
 
 .69909 
 
 S69 
 
 .40091 
 
 20 
 
 50 
 
 .57043 
 
 316 
 
 .96767 
 
 51 
 
 .60276 
 
 367 
 
 .39724 
 
 10 
 
 
 Cksine. 
 
 toiff. 
 
 Sine. 
 
 Difl. 
 
 Cotang. 
 
 Difl. 
 
 Tang. 
 
 Are. 
 
TOPOGHAPHY, MAB EEAMNG, AND EECONNAISSANOE. 
 Table V. — Commwn loQwnihms of circular functions — Continued. 
 
 127 
 
 
 Arc. 
 
 Sine. 
 
 DifE. 
 
 Cosine. 
 
 Difl. 
 
 Tang. 
 
 Difl. 
 
 Cotang. 
 
 
 
 ' / 
 
 22 00 
 
 9.57357 
 
 314 
 
 9.95717 
 
 50 
 
 9.60641 
 
 365 
 
 10.3gf359 
 
 68 00 
 
 
 10 
 
 .57669 
 
 312 
 
 .96665 
 
 52 
 
 .61004 
 
 363 
 
 .38996 
 
 50 
 
 
 20 
 
 .67978 
 
 309 
 
 .96614 
 
 51 
 
 .61364 
 
 360 
 
 .38636 
 
 40 
 
 
 30 
 
 .58284 
 
 306 
 
 .96561 
 
 53 
 
 .61722 
 
 368 
 
 .38278 
 
 30 
 
 
 40 
 
 .58588 
 
 304 
 
 .96509 
 
 52 
 
 .62079 
 
 357 
 
 .37921 
 
 20 
 
 
 SO 
 
 .58889 
 
 301 
 
 .96456 
 
 53 
 
 .62433 
 
 354 
 
 .37567 
 
 10 
 
 
 23 00 
 
 9.59188 
 
 299 
 
 9.96403 
 
 ' 53 
 
 9.62786 
 
 332 
 
 10. 37215 
 
 67 00 
 
 
 10 
 
 .59484 
 
 296 
 
 .96349 
 
 54 
 
 .63135 
 
 360 
 
 .36864 
 
 60 
 
 
 20 
 
 .59778 
 
 294 
 
 .96294 
 
 SS 
 
 .63484 
 
 349 
 
 .36516 
 
 40 
 
 
 30 
 
 .60070 
 
 292 
 
 .96240 
 
 54 
 
 .63830 
 
 346 
 
 .36170 
 
 30 
 
 
 40 
 
 .60369 
 
 289 
 
 .96185 
 
 55 
 
 .64173 
 
 345 
 
 .35823 
 
 20 
 
 
 50 
 
 .60646» 
 
 287 
 
 .96129 
 
 56 
 
 .64517 
 
 342 
 
 .35483 
 
 10 
 
 
 24 00 
 
 9.60931 
 
 285 
 
 9.96073 
 
 56 
 
 9.64858 
 
 341 
 
 10.35142 
 
 66 00 
 
 
 10 
 
 .61214 
 
 283 
 
 .96016 
 
 57 
 
 .65197 
 
 339 
 
 .34803 
 
 50 
 
 
 20 
 
 .61494 
 
 ~ 280 
 
 .95960 
 
 56 
 
 .65535 
 
 338 
 
 .34466 
 
 40 
 
 
 30 
 
 .61773 
 
 279 
 
 .95902 
 
 58 
 
 .65870 
 
 335 
 
 ' .34130 
 
 30 
 
 
 40 
 
 .62049 
 
 276 
 
 .95844 
 
 58 
 
 .66204 
 
 334 
 
 . 33796 
 
 20 
 
 
 50 
 
 .62323 
 
 274 
 
 .95786 
 
 58 
 
 .66537 
 
 333 
 
 . 33463 
 
 10 
 
 
 25 00 
 
 9.62595 
 
 272 
 
 9.95728 
 
 58 
 
 9.66867 
 
 330 
 
 10.33133 
 
 65 00 
 60 
 
 
 10 
 
 .6286b 
 
 270 
 
 .95668 
 
 60 
 
 .67196 
 
 329 
 
 .32804 
 
 
 20 
 
 .63133 
 
 268 
 
 .95609 
 
 39 
 
 .67524 
 
 328 
 
 .32476 
 
 40 
 
 
 30 
 
 . 63398 
 
 265 
 
 .95549 
 
 60 
 
 .67850 
 
 326 
 
 . 32160 
 
 30 ^ 
 
 20 
 
 10 
 
 
 40 
 
 .63662 
 
 264 
 
 .95488 
 
 61 
 
 .68174 
 
 324 
 
 .31826 
 
 
 50 
 
 .63924 
 
 262 
 
 .95427 
 
 61 
 
 .68497 
 
 323 
 
 .31503 
 
 
 26 00 
 10 
 
 20 
 30 
 40 
 50 
 
 9.64184 
 .64442 
 .64698 
 .64953 
 .65203 
 .65456 
 
 260 
 258 
 256 
 255 
 2o2 
 251 
 
 9.95366 
 .96304 
 .95242 
 .95179 
 .95116 
 .95062 
 
 60 
 62 
 62 
 63 
 63 
 64 
 
 9.68818 
 .69138 
 .69457 
 .69774 
 .70089 
 .70404 
 
 321 
 320 
 319 
 ■317 
 315 
 316 
 
 10.31182 
 .30862 
 .30643 
 .30226 
 .29911 
 .29596 
 
 64 00 
 50 
 40 
 30 
 20 
 10 
 
 
 27 00 
 10 
 20 
 30 
 40 
 50 
 
 9.65705 
 .65952 
 .66197 
 .66441 
 .66682 
 .66922 
 
 249 
 247 
 245 
 244 
 241 
 240 
 
 9. 94988 
 .94923 
 .94858 
 . 94793 
 .94727 
 .94660 
 
 64 
 65 
 65 
 65 
 -66 
 67 
 
 9.70717 
 .71028 
 .71339 
 .71648 
 .71955 
 - .72262 
 
 313 
 311 
 311 
 309 
 307 
 307 
 
 10.29283 
 .28972 
 .28661 
 .28332 
 .28044 
 
 . .27738 
 
 63 00 
 60 
 40 
 30 
 20 
 10 
 
 
 28 00 
 10 
 20 
 30 
 40 
 50 
 
 9.67151 
 .67398 
 .67633 
 .67866 
 ' .68098 
 .68328 
 
 239 
 237 
 235 
 233 
 232 
 230 
 
 9.94593 
 .94526 
 .94458 
 .94390 
 .94321 
 .94252 
 
 67 
 67 
 68 
 68 
 69 
 69 
 
 9.72667 
 .72872 
 .73175 
 .73476 
 .73777 
 .74077 
 
 305 
 305 
 303 
 301 
 301 
 300 
 
 10.27433 
 -.27128 
 .26825 
 .26524 
 .26223 
 .25923 
 
 62 00 
 60 
 40 
 30 
 20 
 10 
 
 
 29 00 
 10 
 20 
 30 
 40 
 50 
 
 9.68557 
 .68784 
 .69010 
 .69234 
 .69456 
 .69677 
 
 229 
 227 
 226 
 224 
 222 
 ,221 
 
 9. 94182 
 .94112 
 .94041 
 . 93970 
 .93898 
 .93826 
 
 70 
 70 
 71 
 71 
 72 
 72 
 
 9.74375 
 .74673 
 .74969 
 .76264 
 ,75568 
 .75832 
 
 298 
 298 
 296 
 295 
 294 
 294 
 
 10.25625 
 .25327 
 .25031 
 .24736 
 .24441 
 .24148 
 
 61 00 
 50 
 40 
 30 
 20 
 10 
 
 
 30 00 
 H 
 2C 
 3( 
 4 
 
 9.69897 
 
 .76115 
 
 .70335 
 
 ) .7054' 
 
 ) - .70761 
 
 220 
 218 
 217 
 21J 
 21' 
 
 9.93753 
 .93680 
 .93606 
 .93535 
 .9345' 
 
 73 
 
 73 
 74 
 74 
 
 7£ 
 
 9.76144 
 .76435 
 .76725 
 .77015 
 .77303 
 
 292 
 291 
 290 
 29C 
 28S 
 
 10.23836 
 .23565 
 .23274 
 .2298S 
 .2269' 
 
 60 00 
 50 
 40 
 30 
 20 
 10 
 
 
 6 
 
 5 .709'/. 
 
 i 21' 
 
 ! .9338 
 
 ! 7E 
 
 ,77591 
 
 28? 
 
 .2240E 
 
 
 
 Cosine 
 
 . Difl 
 
 Sine. 
 
 Difl. 
 
 Cotang 
 
 Difl. 
 
 Tang. 
 
 Arc. 
 
128 TOPOGBAPHY, MAP EEAMNG, AND RECONNAISSANCE. 
 
 Table V. — Common logarithms of dreular functions — Continued. 
 
 Arc. 
 
 Sine. 
 
 Difl. 
 
 Cosine. 
 
 Difl. 
 
 Tang. 
 
 Difl. 
 
 Cotang. 
 
 
 31 
 
 00 
 
 9.71184 
 
 211 
 
 9.93307 
 
 75 
 
 9.77877 
 
 286 
 
 10.22123 
 
 59 00 
 
 
 10 
 
 .71393 
 
 209 
 
 .93230 
 
 77 
 
 .78163 
 
 286 
 
 .21837 
 
 50 
 
 
 20 
 
 .71602 
 
 209 
 
 .93154 
 
 76 
 
 .78448 
 
 - 285 
 
 .21552 
 
 40 
 
 
 30 
 
 .71808 
 
 206 
 
 .93077 
 
 77 
 
 .78732 
 
 284 
 
 .21268 
 
 30 
 
 
 40 
 
 .72014 
 
 206 
 
 .92999 
 
 78 
 
 ■ .79015 
 
 283 
 
 .20985 
 
 20 
 
 
 60 
 
 .72218 
 
 204 
 
 .92921 
 
 ■ 78 
 
 .79297 
 
 282 
 
 .20703 
 
 10 
 
 32 
 
 00 
 
 9.72421 
 
 203 
 
 9.92842 
 
 79 
 
 9.79579 
 
 282 
 
 10.20421 
 
 68 00 
 
 
 10 
 
 .72622 
 
 201 
 
 .92763 
 
 79 
 
 .79860 
 
 281 
 
 .20140 
 
 SO 
 
 
 20 
 
 .72823 
 
 201 
 
 .92683 
 
 80 
 
 ,80140 
 
 280 
 
 .19860 
 
 40 
 
 
 30 
 
 .V3022 
 
 199 
 
 .92603 
 
 80 
 
 .80419 
 
 279 
 
 .19581 
 
 30 
 
 
 40 
 
 .73219 
 
 197 
 
 .92522 
 
 81 
 
 ,80697 
 
 278 
 
 , . 19303 
 
 20 
 
 
 50 
 
 .73416 
 
 197 
 
 .92441 
 
 81 
 
 .80975 
 
 278 
 
 .19025 
 
 10 
 
 33 
 
 00 
 
 9.73611 
 
 195 
 
 9.92369 
 
 82 
 
 9.81252 
 
 277 
 
 10. 18748 
 
 57 00 
 
 
 10 
 
 .73805 
 
 194 
 
 .92277 
 
 82 
 
 . 81628 
 
 276 
 
 .18472 
 
 60 
 
 
 20 
 
 .73997 
 
 192 
 
 .92194 
 
 83 
 
 .81803 
 
 275 
 
 ,. 18196 
 
 40 
 
 
 30 
 
 .74189 
 
 192 
 
 .92111 
 
 83 
 
 .82078 
 
 276 
 
 . 17922 
 
 30 
 
 
 40 
 
 .74379 
 
 190 
 
 .92027 
 
 84 
 
 .82352 
 
 274 
 
 .17648 
 
 20 
 
 
 30 
 
 .74588 
 
 189 
 
 .91942 
 
 85 
 
 .82626 
 
 274 
 
 .17374 
 
 10 
 
 34 
 
 00 
 
 9.74756 
 
 188 
 
 9.91867 
 
 85 
 
 9.82899 
 
 273 
 
 10. 17101 
 
 56 00 
 
 
 10 
 
 .74943 
 
 187 
 
 .91772 
 
 85 
 
 .83171 
 
 272 
 
 . 16829 
 
 50 
 
 
 20 
 
 .75128 
 
 185 
 
 .91686 
 
 86 
 
 .83442 
 
 271 
 
 .16557 
 
 40 
 
 
 30 
 
 .75313 
 
 185 
 
 .91599 
 
 87 
 
 .83713 
 
 271 
 
 .16287 
 
 30 
 
 
 40 
 
 .75496 
 
 183 
 
 .91512 
 
 87 
 
 .83984 
 
 271 
 
 .16016 
 
 20 
 
 
 60 
 
 .75678 
 
 182 
 
 .91425 
 
 87 
 
 .84253 
 
 269 
 
 .15746 
 
 10 
 
 35 
 
 00 
 
 9.75859 
 
 181 
 
 9.91336 
 
 S9 
 
 9.84523 
 
 270 
 
 10. 15477 
 
 55 Ob 
 
 
 10 
 
 .76039 
 
 . 180 
 
 .91248 
 
 88 
 
 .84791 
 
 268 
 
 .16209 
 
 50 
 
 
 20 
 
 .76218 
 
 179 
 
 .91158 
 
 90 
 
 .85059 
 
 268 
 
 .14941 
 
 40 
 
 
 30 
 
 .76395 
 
 177 
 
 .91069 
 
 39 
 
 .85327 
 
 268 
 
 .14673 
 
 30 
 
 
 40 
 
 .76572 
 
 177 
 
 .90978 
 
 91 
 
 .86594 
 
 '267 
 
 .14406 
 
 20 
 
 
 SO 
 
 .76747 
 
 175 
 
 .90887 
 
 91 
 
 .85860 
 
 266 
 
 .14140 
 
 10 
 
 36 
 
 00 
 
 9.76922 
 
 175 
 
 9.90796 
 
 91 
 
 9.86126 
 
 266 
 
 10,13874 
 
 54 00 
 
 
 10 
 
 .77095 
 
 173 
 
 .90704 
 
 92 
 
 .86391 
 
 265 
 
 .13608 
 
 50 
 
 
 20 
 
 .77267 
 
 172 
 
 .90511 
 
 93 
 
 .86656 
 
 206 
 
 .13344 
 
 40 
 
 
 30 
 
 .77439 
 
 172 
 
 .90518 
 
 93 
 
 .86921 
 
 265 
 
 . 13079 
 
 30 
 
 
 40 
 
 .77609 
 
 170 
 
 .90424 
 
 94 
 
 .87186 
 
 264 
 
 .12815 
 
 20 
 
 
 60 
 
 .77778 
 
 169 
 
 .90330 
 
 94 
 
 .87448 
 
 263 
 
 .12552 
 
 10 
 
 37 
 
 00 
 
 9.77946 
 
 168 
 
 9.90235 
 
 95 
 
 9.87711 
 
 263 
 
 10.12289 
 
 53 00 
 
 
 10 
 
 .78113 
 
 167 
 
 ,90139 
 
 96 
 
 .87974 
 
 263 
 
 .12026 
 
 50 
 
 
 20 
 
 .78280 
 
 167 
 
 .90043 
 
 96 
 
 .88236 
 
 262 
 
 .11764 
 
 40 
 
 
 30 
 
 .78445 
 
 165 
 
 .89947 
 
 96 
 
 .88498 
 
 262 
 
 ."11502 
 
 30 
 
 
 40 
 
 .78609 
 
 164 
 
 .89849 
 
 98 
 
 .88759 
 
 281 
 
 .11241 
 
 20 
 
 
 60 
 
 .78772 
 
 163 
 
 .89752 
 
 97 
 
 .89020 
 
 261 
 
 .10980 
 
 10 
 
 38 
 
 00 
 
 9.78934 
 
 162 
 
 9.89663 
 
 99 
 
 9.89281 
 
 261 
 
 10. 10719 
 
 52 00 
 
 
 10 
 
 .79095 
 
 161 
 
 .89654 
 
 99 
 
 .89641 
 
 260 
 
 .10469 
 
 50 
 
 
 20 
 
 .79256 
 
 161 
 
 .89455 
 
 99- 
 
 .89801 
 
 - 260 
 
 . 10199 
 
 40 
 
 
 30 
 
 .79415 
 
 159 
 
 .89354 
 
 101 
 
 .90060 
 
 259 
 
 .09939 
 
 30 
 
 
 40 
 
 .79573 
 
 168 
 
 .89254 
 
 100 
 
 ,90320 
 
 260 
 
 .09680 
 
 20 
 
 
 50 
 
 .79731 
 
 158 
 
 .89162 
 
 102 
 
 .90578 
 
 258 
 
 .09421 
 
 10 
 
 39 
 
 00 
 
 9.79887 
 
 156 
 
 9.89050 
 
 102 
 
 9.90837 
 
 259 
 
 10.09163 
 
 61 00 
 
 
 10 
 
 .80043 
 
 156 
 
 .88948 
 
 102 
 
 .91095 
 
 268 
 
 .08905 
 
 50 
 
 
 20 
 
 .80197 
 
 154 
 
 .88844 
 
 104 
 
 .91363 
 
 258 
 
 .08647 
 
 40 
 
 
 30 
 
 .80361 
 
 164 
 
 .88741 
 
 103 
 
 .91610 
 
 267 
 
 .08390 
 
 30 
 
 
 40 
 
 .80604 
 
 153 
 
 .88636 
 
 105 
 
 .91868 
 
 268 
 
 .08132 
 
 20 
 
 
 50 
 
 .80656 
 
 162 
 
 .88531 
 
 105 
 
 .92125 
 
 257 
 
 .07875 
 
 10 
 
 
 Cosine, 
 
 Difl. 
 
 Sine. 
 
 Difl. 
 
 Cotang. 
 
 Difl. 
 
 Tcng. 
 
 Arc. 
 
TOPOGBAPHY, MAP READING, AND RECONNAISSANCE. 129 
 
 Table 'Y. —Common logarithms of circular functions— Goniinued. 
 
 
 
 
 
 
 
 
 
 
 
 Arc. 
 
 . Sine. 
 
 Difl. 
 
 Cosine. 
 
 Difl. 
 
 Tang. 
 
 Difl. 
 
 Cotang. 
 
 
 40 
 
 r 
 
 00 
 10 
 20 
 30 
 40 
 50 
 
 9.80807 
 .80957 
 .81106 
 .81254 
 .81402 
 .81548 
 
 151 
 150 
 149 
 148 
 
 148 
 146 
 
 9.88425 
 .88319 
 .88212 
 .88105 
 .87996 
 .87887 
 
 106 
 106 
 107 
 107 
 109 
 109 
 
 9.92381 
 .92638 
 .92894 
 .93150 
 .93406 
 -.93661 
 
 256 
 257 
 266 
 256 
 266 
 - 255 
 
 10.07618 
 .07362 
 .07106 
 
 ; 06339 
 
 o / 
 
 50 00 
 50 
 40 
 30 
 20 
 10 
 
 41 
 
 00 
 10 
 
 9.81694 
 .81839 
 
 146 
 
 145 
 
 9.87778 
 .87668 
 
 109 
 HO 
 
 9.93916 
 .94171 
 
 -255 
 265 
 
 10.06084 
 .05829 
 
 49 00 
 
 50 
 
 
 20 
 
 .81983 
 
 144 
 
 .87557 
 
 111 
 
 .94426 
 
 265 
 
 .05574 
 
 40 
 
 
 30 
 
 .82126 
 
 143 
 
 .87446 
 
 111 
 
 .94681 
 
 255 
 
 . 05319 
 
 30 
 
 
 40 
 
 .82269 
 
 143 
 
 .87333 
 
 113 
 
 .94935 
 
 254 
 
 .05065 
 
 20 
 
 
 50 
 
 .82410 
 
 141 
 
 .87221 
 
 112 
 
 .95190 
 
 255 
 
 .04811 
 
 10 
 
 ^2 
 
 00 
 
 9.82551 
 
 141 
 
 9.87107 
 
 114 
 
 9.95444 
 
 254 
 
 10.04556 
 
 48 00 
 
 
 10 
 
 .82691 
 
 140 
 
 .86993 
 
 114 
 
 .95698 
 
 254 
 
 .04302 
 
 50 
 
 
 20 
 
 .82830 
 
 139 
 
 .86878 
 
 115 
 
 .96952 
 
 254 
 
 .04048 
 
 40 
 
 
 30 
 
 .82968 
 
 138 
 
 .86763 
 
 115 
 
 .96205 
 
 263 
 
 .03795 
 
 30 
 
 
 40 
 
 .83106 
 
 138 
 
 .86647 
 
 116 
 
 .96459 
 
 254 
 
 .03541 
 
 20 
 
 
 50 
 
 .83242 
 
 136 
 
 .86530 
 
 117 
 
 .96712 
 
 253 
 
 .03288 
 
 10 
 
 43 
 
 00 
 
 9.83378 
 
 136 
 
 9.86413 
 
 117 
 
 9.96966 
 
 254 
 
 10.03034 
 
 47 00 
 
 
 10 
 
 .83513 
 
 135 
 
 .86295 
 
 118 
 
 .97219 
 
 253 
 
 .02781 
 
 50 
 
 
 20 
 
 .83648 
 
 135 
 
 .86176 
 
 119 
 
 .97472 
 
 253 
 
 .02528 
 
 40 
 
 
 30 
 
 .83781 
 
 133 
 
 .86056 
 
 120 
 
 .97725 
 
 253 
 
 .02275 
 
 30 
 
 
 40 
 
 .83914 
 
 133 
 
 .85936 
 
 120 
 
 .97978 
 
 263 
 
 .02022 
 
 20 
 
 
 50 
 
 .84046 
 
 132 
 
 .86815 
 
 121 
 
 .98231 
 
 253 
 
 .01769 
 
 10 
 
 44 
 
 00 
 
 9.84177 
 
 131 
 
 9.85693 
 
 122 
 
 9.98484 
 
 253 
 
 10.01516 
 
 46 00 
 
 
 10 
 
 .84308 
 
 131 
 
 .85671 
 
 122 
 
 .98736 
 
 252 
 
 .01263 
 
 50 
 
 
 20 
 
 .84437 
 
 129 
 
 .85448 
 
 123 
 
 .98989 
 
 253 
 
 .01011 
 
 40 
 
 
 30 
 
 .84566 
 
 129 
 
 .85324 
 
 124 
 
 .99242 
 
 253 
 
 .00758 
 
 30 
 
 
 40 
 
 .84694 
 
 128 
 
 .85200 
 
 124 
 
 .99495 
 
 253 
 
 .00505 
 
 20 
 
 
 50 
 
 .84822 
 
 128 
 
 .85074 
 
 126 
 
 .99747 
 
 252 
 
 .00253 
 
 10 
 
 4S 
 
 -00 
 
 9.84948 
 
 126 
 
 9.84948 
 
 126 
 
 10.00000 
 
 253 
 
 10.00000 
 
 45 00 
 
 
 Cosine. 
 
 Difl. 
 
 Sine. 
 
 Difl. 
 
 Cotang. 
 
 Difl. 
 
 Tang. - 
 
 Arc. 
 
 212. The slide rule is a contrivance for using logs, mechanically. 
 It consists, figure 47, of a rule, in the middle of which is a slide. 
 The edges of the groove and the edges of the slide are graduated, 
 forming four scales called A, B, C, and D. An indicator, which can 
 be set at any point, guides the eye in selecting opposite numbers. 
 The sHde rule deals with mantissas only. Characteristics must be 
 obtained by inspection. 
 
 To multiply. — ^Moves, the slide to the rigJit until 1 on scale B ia 
 opposite the smaller of the two numbers on A; the number on A 
 opposite the latger of the two numbers on B is the product. 
 
 To divide. — ^Move the slide to the left until the _ divisor on B ig 
 under 1 on A. The" number on A opposite the dividend on B ia 
 the quotient desired. To multiply and divide simultaneously, or to 
 solve a proportion, set the divisor on B opposite one of the other 
 numbers on A. The number on A opposite the third number on B 
 is the result desired. 
 
 To find the square of a number. — ^Take the number on A opposite 
 the given number on D. . 
 
 To find the square root. — Take the number on D opposite the 
 given number on A. In taking square roots use only the left half 
 of A, for an odd number of figures in front of the decimal point, 
 and the right halfoulj for even number. 
 
 58740°— 18 9 
 
130 
 
 TOPOGRAPHY, MAP BEADUSTG, AND KECONNAISSANCE. 
 
 To find a cube. — Set 1 on B opposite the gJTen number on D. 
 The number on A opposite the given number onB is the cube desired. 
 
 To find a cube root. — Take the root approximately by inspection. 
 Set this number on B opposite the given number on A. Note whether 
 1 on C is opposite the approximate root on D. If so, the approxi- 
 mate root is the correct one; if not, move the shde slightly one way 
 or the other until the number on B opposite the given number, and 
 the number on D opposite the one on C are the same. Ttds number 
 is the desired cube root. 
 
 Occasional users of the slide rale will do well to adhCTe to the 
 simple operations above described. Kegular users will study the 
 theory and scope of the rule from one of the several treatises on the 
 subject. 
 
 Table VI. — Table of squares, cubes, squcere roots, and cube roots of numbers from 1 
 
 to 1,000. 
 
 No. 
 
 Square. 
 
 Cuba. 
 
 Square 
 root. 
 
 Cuba 
 root. 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Cube 
 root. 
 
 1. 
 2. 
 
 3. 
 i. 
 
 6. 
 6. 
 7. 
 8. 
 9. 
 10.. 
 
 n. 
 
 12.. 
 13.. 
 14.. 
 15.. 
 16.. 
 17.. 
 18.. 
 19-. 
 20.. 
 21.. 
 22.. 
 2S.. 
 
 25.. 
 28.. 
 2J.. 
 
 30.. 
 31.. 
 32.. 
 33.. 
 34.. 
 
 3a.. 
 
 37.. 
 38.. 
 
 41.. 
 
 4a.. 
 
 4S.. 
 44.. 
 45.. 
 46.. 
 47.. 
 48.. 
 49.. 
 SO.. 
 61.. 
 62.. 
 6S.. 
 64.. 
 65.. 
 66.. 
 
 1 
 
 4 
 
 9 
 
 18 
 
 25 
 
 as 
 
 49 
 64 
 81 
 100 
 121 
 144 
 
 27 
 
 64! 
 
 I2S 
 
 216 
 
 343 
 
 512 
 
 729 
 
 lOOO 
 
 1331 
 
 1728 
 
 169 
 
 2197 
 
 19ft 
 
 2744 
 
 235 
 
 3375 
 
 .25S 
 
 4096 
 
 •m 
 
 4913 
 
 324 
 
 6832 
 
 .1«1 
 
 6859 
 
 400 
 
 8G00 
 
 441 
 
 92S1 
 
 484 
 
 t«S4S 
 
 .539 
 
 1ZI67 
 
 576 
 
 I3SH 
 
 625 
 
 15625 
 
 67B 
 
 i 17576 
 
 72» 
 
 19683 
 
 7K4 
 
 21952 
 
 841 
 
 24389 
 
 900 
 
 27SQ0 
 
 901 
 
 29791 
 
 vm 
 
 32768 
 
 1089 
 
 3,W37 
 
 nrifi 
 
 39304 
 
 1225 
 
 42875 
 
 I29S 
 
 46656 
 
 13fi& 
 
 50653 
 
 1444 
 
 S4872 
 
 isai 
 
 58319 
 
 160(1 
 
 64000 
 
 16H1 
 
 68921 
 
 1764 
 
 74088 
 
 1840 
 
 79507 
 
 1936 
 
 85184 
 
 2025 
 
 91125 
 
 2in 
 
 97336. 
 
 2209 
 
 103823 
 
 2304 
 
 110592 
 
 2*)1 
 
 117649 
 
 2SfHI 
 
 125000 
 
 2S01 
 
 132651 
 
 3704 
 
 14aH)8 
 
 2809 
 
 148877 
 
 2916 
 
 157464 
 
 3025 
 
 166375 
 
 3136 
 
 176616 
 
 1. 
 
 1. 4142 
 
 1.7321 
 
 2.0000 
 
 2.2381 
 
 2.4495 
 
 2.6458 
 
 2.8284 
 
 3.000O 
 
 3.1623 
 
 3.3166 
 
 3.4641 
 
 3.S0S& 
 
 3.7417 
 
 3.8730 
 
 4. 
 
 4.1231 
 
 4.2426 
 
 4.4721 
 
 4.582^ 
 
 4.6904 
 
 4.7958 
 
 4.8990 
 
 5, 
 
 S.099O 
 
 5.1962 
 
 S.2915 
 
 5.3852 
 
 5.4772 
 
 5.6678 
 
 5.6569 
 
 5.7446 
 
 5.8310 
 
 6.9161 
 
 6. 
 
 6.0828 
 
 6.1644 
 
 6.2450 
 
 6.3246 
 
 6.40S1 
 
 6.4807 
 
 6.5574 
 
 6. 6332 
 
 6. 7082 
 
 6.7833 
 
 6. 8557 
 
 6.9282 
 
 J. 
 
 7.Q7U 
 
 7.1414 
 
 7.2111 
 
 7.2801 
 
 7.3485 
 
 7.4162 
 
 7,4833 ' 
 
 1. 
 
 1.2599 
 1.4422 
 1.5874 
 1.7100 
 1.8171 
 1.9129 
 2; 0000 
 2.0801 
 2.1544 
 2.2240 
 2.2894 
 2.3513 
 2. 4101 
 2. 4062 
 2.5198 
 2.5713 
 2.6207 
 2.6684 
 2.7144 
 2.7589 
 2.8020 
 2.8439 
 2.8845 
 2.%40 
 2-9625 
 3.000O 
 3.0366 
 3.0723 
 3.1072 
 3.1414 
 3.1748 
 S.2075 
 3.2396 
 3.2711 
 3.3019 
 a. 3322 
 3.3620 
 3.3912 
 3.4200 
 3.4482 
 3.4760 
 3.5034 
 3.6303 
 3.5569 
 3.5830 
 3.6088 
 3.6342 
 3.6593 
 3.6840 
 3.70S4 
 3.7325 
 3.7563 
 3.7798 
 3.8030 
 3,8259 
 
 90.. 
 91.. 
 92.. 
 
 9a.. 
 
 94.. 
 95.. 
 
 97.. 
 
 100.. 
 101. 
 102- 
 103. 
 104. 
 105. 
 lOfl., 
 107.. 
 108.. 
 109.. 
 110.. 
 111.. 
 112., 
 
 3249 
 
 3364 
 3481 
 3600 
 3721 
 S44 
 
 43.56 
 4489 
 4624 
 4761 
 4900 
 5Mt 
 6184 
 5329 
 5476 
 5625 
 5776 
 5929 
 60S4 
 6241 
 
 emi 
 
 6561 
 6724 
 6889 
 7056 
 722S 
 7396 
 7569 
 7744 
 7921 
 
 185193 
 195112 
 205379 
 216000 
 
 K8328 
 238047 
 262144 
 274625 
 287496 
 300763 
 3U^2 
 328509 
 343000 
 
 ssraai 
 
 373248 
 389017' 
 405224 
 421875 
 
 436533 
 474552 
 
 .6il;»)0O 
 531441 
 
 551368 
 571787 
 592704 
 614125 
 636056 
 658503 
 681472 
 
 810O 
 
 729000 
 
 8281 
 
 753571 
 
 8464 
 
 778688 
 
 8649 
 
 804357 
 
 8836 
 
 830584 
 
 0025 
 
 8S737& 
 
 9216 
 
 884736 
 
 9409 
 
 912673 
 
 9604 
 
 941192 
 
 9801 
 
 970299 
 
 10000 
 
 loeoooo 
 
 10201 
 
 1030301 
 
 10404 
 
 1061268 
 
 10809 
 
 1092727 
 
 10816 
 
 11^864 
 
 U025 
 
 1157625 
 
 11236 
 
 1191016 
 
 11449 
 
 1225(M3 
 
 11664 
 
 1259712 
 
 11881 
 
 1295029 
 
 12100 
 
 1331000 
 
 12321 
 
 1367631 
 
 lii644 
 
 1401928 
 
 7.5498 
 7.6158 
 7.6811 
 7.7460 
 7. 8102 
 7.8740 
 ?.9373 
 S. 
 
 8.0623 
 8. 1240 
 8.1854 
 8.2462 
 8.3066 
 8.3666 
 8.4261 
 8.4853 
 8.5440 
 8.6023 
 8.66IB 
 8.7178 
 a 7750 
 a. 8318 
 8.8S82 
 S.94^ 
 9. 
 
 9.05S4 
 9.1104 
 9.1SS2 
 ».2I95 
 9.2736 
 9L3a74 
 9>.3808 
 9.4340 
 9.4868 
 9.6394 
 91.5917 
 9.6437 
 ft 6954 
 9^7468 
 9.7980 
 9.8489 
 9.8996 
 9.9499 
 la 
 
 10.0499 
 10.0985 
 10.1489 
 10.1980 
 10.2470 
 10.2956 
 10.3441 
 10.3923 
 ia440S 
 10.4881 
 I0'.5357 
 10.5830 
 
 3.8485 
 
 3.8709 
 
 3.8930 
 
 3.9149 
 
 3.9365 
 
 3.9579 
 
 3.9791 
 
 4. 
 
 4.0207 
 
 4. 0412 
 
 4.0615 
 
 4.0817 
 
 4. 1016 
 
 4. 1213 
 
 4. 1408 
 
 4. 1602 
 
 4.1793 
 
 4.1983 
 
 4.2172 
 
 4.2358 
 
 4-25e 
 
 4.2727 
 
 4.2908 
 
 4^3089 
 
 4.3267 
 
 4.3445 
 
 4.3795 
 4.3968 
 4,4140 
 4.4310 
 4.4480 
 4.4647 
 4.4814 
 4.4979 
 4.5144 
 4.5302 
 4.5468 
 4.56391 
 4,5789 
 4.59«n 
 4.6104 
 4.6261 
 4.64le 
 4.6570 
 4.6725 
 4.687S 
 4.7027 
 4.7177 
 4.7326 
 4.7475 
 4.7622 
 4.7769 
 4.7314 
 4. 8059 
 4.8203' 
 
TOPOGRAPHY, MAP BEADING, AND EECONNAISSANCE. 131 
 
 Tab^e YI.-Table of squares, cubes square rooU, and cube roots of numbers from 1 to 
 
 1,000 — Contmued. 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 113 
 
 12769 
 
 12996 
 
 13225 
 
 13456 
 
 13689 
 
 13924 
 
 14161 
 
 14400 
 
 14641 
 
 14884 
 
 15129 
 
 15376 
 
 15625 
 
 15876 
 
 16129 
 
 16384 
 
 16641 
 
 16900 
 
 17161 
 
 17424 
 
 17689 
 
 17956 
 
 18225 
 
 18496 
 
 18769 
 
 19044 
 
 19321 
 
 19600 
 
 19881 
 
 20164 
 
 20449 
 
 20736 
 
 21025 
 
 21316 
 
 21609 
 
 21904 
 
 22201 
 
 22500 
 
 22801 
 
 23104 
 
 23409 
 
 23716 
 
 24025 
 
 24336 
 
 24649 
 
 24964 
 
 25281 
 
 2560O- 
 
 25921 
 
 26244 
 
 26569 
 
 26896 
 
 27225 
 
 27556 
 
 27889 
 
 28224 
 
 28561 
 
 28900 
 
 29241 
 
 29584 
 
 29929 
 
 30276 
 
 3062S 
 
 30976 
 
 31329 
 
 31684 
 
 32041 
 
 32400 
 
 32761 
 
 33124 
 
 33489 
 
 33856 
 
 34225 
 
 34596 
 
 34969 
 
 3S344 
 
 35721 
 
 36100 
 
 36481 
 
 1442897 
 1481544 
 1520875 
 1560896 
 1601613 
 1643032 
 1685159 
 1728000 
 1771561 
 1815848 
 1860867 
 1906624 
 1953125 
 2000376 
 2048383 
 2097152 
 2146689 
 2197000 
 2248091 
 2299968 
 2352637 
 2406104 
 2460375 
 2515456 
 2571353 
 2628072 
 2685619 
 2744000 
 2803221 
 2863288 
 2924207 
 2985984 
 3048625 
 3112136 
 3176523 
 3241792 
 3307949 
 3375000 
 3442951 
 3511808 
 3581577 
 3652264 
 3723875 
 3796416 
 3869893 
 3944312 
 4019679 
 4096000 
 4173281 
 4251528 
 4330747 
 4410944 
 4492125 
 4574296 
 4657463 
 4741632 
 4826809 
 4913000 
 5000211 
 5088448 
 5177717 
 S268024 
 5359375 
 5ai776 
 5546233 
 5639752 
 5735339 
 5S32000 
 5929741 
 60285C8 
 6128487 
 6229504 
 6331625 
 64348S6 
 6539203 
 6644672 
 6761269 
 6859000 
 . 6967871 
 
 10.6301 
 
 10.6771 
 
 10.7238 
 
 10.7703 
 
 10.8167 
 
 10.8628 
 
 10.9087 
 
 10.9545 
 
 11. 
 
 11.0454 
 
 11.0905 
 
 11.1355 
 
 11. 1803 
 
 11.2250 
 
 11.2694 
 
 11.3137 
 
 11.3578 
 
 11.4018 
 
 11.4455 
 
 11.4891 
 
 11.5326 
 
 11.5758 
 
 11.6190 
 
 11.6619 
 
 11. 7047 
 
 11. 7473 
 
 11. 7898 
 11.8322 
 11.8743 
 11.9164 
 11.9583 
 12. 
 
 12. 0416 
 12.0830 
 12.1244 
 12.1665 
 12.2066 
 12.2474 
 12.2882 
 12.3288 
 12.3693 
 12.4097 
 12.4499 
 12.4900 
 12.5300 
 12. 5698 
 12.6095 
 12. 6491 
 12.6886 
 12.V2'/9 
 12.7671 
 12.8062 
 12.8452 
 12.8841 
 12.9228 
 12.9615 
 13. 
 
 13.0384 
 13.0767 
 13. 1149 
 13. 1629 
 13.1909 
 13.2288 
 13.2665 
 13.3041 
 13.3417 
 13.3791 
 13.4164 
 13.4536 
 13.4907 
 13.5277 
 13. 5647 
 13.6015 
 13.6382 
 13.6748 
 13.7113 
 13.7477 
 13.7840 
 13.8203 
 
 4.8346 
 4.8488 
 4.8629 
 4.8770 
 4.8910 
 4.9049 
 4.9187 
 4.9324 
 4.9461 
 4.9597 
 4.9732 
 4.9866 
 5. 
 
 5.0133 
 5.0265 
 5.0397 
 5.0528 
 5.0658 
 5.0788 
 ,5.0916 
 5.1045 
 5.1172 
 5.1299 
 5.1426 
 5. 1551 
 5. 1676 
 5.1801 
 6.1925 
 5.2048 
 5.2171 
 5.2293 
 5.2415 
 5.2536 
 5.26.56 
 5.2776 
 5.2896 
 5.3015 
 5.3133 
 5.3251 
 5.3368 
 5.3485 
 5.3601 
 5.3717 
 5.3832 
 5.3947 
 5.4081 
 5.4175 
 6.4288 
 5.4401 
 S.4S14 
 5.4626 
 5.4737 
 5.4848 
 5.4959 
 5.5069 
 5.5178 
 5.5288 
 5.5397 
 5.5505 
 5.5613 
 5.5721 
 5.5828 
 5.6934 
 5.6041 
 5.6147 
 5.6252 
 5.6357 
 S.«462 
 5.6567 
 5.6671 
 5.6774 
 6.6877 
 5.6980 
 5.7083 
 5.7186 
 5.7287 
 5.7388 
 5.7489 
 5.7580 
 
 192 
 
 36864 
 37249 
 37636 
 38025 
 38416 
 38809 
 39204 
 39601 
 40000 
 40401 
 40804 
 41209 
 41616 
 42026 
 42436 
 42849 
 43264 
 43681 
 44100 
 44521 
 44944 
 45369 
 45796 
 4622S 
 46656 
 47089 
 47524 
 47961 
 48400 
 48841 
 49284 
 49729 
 50176 
 S0625 
 51076' 
 51529 
 61984 
 52441 
 52900 
 53361 
 53824 
 54289 
 54756 
 56225 
 55696 
 56169 
 
 67121 
 57600 
 58081 
 58564 
 59049 
 59536 
 60025 
 60516 
 61009 
 61504 
 62001 
 62500 
 63091 
 63504 
 64009 
 64516 
 65025 
 65636 
 66049 
 66564 
 67081 
 67600 
 68121 
 68644 
 69169 
 69696 
 70225 
 70756 
 71289 
 71824 
 72361 
 72900 
 
 7077888 
 7189057 
 7301384 
 7414875 
 7529636 
 7645373 
 7762392 
 7880599 
 8000000 
 8130601 
 8242408 
 8365427 
 8489664 
 8615125 
 8741816 
 8869743 
 8998912 
 9129329 
 9261000 
 9393931 
 9528128 
 9883597 
 9800344 
 9938375 
 10077696 
 10218313 
 10360232 
 10503459 
 10648000 
 10793861 
 10941048 
 110S9S67 
 11239424 
 11390625 
 11543176 
 11697083 
 11862352 
 12008989 
 12167000 
 12326391 
 12487168 
 12649337 
 12812904 
 12977875 
 13144256 
 13312053 
 13481272 
 13651919 
 13824000 
 13997521 
 14172488 
 14348907 
 14526784 
 14706125 
 14886936 
 15069223 
 16262992 
 15438249 
 16625000 
 15813251 
 16003008 
 16194277 
 16387064 
 16581375 
 16777216 
 16974593 
 17173512 
 17373979 
 17576000 
 17779581 
 17984728 
 18191447 
 18399744 
 18609625 
 18821096 
 19034163 
 19248832 
 19465109 
 19683000 
 
 13.8564 
 13.8924 
 13.9284 
 13.9642 
 14. , 
 ' 14.0357 
 -14.0712 
 14.1067 
 14.1421 
 14. 1774 
 14.2127 
 14.2478 
 14.2829 
 14.3178 
 14.3527 
 14.3875 
 14.4222 
 14.4568 
 14.4914 
 14. 5268 
 14.5602 
 14. 5945 
 14. 6287 
 14. 6629 
 14.6969 
 14. 7309 
 14.7648 
 14. 7986 
 14.8324 
 14. 8661 
 14. 8997 
 
 14. 9332 
 14.9666 
 IS. 
 
 15.0333 
 16.0665 
 15.0997 
 15. 1327 
 15. 1658 
 15. 1987 
 
 ■ 15.2315 
 15.2643 
 15.2971 
 15.3297 
 15. 3623 
 15.3948 
 15.4272 
 
 15. 4596 
 IS. 4919 
 15.5242 
 15.5563 
 15.5885 
 15. 6205 
 15. 6525 
 15.6844 
 15.7162 
 15. 7480 
 15. 7797 
 15. 8114 
 15.8430 
 15.8745 
 15.9060 
 15.9374 
 15.9687 
 16 
 
 16.0312 
 16.0624 
 16.0935 
 16.1246 
 16. 1555 
 16. 1864 
 16.2173 
 16.2481 
 16.2788 
 16.3095 
 16.3401 
 16.3707 
 16.4012 
 IS. 4317 
 
 
 114 
 
 5 7690 
 
 
 193 
 
 
 11R 
 
 194 
 
 5 7890 
 
 
 195 
 
 
 
 1S6 
 
 5. 8088 
 
 110 
 
 197 
 
 5.8186 
 
 
 198 
 
 5.8285 
 
 121 
 
 199 
 
 5.8383 
 
 122 
 
 201 
 
 6.8578 
 
 
 202 
 
 
 5.8675 
 
 
 203.. . 
 
 
 - 5. 8771 
 
 125 
 
 
 126 
 
 
 
 127 
 
 
 5.9059 
 5.9165 
 
 128 
 
 
 129 
 
 208 
 
 130 
 
 
 6.9345 
 5.9439 
 5.9633 
 5.9627 
 5.9721 
 6.9814 
 
 5 9907 
 6 
 
 6.0092 
 6.0185 
 6. 02VV 
 
 6 0368 
 
 131 
 
 210 
 
 132 
 
 
 133 
 
 212 
 
 134 
 
 
 135 
 
 214 
 
 136 
 
 
 137 
 
 216 
 
 138 
 
 
 139 
 
 218 
 
 140 
 
 
 141 
 
 220 
 
 142 
 
 22l 
 
 
 143 
 
 222 
 
 6 0550 
 
 144 
 
 
 
 145 
 
 224 
 
 6 0732 
 
 146 
 
 
 6 0822 
 
 147 
 
 226 
 
 6 0912 
 
 148 ., 
 
 227 
 
 6.1002 
 
 14fl ' 
 
 228 
 
 6 1091 
 
 ISO 
 
 229 . ;... 
 
 6.1180 
 
 151 
 
 230 
 
 231 
 
 6.1269 
 
 152 
 
 6.1358 
 
 153 
 
 232 
 
 6.1446 
 
 154. . 
 
 233 
 
 6.1534 
 
 155 
 
 234 
 
 6.1622 
 
 156 
 
 235 
 
 6. 1710 
 
 157 
 
 236 
 
 6. 1797 
 
 158 
 
 237 
 
 6.1885 
 
 159 
 
 238 ,-. 
 
 6. 1972 
 
 160 
 
 239 
 
 6.205S 
 
 161 
 
 240 
 
 6. 2145 
 
 162 
 
 241 
 
 6.2231 
 
 163 
 
 242 
 
 6.2317 
 
 164 
 
 243 
 
 6.2403 
 
 IfiS 
 
 244 
 
 6.2488 
 
 166 
 
 245 
 
 6.257S 
 
 167 
 
 246 
 
 6.2658 
 
 168 
 
 247 
 
 6.2743 
 
 169 
 
 248 
 
 6.2828 
 
 170 
 
 249 
 
 6.2912 
 
 171 
 
 250 
 
 6.2996 
 
 172 
 
 251 
 
 6.3080 
 
 173 
 
 252 
 
 6.3164 
 
 174 
 
 233 
 
 6.3247 
 
 175 
 
 254 
 
 6. 3!'30 
 
 176 
 
 265 . 
 
 6.3413 
 
 177 
 
 256 . 
 
 6 349fi 
 
 178 
 
 257 
 
 6 3579 
 
 179 
 
 258 
 
 6.3661 
 
 180 . .. 
 
 259 
 
 6 3743 
 
 181 
 
 260 
 
 6.3825 
 
 182 
 
 261 
 
 6 3907 
 
 183 
 
 262 
 
 6.3988 
 
 184 
 
 263 
 
 6.4070 
 
 ISS 
 
 264 
 
 6 4151 
 
 
 265 
 
 6.4232 
 
 187 
 
 266 
 
 6.4312 
 
 188 
 
 267 ,. 
 
 268 
 
 6.4393 
 
 1AQ 
 
 6.447i 
 
 
 269 
 
 6.4553 
 
 191 
 
 270 
 
 6.4633 
 
132 TOPOGEAPHY, MAP KEADING, AND EECONNAISSANOE. 
 
 Table VI. — Tabk of squares, cubes, square roots, and cube roots of numbers from 1 to 
 
 i.OOO— Contiaued. 
 
 No. 
 
 Square. 
 
 Cute. 
 
 Square 
 ' root. 
 
 Cube 
 root. 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 271 
 
 73441 
 
 73984 
 
 74529 
 
 75076 
 
 75625 
 
 76176 
 
 76729 
 
 77284 
 
 77841 
 
 78400 
 
 78961 
 
 79524 
 
 80089 
 
 80656 
 
 81225 
 
 81796 
 
 82369 
 
 82944 
 
 83521 
 
 84100 
 
 84681 
 
 85264 
 
 85849 
 
 86436 
 
 87025 
 
 87616 
 
 88209 
 
 88804 
 
 89401 
 
 90000 
 
 90601 
 
 91204 
 
 91809 
 
 92416 
 
 93025 
 
 93636 
 
 94249 
 
 94864 
 
 95481 
 
 96100 
 
 96721 
 
 97344 
 
 97969 
 
 98596 
 
 99225 
 
 99856 
 
 100489 
 
 101124 
 
 101761 
 
 102400 
 
 103041 
 
 103684 
 
 104329 
 
 104976 
 
 10S62S 
 
 106276 
 
 106929 
 
 107584 
 
 108241 
 
 108900 
 
 109561 
 
 110224 
 
 110889 
 
 111556 
 
 112225 
 
 112896 
 
 113569 
 
 114244 
 
 114921 
 
 115600 
 
 116281 
 
 116964 
 
 117649 
 
 118336 
 
 119025 
 
 119716 
 
 120409 
 
 121104 
 
 121801 
 
 19902511 
 20123648 
 20346417 
 20570824 
 20796875 
 21024576 
 21253933 
 21484952 
 21717639 
 21952000 
 22188041 
 22425768 
 22665187 
 22906304 
 23149125 
 23393666 
 23639903 
 23887872 
 24137569 
 24389000 
 24642171 
 24897088 
 25153757 
 25412184 
 25672375 
 25934336 
 26198073 
 26463692 
 26730899 
 27000000 
 27270901 
 27643608 
 27818127 
 28094464 
 28372625 
 28652616 
 28934443 
 29218112 
 29503629 
 29791000 
 30080231 
 30371328 
 30664297 
 30959144 
 31255875 
 31554496 
 31856013 
 32157432 
 32461759 
 32768000 
 33076161 
 33386248 
 33698267 
 34012224 
 34328125 
 34645976 
 ■34965783 
 35287552 
 35611289 
 35937000 
 36264691 
 36594368 
 36926037 
 37269704 
 37595375 
 37933056 
 38272763 
 38614472 
 38958219 
 39304000 
 39661821 
 40001688 
 40353607 
 40707S84 
 41063625 
 41421736 
 41781923 
 42144192 
 42608649 
 
 16.4621 
 16.4924 
 16. 5227 
 16. 6529 
 16. 6831 
 16.6132 
 16. 6433 
 16. 6733 
 16. 7033 
 16.7332 
 16.7631 
 16.7929 
 16. 8226 
 16. 8623 
 
 16. 8819 
 16.9115 
 16.9411 
 16.9706 
 17 
 
 17.0294 
 17.0587 
 17.0880 
 17.1172 
 17.1464 
 17.1766 
 
 17. 2047 
 17. 2337 
 17. 2627 
 17.2916 
 17.3205 
 17.3494 
 17.3781 
 17.4069 
 17.4356 
 17.4642 
 17.4929 
 17.5214 
 17.5499 
 17.5784 
 17.6068 
 17.6362 
 17.6636 
 17.6918 
 17. 7200 
 17.7482 
 17.7764 
 17.8045 
 17. 8326 
 17.8606 
 17.8885 
 17.9166 
 17.9444 
 17.9722 
 18. 
 
 18.0278 
 18.0556 
 18.0831 
 18. 1108 
 18. 1384 
 18. 1659 
 18. 1934 
 18.2209 
 18.2483 
 18. 2757 
 18.3030 
 18.3303 
 18.3676 
 18.3848 
 18.4120 
 18.4391 
 18.4662 
 18.4932 
 18.6203 
 18.5472 
 18.6742 
 18.6011 
 18.6279 
 18.6648 
 18.«815 
 
 6.4713 
 6.4792 
 6.4872 
 6.4951 
 6.5030 
 6.5108 
 6.5187 
 . 6.5265 
 6.5343 
 6.6421 
 6.6499 
 6.5577 
 6.6654 
 6.5731 
 6.5808 
 6. 5885 
 6.5962 
 6.6039 
 6.6115 
 6.6191 
 6.6267 
 6.6343 
 6.6419 
 6.6494 
 6. 6569 
 6.6644 
 6.6719 
 6.6794 
 6.6869 
 6.6943 
 6.7018- 
 6.7092 
 6.7166 
 6.7240 
 6.7313 
 6.7387 
 6. 7460 
 6.7533 
 6.7606 
 6.7679 
 6.7752 
 6.7824 
 6.7897 
 6.7969 
 6.8041 
 6.8113 
 6.8185 
 6.8266 
 6.8328 
 6.8399 
 6.8470 
 6.8541 
 6.8612 
 6.8683 
 6.8763 
 6. 8824 
 6.8894 
 6.8964 
 6.9034 
 6.9104 
 6.9174 
 6.9244 
 6.9313 
 6.9382 
 6.9451 
 6.9621 
 6.9689 
 6.9668 
 6.9727 
 6.9795 
 6.9864 
 6.9932 
 7. 
 
 7.0068 
 7.0136 
 7.0203 
 7.0271 
 7.0338 
 7,0408 
 
 350 
 
 122500 
 123201 
 123904 
 124609 
 125316 
 126025 
 126736 
 127449 
 128164 
 128881 
 129600 
 130321 
 131044 
 131769 
 132496 
 133225 
 133956 
 134689 
 136424 
 136161 
 136900 
 137641 
 138384 
 139129 
 139876 
 140626 
 141376 
 142129 
 142884 
 143641 
 144400 
 145161 
 145924 
 146689 
 147456 
 148225 
 148996 
 149769 
 150544 
 161321 
 152100 
 162881 
 153664 
 164449 
 155236 
 156025 
 166816 
 157609 
 158404 
 169201 
 160000 
 160801 
 161604 
 162409 
 163216 
 164025 
 164836 
 165649 
 166464 
 167281 
 ■ 168100 
 168921 
 169744 
 170569 
 171396 
 172225 
 173056 
 173889 
 174724 
 176561 
 176400 
 177241 
 178084 
 178929 
 179776 
 180625 
 181476 
 182329 
 1831S4 
 
 42875000 
 43243651 
 43614208 
 43986977 
 44361864 
 44738875 
 45118016- 
 45499293 
 45882712 
 46268279 
 46666000 
 47045881 
 47437928 
 47832147 
 48228544 
 48627125 
 49027896 
 49430863 
 49836032 
 50243409 
 60653000 
 51064811 
 61478848 
 51895117 
 62313624 
 52734375 
 53167376 
 53582633 
 54010152 
 54439939 
 54872000 
 66306341 
 55742968 
 56181887 
 66623104 
 57066625 
 67512456 
 57960603 
 58411072 
 68863869 
 59319000 
 59776471 
 60236288 
 60698457 
 611629S4 
 61629875 
 62099136 
 62670773 
 63044792 
 63521199 
 64000000 
 64481201 
 64964808 
 65460827 
 66939264 
 66430125 
 66923416 
 67419143 
 67917312 
 68417929 
 68921000 
 69426531 
 69934628 
 70444997 
 70957944 
 71473375 
 71991296 
 72611713 
 73034632 
 '73560059 
 74088000 
 74618461 
 75151448 
 76686967 
 76226024 
 76765625 
 77308776 
 77854483 
 78402762 
 
 18. 7083 
 
 18.7350 
 
 18.7617 
 
 18. 7883 
 
 18. 8149 
 
 18.8414 
 
 18.8680 
 
 18. 8944 
 
 18.9209 
 
 18.9473 
 
 18.9737 
 
 19. 
 
 19.0263 
 
 19.0526 
 
 19. 0788 
 
 19.1050 
 
 19.1311 
 
 19. 1575 
 
 19.1833 
 
 19.2094 
 
 19.2354 
 
 19.2614 
 
 19.2873 
 
 19.3132 
 
 19.3391 
 
 19.3649 
 
 19.3907 
 
 19.4165 
 
 19.4422 
 
 19.4679 
 
 19.4936 
 
 19.6192 
 
 19.6448 
 
 19.5704 
 
 19.5959 
 
 19.6214 
 
 19.6469 
 
 19.6723 
 
 19.6977 
 
 19.7231 
 
 19.7484 
 
 19.7737 
 
 19.7990 
 
 19. 8242 
 
 19.8494 
 
 19.8746 
 
 19.8997 
 
 19.9249 
 
 19. 9499 
 
 19.9750 
 
 20. 
 
 20.0250 
 
 20.0499 
 
 20.0749 
 
 20.0998 
 
 20.1246 
 
 20. 1494 
 
 20. 1742 
 
 20.1990 
 
 20. 2237 
 
 20.2485 
 
 20.2731 
 
 20.2978 
 
 20.3224 
 
 20.3470 
 
 20.3715 
 
 20.3961 
 
 20.4206 
 
 20.4460 
 
 20.4695 
 
 20. 4939 
 
 20.5183 
 
 20.5426 
 
 20.5670 
 
 20.5913 
 
 20.6165 
 
 20. 6398 
 
 20.6640 
 
 20.6882 
 
 7.0473 
 
 272 
 
 351 
 
 7.0540 
 
 273 
 
 
 7.0607 
 
 274 
 
 353 
 
 7.0674 
 
 275 
 
 354 
 
 7.0740 
 
 276 
 
 
 7.0807 
 
 277 
 
 356 
 
 7.0873 
 
 278 
 
 357.. 
 
 7.0940 
 
 279 
 
 358 
 
 7.1006 
 
 280 
 
 359 
 
 7. 1072 
 
 281 
 
 
 7.1138 
 
 282 
 
 361' 
 
 7.1204 
 
 283... 
 
 
 7. 1269 
 
 284 
 
 363 
 
 7. 1335 
 
 285 
 
 364 
 
 7. 1400 
 
 286 
 
 365 
 
 7. 1466 
 
 287 
 
 366 
 
 7. 1631 
 
 288 
 
 
 7. 1596 
 
 289 
 
 368 
 
 7. 1661 
 
 290 
 
 369 .. . 
 
 7.1726 
 
 291 
 
 370 
 
 7. 1791 
 
 292 
 
 371 
 
 7.1855 
 
 293 
 
 372 . 
 
 7.1920 
 
 294 
 
 373 
 
 7. 1984 
 
 295 
 
 374 
 
 7.2048 
 
 296 
 
 375 
 
 7. 2112 
 
 297 
 
 376 
 
 7. 2177 
 
 298 
 
 377 
 
 7.2240 
 
 299 
 
 378 
 
 7.2304 
 
 300 
 
 379 
 
 7.2368 
 
 301 
 
 380 
 
 7. 2432 
 
 302. 
 
 381 
 
 7. 2495 
 
 303 
 
 382 
 
 7. 2558 
 
 304 
 
 383 
 
 7.2622 
 
 305 
 
 384 . 
 
 7.2685 
 
 306 
 
 385 
 
 7.2748 
 
 307 
 
 386 
 
 7. 2811 
 
 308 
 
 387 
 
 7.2874 
 
 309 
 
 388 
 
 7.2936 
 
 310 
 
 389 
 
 7. 2999 
 
 3U 
 
 390 
 
 7.3061 
 
 312 
 
 391 
 
 7. 3124 
 
 313 
 
 392 
 
 7.3186 
 
 314 . 
 
 393 .. 
 
 7. 3248 
 
 315 
 
 394 
 
 7. 3310 
 
 316 
 
 396 
 
 7. 3372 
 
 317 
 
 396. . 
 
 7 3434 
 
 318 
 
 397 i . . 
 
 
 319 
 
 398 
 
 7 3558 
 
 320 
 
 399 -. 
 
 7 3619 
 
 321 
 
 400 
 
 7.3681 
 
 322 
 
 401 
 
 323 
 
 ■402 
 
 7.3803 
 
 324 
 
 403 
 
 325 
 
 404 
 
 
 326 
 
 405 
 
 7 3986 
 
 327 
 
 406 
 
 7.4047 
 7.4108 
 7.4169 
 7.4229 
 7.4290 
 7.436Q 
 7.4410 
 7. 4470 
 7.4630 
 7.4690 
 7.4650 
 7.4710 
 7. 4770 
 7.4829 
 7.4889 
 7.4948 
 7.5007 
 7.5067 
 7.6126 
 7.5185 
 7.5244 
 7.5302 
 
 7-!i9«1 
 
 328 
 
 407 
 
 329 
 
 408 .. . . 
 
 330r. 
 
 409 
 
 331 
 
 410 
 
 332 
 
 411. .. . 
 
 333 
 
 412 
 
 334 
 
 413 
 
 335 
 
 414 
 
 336 
 
 415 
 
 337 
 
 416 
 
 338 
 
 417 
 
 339 
 
 418 
 
 340 
 
 419 
 
 341 
 
 420 
 
 342 
 
 421 
 
 343 
 
 422 
 
 S44 
 
 423 
 
 345 
 
 424 
 
 346 
 
 426 
 
 347 
 
 426 
 
 348 
 
 427 
 
 349 
 
 428, 
 
TOPOGEAPHY, MAP EEAMNG, AKD RECONNAISSANCE. 133 
 
 Table Yl.—Tabk of squares, cubes, square roots, and cube roots of numbers from 1 to 
 
 ijOOO— Continued. 
 
 No. 
 
 Square. 
 
 Cute. 
 
 Square 
 root. 
 
 Cute 
 root. 
 
 No. 
 
 3quare. 
 
 Cute. 
 
 Square 
 root. 
 
 Cute 
 root. 
 
 429..... 
 
 184041 
 184900 
 1?5761 
 186624 
 187489 
 188356 
 189225 
 190096 
 190969 
 191844 
 192721 
 193600 
 194481 
 195364 
 196249 
 197136 
 198025 
 198916 
 199809 
 200704 
 201601 
 202500 
 203401 
 204304 
 205209 
 206116 
 207025 
 207936 
 208849 
 209764 
 210681 
 211600 
 212621 
 213444 
 214369 
 215296 
 216225 
 217156 
 218089 
 219024 
 219961 
 220900 
 221841 
 222784 
 223729 
 224676 
 225625 
 226576 
 227629 
 228484 
 229441 
 230400 
 231361 
 232324 
 233289 
 234266 
 235225 
 236196 
 237169 
 238144 
 239121 
 240100 
 241081 
 242064 
 243049 
 244036 
 245025 
 246016 
 247009 
 248004 
 249001 
 250000 
 251001 
 252004 
 253009 
 254016 
 256025 
 256038 
 257049 
 258064 
 
 78953589 
 79507000 
 80062991 
 80621568 
 81182737 
 81746504 
 82312875 
 82881856 
 83453453 
 84027672 
 84604519 
 86184000 
 85766121 
 86350888 
 86938307 
 87528384 ' 
 88121125 
 88716536 
 89314623 
 89915392 
 90518849 
 91125000 
 91733861 
 92345408 
 92959677 
 93576664 
 94196375 
 94818816 
 95443993 
 96071912 
 96702579 
 97336000 
 97972181 
 98611128 
 99252847 
 99897344 
 100644625 
 101194696 
 101847563 
 102503232 
 103161709 
 103823000 
 104487111 
 105154048 
 106823817 
 106496424 
 107171875 
 107850176 
 108631333 
 109215362 
 109902239 
 110592000 
 111284641 
 111980168 
 112678587 
 113379904 
 114084125 
 114791256 
 116601303 
 116214272 
 116930169 
 117649000 
 118370771 
 119095488 
 119823167 
 120553784 
 121287375 
 122023936 
 122763473 
 123505992 
 124251499 
 125000000 
 125751501 
 126506008 
 127263527 
 128024064 
 128787626 
 129554216 
 130323843 
 131096512 
 
 20.7123 
 
 20.7364 
 
 20.7605 
 
 20.7846 
 
 20.8087 
 
 20.8327 
 
 20.8567 
 
 20.8806 
 
 20.9046 
 
 20;92S4 
 
 20.9623 
 
 20.9762 
 
 21. 
 
 21.0238 
 
 21.0476 
 
 21.0713 
 
 21.0950 
 
 21.1187 
 
 21.1424 
 
 21. 1660 
 
 21.1896 
 
 21.2132 
 
 21.2368 
 
 21.2603 
 
 21.2838 
 
 21.3073 
 
 21.3307 
 
 21.3542 
 
 21.3776 
 
 21.4009 
 
 21.4243 
 
 21. 4476 
 
 21.4709 
 
 21.4942 
 
 21.5174 
 
 21.5407 
 
 21.5639 
 
 21.5870 
 
 21.6102 
 
 21.6333 
 
 21.6564 
 
 21.6795 
 
 21.7025 
 
 21.7256 
 
 21.7486 
 
 21.7715 
 
 21. 7945 
 21.8174 
 21.8403 
 21.8632 
 21.8861 
 21.9089 
 21.9317 
 21.9545 
 21.9773 
 22. 
 22.0227 
 
 .22.0454 
 22.0681 
 22.0907 
 22. 1133 
 22. 1369 
 22.1585 
 22. 1811 
 22.2036 
 22.2261 
 22.2486 
 22.2711 
 22.2935 
 22.3159 
 22.3383 
 22.3607 
 22.3830 
 22.4064 
 22.4277 
 
 22. 4499 
 22. 4722 
 22.4944 
 22. 5167 
 22.6389 
 
 7.5420 
 7.5478 
 7.5537 
 7.5596 
 7.5654 
 7.5712 
 7.6770 
 7.5828 
 7.5886 
 7.5944 
 7.6001 
 7.6059 
 7.6117 
 7.6174 
 7.6232 
 7.6289 
 7.6346 
 7. 6403 
 7.6460 
 7.6617 
 7.6574 
 7.6631 
 7.6888 
 7.6744 
 7.6801 
 7.6867 
 7.6914 
 7.6976 
 7.7026 
 7.7082 
 7.7138 
 7.7194 
 7.7250 
 7.7306 
 7.7362 
 7.7418 
 7.7473 
 7.7529 
 7.7584 
 7.7639 
 7.7695 
 7.7760 
 7.7805 
 7.7860 
 7.7916 
 7.7CI70 
 7.8025 
 7.8079 
 7.8134 
 7.8188 
 7.8243 
 7.8297 
 7.8352 
 7.8406 
 7.8460 
 7.8514 
 7.8568 
 7.8622 
 7.8676 
 7.8730 
 7.8784 
 7.8837 
 7.8891 
 7.8944 
 7.8998 
 7.9051 
 7.9105 
 7.9168 
 7.9211 
 7.9264 
 7.9317 
 7.9370 
 7.9423 
 7.9476 
 7.9628 
 7.9581 
 7.9634 
 7.9686 
 7.9739 
 7.9791 
 
 
 346744 
 259081 
 260100 
 261121 
 262144 
 263169 
 264196 
 265226 
 266256 
 267289 
 268324 
 269361 
 270400 
 271441 
 272484 
 273629 
 274576 
 275625 
 276676 
 277729 
 278784 
 279841 
 280900 
 281961 
 283024 
 284089 
 285156 
 286225 
 287296 
 288369 
 289444 
 290521 
 291600 
 292681 
 293764 
 294849 
 296936 
 297025 
 298116 
 299209 
 300304 
 301401 
 302500 
 303601 
 304704 
 305809 
 306916 
 308025 
 309136 
 310249 
 311364 
 312481 
 313600 
 314721 
 315844 
 316969 
 318096 
 319225 
 320366 
 321489 
 322624 
 323761 
 324900 
 326041 
 327184 
 328329 
 329476 
 330625 
 331776 
 332929 
 334084 
 335241 
 336400 
 337661 
 338724 
 339889 
 341066 
 342225 
 343396 
 344569 
 
 203297472 
 
 131872229 
 
 132651000 
 
 133432831 
 
 134217728 
 
 135006697 
 
 135796744 
 
 136590875' 
 
 137388096 
 
 138188413 
 
 138991832 
 
 139798359 
 
 140608000 
 
 141420761 
 
 142236848 
 
 143056667 
 
 143877824 
 
 144703125 
 
 146531676 
 
 146363183 
 
 147197952 
 
 148035889 
 
 148877000 
 
 149721291 
 
 150568768 
 
 151419437 
 
 152273304 
 
 153130375 
 
 163990656 
 
 154864163 
 
 155720872 
 
 166590819 
 
 167464000 
 
 158340421 
 
 159220088 
 
 160103007 
 
 160989184 
 
 161878625 
 
 162771336 
 
 163667323 
 
 164666592 
 
 166469149 
 
 166375000 
 
 167284161 
 
 168196608 
 
 169112377 
 
 170031464 
 
 170963875 
 
 171879616 
 
 172808693 
 
 173741112 
 
 174676879 
 
 175616000 
 
 176558481 
 
 177504328 
 
 178453647 
 
 179406144 
 
 180362125 
 
 181321406 
 
 182284263 
 
 183250432 
 
 184220009 
 
 185193000 
 
 186169411 
 
 187149248 
 
 188132517 
 
 189119224 
 
 190109375 
 
 191102976 
 
 192100033 
 
 193100552 
 
 194104539 
 
 195112000 
 
 196122941 
 
 197137368 
 
 198155287 
 
 199176704 
 
 200201626 
 
 201230056 
 
 202262003 
 
 24.2487 
 22.6610 
 22. 5832 
 22.6053 
 
 22. 6274 
 22.6495 
 22.6716 
 22.6936 
 22.7156 
 22.7376 
 22.7596 
 22.7816 
 22.8035 
 22.8254 
 22.8473 
 22.8692 
 22. 8910 
 22. 9129 
 22.9347 
 22.9685 
 22.9783 
 23. 
 
 23.0217 
 23.0434 
 23.0651 
 23.0868 
 23.1084 
 23. 1301 
 23.1517 
 23.1733 
 25. 1948 
 23.2164 
 23.2379 
 23.2594 
 23.2809 
 23. 3024 
 
 23. 3238 
 23. 3462 
 23. 3666 
 23. 3880 
 23. 4094 
 23.4307 
 23.4521 
 23. 4734 
 23. 4947 
 23.5160 
 23.5372 
 23.5684 
 23. 5797 
 23.6008 
 23.6220 
 23.6432 
 23.6643 
 23. 6854 
 23.7066 
 23.7276 
 23.7487 
 23. 7697 
 
 ■23.7908 
 23.8118 
 
 -23.8328 
 23,8637 
 23.8747 
 
 23. 8956 
 23.9165 
 23.9374 
 23.9583 
 23.9792 
 24. 
 
 24.0208 
 24.0416 
 
 24. 0624 
 24.0832 
 24. 1039 
 24.1247 
 
 - 24. 1464 
 24.1661 
 24. 1868 
 24.2074 
 24.2281 
 
 8.37VV 
 7.9843 
 7.9896 
 7.9948 
 8. ' 
 8.0052 
 8.0104 
 8.0156 
 8.0208 
 8.0260 
 8. 0311 
 
 430 
 
 509 
 
 431 
 
 
 432 
 
 511 
 
 433 
 
 
 434 
 
 513 
 
 435.. 
 
 
 436 
 
 615 
 
 437 
 
 
 438.. 
 
 617 
 
 439 
 
 618 
 
 440 
 
 619 
 
 441 
 
 520. . 
 
 8.0415 
 8.0466 
 8.0617 
 8 0669 
 
 442 
 
 521 
 
 443 
 
 522... 
 
 444 
 
 623 
 
 446 
 
 624 
 
 
 446 
 
 525.. 
 
 8 0671 
 
 447 
 
 626 
 
 
 448 
 
 627... 
 
 8 0774 
 
 449 
 
 528 
 
 
 450.. . . 
 
 529.^ 
 
 530 
 
 8 0876 
 
 451 
 
 8.0927 
 
 452.. 
 
 531 
 
 8 0978 
 
 453 
 
 532 
 
 
 454.. 
 
 533 
 
 8 1079 
 
 455 
 
 534 
 
 
 456.. 
 
 535 
 
 8 1180 
 
 457 
 
 636 
 
 
 458 . 
 
 537 
 
 8 1281 
 
 469 
 
 538 
 
 
 460 
 
 539 
 
 8 1382 
 
 461. . 
 
 540 
 
 
 462 
 
 541 
 
 8 1483 
 
 463.. 
 
 542 
 
 8 1533 
 
 464 
 
 543 
 
 8 1683 
 
 465.- 
 
 544 
 
 8.1633 
 
 466 
 
 545 
 
 8 1683 
 
 467. 
 
 546 
 
 8. 1733 
 
 468 
 
 547 
 
 8 1783 
 
 469. 
 
 548 
 
 8.1833 
 
 470 
 
 549 
 
 8.1882 
 
 471. 
 
 550 
 
 8. 1932 
 
 472 
 
 551 
 
 8. 1982 
 
 473.. 
 
 652 
 
 8.2031 
 
 474 
 
 553 
 
 8. 2081 
 
 475 
 
 554 
 
 8. 213C 
 
 476 
 
 566 
 
 8. 218C 
 
 477 
 
 556 
 
 8.2229 
 
 478.. 
 
 557 
 
 8. 227S 
 
 479 
 
 568 
 
 8. 2325 
 
 480.. 
 
 659 
 
 8. 2375 
 
 481 
 
 560 
 
 8.242e 
 
 482 
 
 561 
 
 r 8.247J 
 
 483.. 
 
 662 
 
 8.2624 
 
 484 
 
 663 
 
 8. 267S 
 
 485. 
 
 564 
 
 8. 262] 
 
 486 
 
 565 
 
 8. 267( 
 
 487 
 
 566. 
 
 8. 271£ 
 
 488 
 
 567 
 
 8. 276S 
 
 489 
 
 568 
 
 8.28U 
 
 490 
 
 869 
 
 8. 286; 
 
 491 
 
 570. 
 
 8 291; 
 
 492 
 
 671 
 
 8 296S 
 
 493 
 
 572 
 
 
 494 
 
 573 
 
 8 306C 
 
 495 
 
 574 
 
 8. 310' 
 
 496 
 
 575. .» 
 
 8 315t 
 
 497 
 
 576 
 
 8 320' 
 
 498 
 
 577 
 
 
 499 
 
 578 
 
 8 330C 
 
 500 
 
 579 
 
 
 
 580 
 
 
 
 581 
 
 8. 344: 
 
 
 582 
 
 
 
 583 
 
 8. 353£ 
 
 605 
 
 684 :. 
 
 8. 368' 
 
 
 585 
 
 8.363^ 
 
 fi07 
 
 686 
 
 8.3682 
 
 fiPS 
 
 687 
 
 8.373C 
 
134 TOPOGRAPHY, MAP HEADING, AND RECONNAISSANCE. 
 
 Table \l.— Table of squares, cubes, square roots, and cube roots of nambers from 1 to 
 
 i, 000— Continued. 
 
 No. 
 
 1 
 
 Square. 
 
 Cube. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 589 
 
 346921 
 348100 
 349281 
 360464 
 351649 
 352836 
 354025 
 855216 
 356409 
 357604 
 358801 
 360000 
 361201 
 362404 
 363609 
 364816 
 366025 
 367236 
 368449 
 369664 
 370881 
 372100 
 373321 
 374544 
 375769 
 376996 
 378225 
 379456 
 380689 
 381924 
 383161 
 384400 
 386641 
 386884 
 388129 
 389376 
 390625 
 391876 
 393129 
 394384 
 395641 
 396900 
 398161 
 399424 
 400689 
 401956 
 403225 
 404496 
 405769 
 407044 
 408321 
 409600 
 410881 
 412164 
 413449 
 414736 
 416025 
 417316 
 418609 
 419904 
 421201 
 422500 
 423801 
 426104 
 426409 
 427716 
 429025 
 430336 
 431649 
 432964 
 434281 
 435600 
 436921 
 438244 
 439569 
 440896 
 442225 
 443566 
 444889 
 
 204336469 
 205379000 
 208425071 
 207474688 
 208527857 
 209584584 
 210644876 
 211708736 
 212776173 
 213847192 
 214921799 
 216000000 
 217081801 
 218167208 
 219256227 
 220348864 
 221445126 
 222545016 
 223648543 
 224755712 
 226866529 
 226981000 
 228099131 
 229220928 
 230346397 
 231475544 
 232008375 
 233744896 
 234885113 
 236029032 
 237176659 
 238328000 
 239483061 
 240641848 
 2418043B7 
 242970624 
 244140626 
 245314376 
 246491883 
 247673152 
 248888189 
 250047000 
 251239691 
 252435968 
 253636137 
 254840104 
 256047875 
 257259456 
 258474853 
 269694072 
 260917119 
 262144000 
 263374721 
 264609288 
 266847707 
 267089984 
 268336125 
 269586136 
 
 272097792 
 2V3359449 
 274625000 
 276894451 
 277167808 
 278446077 
 279726264 
 281011375 
 282300416 
 283593393 
 284890312 
 286191179 
 287496000 
 288804781 
 290117628 
 291434247 
 292754944 
 294079625 
 295408296 
 296740963 
 
 24.2693 
 
 24.2899 
 
 24.3105 
 
 24.3311 
 
 24.3516 
 
 24.3721 
 
 24. 8926 
 
 24.4131 
 
 24.4336 
 
 24.4640 
 
 24.4745 
 
 24.4949 
 
 24.5163 
 
 24.5357 
 
 24.5561 
 
 24.6764 
 
 24.5967 
 
 24.6171 
 
 24.6374 
 
 24.6577 
 
 24.6779 
 
 24.6982 
 
 24.7184 
 
 24.7386 
 
 24.7588 
 
 24.7790 
 
 24.7992 
 
 24.8193 
 
 24. 8396 
 
 24.8596 
 
 24.8797 
 
 24.8998 
 
 24.9199 
 
 24.9399 
 
 24.9600 
 
 24.9800 
 
 25. 
 
 25.0200 
 
 26.0400 
 
 25.0599 
 
 26.0799 
 
 26.0998 
 
 25.1197 
 
 25.1396 
 
 25. 1596 
 
 26.1794 
 
 25.1992 
 
 25.2190 
 
 26.2389 
 
 26.2687 
 
 25.2784 
 
 25.2982 
 
 25.3180 
 
 25.3377 
 
 26.3674 
 
 25.3772 
 
 25.3969 
 
 25.4165 
 
 25.4362 
 
 25.4658 
 
 26.4755 
 
 25.4951 
 
 26.5147 
 
 26.8343 
 
 2S.5539 
 
 25.5734 
 
 25.5930 
 
 25.6125 
 
 25.6320 
 
 26.6615 
 
 26.6710 
 
 26.6905 
 
 26.7099 
 
 26. 7294 
 
 26.7488 
 
 26.7682 
 
 26.7876 
 
 25.8070 
 
 25.8263 
 
 8.3825 
 8.3872 
 8. 3919 
 8. 3967 
 8.4014 
 8.4061 
 8.4108 
 8.4156 
 8.4202 
 8.4249 
 8.4296 
 8. 4343 
 8.4390 
 8.4437 
 8.4484 
 8.4630 
 8.4577 
 8.4623 
 8.4670 
 8.4716 
 8.4763 
 8. 4809 
 8.4856 
 8.4902 
 8.4948 
 8.4994 
 8.6040 
 8.5086 
 8.5132 
 8.5178 
 8.5224 
 8.6270 
 8.5316 
 8.5362 
 8.5408 
 8.5453 
 8.5499 
 8.5544 
 8.6690 
 8.5635 
 8.5681 
 8.5726 
 8. 5772 
 8.5817 
 8.5862 
 8.6907 
 8.5952 
 8.5997 
 8.6043 
 8.6088 
 8.6132 
 8.6177 
 8.6222 
 8.6267 
 8.6312 
 8.6357 
 8.6401 
 8.6446 
 8.6490 
 8.6535 
 8.6579 
 8.6624 
 8.6668 
 8.6713 
 8.6767 
 8.6801 
 8.6845 
 8.6890 
 8.6934 
 8.6978 
 8.7022 
 8.7066 
 8.7110 
 8.7154 
 8.7198 
 8.7241 
 8.7285 
 8.7329 
 8.7373 
 
 668 
 
 446224 
 447761 
 448900 
 450241 
 461684 
 462929 
 454276 
 455625 
 466976 
 468329 
 459684 
 461041 
 462400 
 463761 
 465124 
 466489 
 467866 
 469225 
 470596 
 471969 
 473344 
 474721 
 476100 
 477481 
 478864 
 480249 
 481636 
 483025 
 484416 
 485809 
 487204 
 488601 
 490000 
 491401 
 492804 
 494209 
 495616 
 497025 
 498436 
 499849 
 501264 
 502681 
 504100 
 505521 
 606944 
 608369 
 509796 
 511225 
 512656 
 514089 
 515524 
 516961 
 518400 
 519841 
 521284 
 622729 
 524176 
 625625 
 527076 
 528529 
 629984 
 631441 
 532900 
 534361 
 536824 
 537289 
 538766 
 540225 
 541696 
 543169 
 544644 
 546121 
 647600 
 649081 
 650564 
 662049 
 653536 
 655025 
 6S6616 
 
 298077632 
 
 299418309 
 
 300763000 
 
 302111711 
 
 303464448 
 
 304821217 
 
 306182024 
 
 307546875 
 
 308915776 
 
 310288733 
 
 311666762 
 
 313046839 
 
 314432000 
 
 315821241 
 
 317214568 
 
 318611987 
 
 320013504 
 
 321419126 
 
 322828856 
 
 324242703 
 
 325660672 
 
 327082769 
 
 328609000 
 
 329939371 
 
 331373888 
 
 332812567 
 
 334265384 
 
 335702375 
 
 337153636 
 
 338608873 
 
 340068392 
 
 341532099 
 
 343000000 
 
 344472101 
 
 346948408 
 
 347428927 
 
 348913664 
 
 350402625 
 
 351895816 
 
 353393243 
 
 354894912 
 
 356400829 
 
 357S11000 
 
 359425431 
 
 360944128 
 
 362467097 
 
 363994344 
 
 365525875 
 
 367061696 
 
 368601813 
 
 370146232 
 
 371694959 
 
 373248000 
 
 374806361 
 
 376367048 
 
 377933067 
 
 3'9503424 
 
 381078126 
 
 382657176 
 
 384240583 
 
 385828352 
 
 387420489 
 
 389017000 
 
 390617891 
 
 392223168 
 
 393832837 
 
 395446904 
 
 397066375 
 
 398688266 
 
 400315653 
 
 401947272 
 
 403583419 
 
 406224000 
 
 406869021 
 
 408518488. 
 
 410172407 
 
 411830784 
 
 413493625 
 
 115160936 
 
 25.8467 
 
 25.8650 
 
 25.8844 
 
 26.9037 
 
 25.9230 
 
 25.9422 
 
 26.9615 
 
 25.9808 
 
 26-, 
 
 26.0192 
 
 26.0384 
 
 26.0576 
 
 26.0768 
 
 26.0960 
 
 26.1161 
 
 26.1343 
 
 26.1534 
 
 26.1725 
 
 26.1916 
 
 26.2107 
 
 26.2298 
 
 26.2488 
 
 26.2679 
 
 26.2869 
 
 26.3059 
 
 26.3249 
 
 26.3439 
 
 26.3629 
 
 26.3818 
 
 26.4008 
 
 26. 4197 
 
 26.4386 
 
 26.4575 
 
 26.4764 
 
 26. 4963 
 
 26.5141 
 
 26.5330 
 
 26.5518 
 
 26.5707 
 
 26.5896 
 
 26.6083 
 
 26.6271 
 
 26.64E8 
 
 26.6646 
 
 26.6833 
 
 26.7021 
 
 26.7208 
 
 26. 7396 
 
 26.7582 
 
 26.7769 
 
 26.7955 
 
 26.8142 
 
 26.8328 
 
 26. 8514 
 
 26.8701 
 
 26.8887 
 
 26.9072 
 
 26.9258 
 
 26.9444 
 
 26.9629 
 
 26.9815 
 
 27. 
 
 27.0185 
 
 27.0370 
 
 27.0555 
 
 27.0740 
 
 27.0924 
 
 27.1109 
 
 27.1293 
 
 27.1477 
 
 27.1662 
 
 27.1846 
 
 27.2029 
 
 27.2213 
 
 27.2397 
 
 27.2680 
 
 27.2764 
 
 27.2947 
 
 27.3130 
 
 8.7416 
 
 690 
 
 669 
 
 8.7460 
 
 691 
 
 670 
 
 8.7503 
 
 692... 
 
 671 
 
 8.7547 
 
 593 
 
 672 
 
 8.7590 
 
 594 
 
 673 
 
 8.7634 
 
 595 
 
 674 
 
 8.7677 
 
 696 
 
 675 
 
 8.7721 
 
 697. 
 
 676 
 
 8.7764 
 
 598 
 
 677 
 
 8.7807 
 
 S99 
 
 678 
 
 8.7850 
 
 600 
 
 679 
 
 8.7893 
 
 601 
 
 680 
 
 8.7937 
 
 602 
 
 681 
 
 8.7980 
 
 603 
 
 682 
 
 8.8023 
 
 664. . . 
 
 683 
 
 8.8066 
 
 605 
 
 664 
 
 8.8109 
 
 606 
 
 685 
 
 8.8152 
 
 607 
 
 686 
 
 8. 8194 
 
 608 
 
 687 
 
 8.8237 
 
 609.. 
 
 688 
 
 8.8280 
 
 610 
 
 689 
 
 8.8323 
 
 611 
 
 690 
 
 8.8366 
 
 612 
 
 691 
 
 8.8408 
 
 613 
 
 692 
 
 8.8451 
 
 614. 
 
 693 
 
 8.8492 
 
 616 
 
 694 
 
 8.8536 
 
 616 
 
 695 
 
 8.8678 
 
 617 
 
 695 
 
 8.8621 
 
 618 
 
 697 
 
 8.8663 
 
 619 . 
 
 69S 
 
 8.8706 
 
 620 
 
 699 
 
 8. 8748 
 
 621.. , . . 
 
 700 
 
 8.8790 
 
 622 
 
 701 
 
 8.8833 
 
 623 
 
 702 
 
 8.8875 
 
 624.. 
 
 703 
 
 8. 8917 
 
 626 
 
 704. 
 
 8.8969 
 
 626 
 
 706 
 
 8.9001 
 
 627 
 
 706 
 
 8.9043 
 
 628 
 
 707 
 
 8.9085 
 
 629 
 
 708 
 
 8.'9127 
 
 630 
 
 709 
 
 8.9169 
 
 631 
 
 710 
 
 8. 9211 
 
 632 
 
 711 
 
 8.9253 
 
 633 
 
 712 
 
 8.9295 
 
 634 
 
 713 
 
 8.9337 
 
 636 
 
 714 
 
 8. 9378 
 
 636 
 
 716 
 
 8.9420 
 
 637 
 
 716 
 
 717 
 
 , 8. 9462 
 
 638 
 
 8 9503 
 
 639 
 
 718 
 
 8.954S 
 
 640 
 
 719 
 
 8 9587 
 
 641 
 
 720 
 
 8.9628 
 
 642 
 
 721 
 
 8.9670 
 
 643 
 
 722 . .. . 
 
 8 9711 
 
 644 
 
 723 
 
 8. 9752 
 
 645 
 
 724 
 
 S 9794 
 
 646 
 
 725 
 
 8.9835 
 
 647 
 
 726 
 
 8 9876 
 
 648 
 
 727 
 
 8.9918 
 
 649 
 
 728 
 
 8 9959 
 
 650 
 
 729 
 
 9. 
 
 661 
 
 730 
 
 9 0041 
 
 652 
 
 731 
 
 9 0082 
 
 663 . ... 
 
 732 
 
 
 654 
 
 733 
 
 9 0164 
 
 666 
 
 734 
 
 
 666 
 
 736 
 
 9 0246 
 
 657 
 
 736 
 
 
 658 
 
 737 
 
 9 0328 
 
 659 
 
 738 
 
 
 660 
 
 739 
 
 9 0410 
 
 661 
 
 740 
 
 
 662 
 
 741 
 
 9 0491 
 
 663 
 
 742 
 
 
 664 
 
 743 
 
 9.0572 
 
 
 744 
 
 666 
 
 745 
 
 9 0654 
 
 867 
 
 746 
 
 9.0394 
 
TOPOGRAPHY, MAP EEAMNG, AND EECONKAISSANCE. 
 
 135 
 
 Table VI. — Table of squares, cubes, square roots, and cube roots of numbers from 1 to 
 
 i, 000— Continued. 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 747.. 
 748.. 
 749.. 
 750.. 
 751.. 
 752.. 
 753.. 
 754.. 
 755.. 
 756.. 
 757.. 
 758.. 
 759.. 
 760.. 
 761., 
 762. 
 763. 
 764. 
 765. 
 766. 
 767. 
 
 769... 
 770... 
 771... 
 772... 
 773... 
 774... 
 775... 
 776... 
 777... 
 778.. 
 779.. 
 780.. 
 781.. 
 782.. 
 783.. 
 784.. 
 785.. 
 786.. 
 787^. 
 788.. 
 
 790.. 
 791.. 
 792.. 
 793.. 
 794.. 
 796.. 
 796.. 
 797.. 
 798.. 
 799. 
 800. 
 801. 
 
 558009 
 569504 
 561001 
 562500 
 564001 
 565504 
 567009 
 668516 
 570025 
 S71536 
 573049 
 574564 
 576081 
 577600 
 679121 
 580644 
 682169 
 583696 
 585226 
 S86756 
 
 589824 
 591361 
 592900 
 594441 
 595984 
 S97529 
 599076 
 600625 
 602176 
 603729 
 605284 
 606841 
 608400 
 609961 
 611524 
 613089 
 614656 
 616225 
 617796 
 619369 
 620944 
 622521 
 624100 
 625681 
 627264 
 
 416832723 
 418508992 
 10189749 
 !1875000 
 423564761 
 425259008 
 426957777 
 428661064 
 430368876 
 432081216 
 433798093 
 435519512 
 437245479 
 438976000 
 440711081 
 442450728 
 444194947 
 445943744 
 447697125 
 449455096 
 451217663 
 452984832 
 454756609 
 456533000 
 458314011 
 460099648 
 461889917 
 463684824 
 465484375 
 467288576 
 469097433 
 470910952 
 472729139 
 474552000 
 476379541 
 478211768 
 480048687 
 481890304 
 483736625 
 485587656 
 487444403 
 
 630436 
 632025 
 633616 
 635209 
 636804 
 638401 
 640000 
 641601 
 643204 
 
 804 .'..- 646416 
 
 648025 
 640636 
 651249 
 652864 
 654481 
 656100 
 657721 
 659344 
 
 807.. 
 808.. 
 809. 
 810. 
 811. 
 812. 
 813. 
 814. 
 815. 
 816. 
 817. 
 818. 
 819. 
 820. 
 821. 
 
 824. 
 825. 
 
 662596 
 664225 
 665856 
 667489 
 ,669124 
 670761 
 672400 
 674041 
 676684 
 677329 
 
 680625 
 
 491169069 
 
 493039000 
 
 494913671 
 
 496793088 
 
 498677257 
 
 500666184 
 
 502459875 
 
 504358336 
 
 506261573 
 
 508169592 
 
 510082399 
 
 512000000 
 
 513922401 
 
 515849608 
 
 517781627 
 
 519718464 
 
 521660125 
 
 623606616 
 
 526557943 
 
 627514112 
 
 529475129 
 
 S31441000 
 
 533411731 
 
 635387328. 
 
 537367797 
 
 539353144 
 
 54134337S 
 
 543338496 
 
 545338513 
 
 647343432 
 
 549353259 
 
 551368000 
 
 553387661 
 
 555412248 
 
 557441767 
 
 659476224 
 
 561515625 
 
 27. 3313 
 
 27. 3496 
 
 27.3679 
 
 27. 3861 
 
 27. 4044 
 
 27.4226 
 
 27. 4408. 
 
 27. 4591 
 
 27. 4773 
 
 27.4955 
 
 27.5136 
 
 27.5318 
 
 27.5500 
 
 27.5681 
 
 27.5862 
 
 27.6043 
 
 27.6225 
 
 27.6405 
 
 27.6586 
 
 27.6767 
 
 27.6948 
 
 27.7128 
 
 27. 7308 
 
 27.7489 
 
 27.7669 
 
 27.7849 
 
 27.8029 
 
 27.8209 
 
 27.8388 
 
 27.8668 
 
 27.8747 
 
 27.8927 
 
 27.9106 
 
 27.9285 
 
 27.9464 
 
 27.9643 
 
 27.9821 
 
 28. 
 
 28.0179 
 
 28. 0367 
 
 28. 0535 
 
 28.0713 
 
 28:0891 
 
 28. 1069 
 
 28. 1247 
 
 28. 1426 
 
 28.1603 
 
 28. 1780 
 
 28.19S7 
 
 28.2135 
 
 28.2312 
 
 28. 2489 
 
 28. 2666 
 
 28.2843 
 
 28. 3019 
 
 28.3196 
 
 28.3373 
 
 28.3549 
 
 28.3725 
 
 28.3901 
 
 28.4077 
 
 28. 4253 
 
 28.4429 
 
 28.4605 
 
 28. 4781 
 
 28.4968 
 
 28. 5132 
 
 28.5307 
 
 28. 5482 
 
 28. 5657 
 
 28.5832 
 
 28.6007 
 
 28. 6182 
 
 28. 6356 
 
 28. 6631 
 
 28. 6705 
 
 28.6880 
 
 28.7054 
 
 28.7228 
 
 9.0735 
 
 9.0775 
 
 9. 0816 
 
 9.0856 
 
 9.0896 
 
 9.0937 
 
 9.0977 
 
 9. 1017 
 
 9.1057 
 
 9.1098 
 
 9.1138 
 
 9. 1178 
 
 9. 1218 
 
 9.1258 
 
 9.1298 
 
 9. 1338 
 
 9.1S7S 
 
 9. 1418 
 
 9.1458 
 
 9.1498 
 
 9. 1537 
 
 9.1577 
 
 9. 1617 
 
 9.1657 
 
 9.1696 
 
 9. 1736 
 
 9. 1775 
 
 9.1815 
 
 9.1855 
 
 9.1894 
 
 9.1933 
 
 9.1973 
 
 9,2012 
 
 9.2052 
 
 9.2091 
 
 9.2130 
 
 9.2170 
 
 9.i2209 
 
 9.2248 
 
 9.2287 
 
 9.2326 
 
 9.2366 
 
 '9.2404 
 
 9.2443 
 
 9.2482 
 
 9.2521 
 
 9. 2560 
 
 9.2599 
 
 9.2638 
 
 9.2677 
 
 9. 2716 
 
 9.2764 
 
 9.2793 
 
 9.2832 
 
 9.2870 
 
 9.2909 
 
 9.2948 
 
 9. 2986 
 
 9.3025 
 
 9.3063 
 
 9.3102 
 
 9.3140 
 
 9.3179 
 
 9.3217 
 
 9.3255 
 
 9.3294 
 
 9.3332 
 
 9.3370 
 
 9.3408 
 
 9.3447 
 
 9.3485 
 
 9.3523 
 
 9. 3561 
 
 9.3599 
 
 9.3637 
 
 9.3675 
 
 9.3713 
 
 9.3751 
 
 9.3789 
 
 827.. 
 828.. 
 829.. 
 830.- 
 831.. 
 832.. 
 833.. 
 834.. 
 835-. 
 836.. 
 837.. 
 838.. 
 839.. 
 840.. 
 841.. 
 842.. 
 843.. 
 844.. 
 845.. 
 846.. 
 847-. 
 848.. 
 849.. 
 850.. 
 861.. 
 852-. 
 853.. 
 854.. 
 855-. 
 856.. 
 867. 
 858. 
 869. 
 860. 
 861. 
 862- 
 863. 
 864. 
 865- 
 866. 
 867. 
 
 870., 
 871., 
 872., 
 873- 
 874- 
 876- 
 876. 
 877. 
 878. 
 879- 
 880- 
 881. 
 
 885. 
 886- 
 887- 
 
 900- 
 901. 
 902- 
 903. 
 904. 
 
 682276 
 683929 
 685584 
 687241 
 
 663559976 
 665609283 
 567663652 
 
 690561 
 692224 
 
 695566 
 597225 
 
 700569 
 
 702244 
 
 703921 
 
 705600 
 
 707281 
 
 708964 
 
 710649 
 
 712336 
 
 714025 
 
 715716 
 
 717409 
 
 719104 
 
 720801 
 
 722600 
 
 724201 
 
 725904 
 
 727609 
 
 729316 
 
 731025 
 
 732736 
 
 724449 
 
 736164 
 
 737881 
 
 739600 
 
 741321 
 
 743844 
 
 744769 
 
 746496 
 
 748225 
 
 749966 
 
 751689 
 
 763424 
 
 765161 
 
 766900 
 
 758641 
 
 760384 
 
 762129 
 
 763876 
 
 765625 
 
 767376 
 
 769129 
 
 770884 
 
 772641 
 
 774400 
 
 776161 
 
 777924 
 
 779689 
 
 781466 
 
 783226 
 
 784996 
 
 786769 
 
 788544 
 
 790321 
 
 792100 
 
 793881 
 
 795664 
 
 797449 
 
 799236 
 
 801025 
 
 802816 
 
 804609 
 
 806404 
 
 571787000 
 573866191 
 575930368 
 578009537 
 580093704 
 582182875 
 584277056 
 586376253 
 588480472 
 590689719 
 592704000 
 594823321 
 
 599077107 
 601211584 
 603351125 
 605495736 
 607645423 
 609800192 
 611960049 
 614125000 
 616296051 
 618470208 
 620650477 
 622836864 
 625026376 
 627222016 
 629422793 
 631628712 
 633839779 
 636056000 
 638277381 
 640503928 
 612736647 
 644972644 
 647214625 
 
 651714363 
 653972032 
 656234909 
 668603000 
 660776311 
 663054848 
 666338617 
 667627624 
 669921876 
 672221376 
 674526133 
 676836152 
 679151439 
 681472000 
 683797841 
 
 810000 
 811801 
 813604 
 815409 
 817216 
 
 688466387 
 690807104 
 693164125 
 695506456 
 697864103 
 700227072 
 702595369 
 704969000 
 707347971 
 709732288 
 712121957 
 714516984 
 716917376 
 719323136 
 721734273 
 724150792 
 726572699 
 729000000 
 731432701 
 
 736314327 
 738763264 
 
 28.7402 
 
 28. 7676 
 
 28. 7750 
 
 28.7924 
 
 28. 8097 
 
 28. 8271 
 
 28.8444 
 
 28.8617 
 
 28. 8791 
 
 28.8964 
 
 28. 9137 
 
 28. 9310 
 
 28. 9482 
 
 28. 9666 
 
 28.9828 
 
 29. 
 
 29.0172 
 
 29.0345 
 
 29. 0517 
 
 29.0689' 
 
 29.0861 
 
 29. 1033 
 
 29. 1204 
 
 29.1376 
 
 29.1548 
 
 29. 1719 
 
 29.1890 
 
 29.2062 
 
 29.2233 
 
 29.2404 
 
 29.2575 
 
 29. 2746 
 
 29.2916 
 
 29. 3087 
 
 29.3258 
 
 29.3428 
 
 29.3598 
 
 29. 3769 
 
 29.3939 
 
 29.4109 
 
 29.4279 
 
 29.4449 
 
 29.4618 
 
 29.4788 
 
 29.4958 
 
 29.5127 
 
 29. 5296 
 
 29. 5466 
 
 29; 6635 
 
 29.5804 
 
 29. 5973 
 
 29. 6142 
 
 29. 6311 
 
 29.6479 
 
 29.6648 
 
 29. 6816 
 
 29. 6985 
 
 29.7163 
 
 29. 7321 
 
 29. 7489 
 
 29. 7658 
 
 29. 7825 
 
 29.7993 
 
 29. 8161 
 
 29. 8329 
 
 29.8496 
 
 29.8664 
 
 29.8831 
 
 29.8998 
 
 29. 9166 
 
 29.9333 
 
 29.9500 
 
 29. 9666 
 
 29. 9833 
 
 30. 
 
 30.0167 
 
 30.0333 
 
 30. 0500 
 
 30.0666 
 
 9.3827 
 
 9.3865 
 
 9.3902 
 
 9.3940 
 
 9.3978 
 
 9. 4016 
 
 9.4053 
 
 9.4091 
 
 9.4129 
 
 9.4166 
 
 9.4204 
 
 9.4241 
 
 9.4279 
 
 9.4318 
 
 9.4354 
 
 9.4391 
 
 9.4429 
 
 9.4466 
 
 9.4603 
 
 9.4541 
 
 9. 4678 
 
 9.4615 
 
 9.4662 
 
 9.4690 
 
 9.4727 
 
 9.475J 
 
 9.4801 
 
 9.4838 
 
 9.4875 
 
 9. 4913 
 
 9.4949 
 
 9.4986 
 
 9.5023 
 
 9.6060 
 
 9.5097 
 
 9.613t 
 
 9.5171 
 
 9.5207 
 
 9.6244 
 
 9.62^ 
 
 9.5317 
 
 9.5354 
 
 9.6391 
 
 9.6427 
 
 9.5464 
 
 9.5501 
 
 9. 5537 
 
 9.5674 
 
 9.6610 
 
 9.5647 
 
 9.6683 
 
 9.5719 
 
 9.6756 
 
 9.5792 
 
 9. 6828 
 
 9.5865 
 
 9.6901 
 
 9. 5937 
 
 9.5973 
 
 9.6010 
 
 9.6046 
 
 9. 6082 
 
 9. 6118 
 
 9.6154 
 
 9.6190 
 
 9.6226 
 
 9. 6262 
 
 9. 6298 
 
 9.6334 
 
 9.6370 
 
 9.6406 
 
 9.6442 
 
 9.6477 
 
 9.6513 
 
 9.6549 
 
 9.6685 
 
 9.6620 
 
 9. 6656 
 
136 TOPOGEAPHY, MAP KEADING, AND KECONNAISSANCB. 
 
 Table Yl.— Table of squares, cubes, square roots, and cube roots of numbers from 1 to 
 
 1,000 — Continued. 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Square 
 root. 
 
 Cube 
 root. 
 
 
 819025 
 820836 
 822649 
 824464 
 826281 
 828100 
 829921 
 831744 
 833669 
 835396 
 837225 
 839056 
 840889 
 842724 
 844661 
 846400 
 848241 
 860084 
 851929 
 863776 
 855626 
 857476 
 859329 
 861184 
 863041, 
 864900 
 866761 
 868824 
 870489 
 872356 
 874225 
 876096 
 877969 
 879844 
 881721 
 883600 
 885481 
 887364 
 889249 
 891136 
 893025 
 894916 
 896809 
 898704 
 900601 
 902500 
 904401 
 906304 
 
 741217625 
 743677416 
 746142643 
 748613312 
 751089429 
 753571000 
 756058031 
 758550528 
 761048497 
 763551944 
 766060875- 
 768575296 
 771095213 
 773620632 
 776151559 
 778688000 
 781229961 
 783777448' 
 786330467 
 788889024 
 791453125 
 794022776 
 796697983 
 799178762 
 801766089 
 804357000 
 806954491 
 809557568 
 812166237 
 814780504 
 817400375 
 820025866 
 822656963 
 826293672 
 827936019 
 830684000 
 833237621 
 836896888 
 838661807 
 841232384 
 843908625 
 846590636 
 849278123 
 861971392 
 864670349 
 857375000 
 860085351 
 862801408 
 
 30.0832 
 30.0998 
 30.1164 
 30.1330 
 30.1496 
 30.1662 
 30. 1828 
 30. 1993 
 30.2159 
 30.2324 
 30.2490 
 30.2655 
 . 30.2820 
 30.2985 
 30.3160 
 30.3315 
 3013480 
 30.3645 
 30.3809 
 30.3974 
 30.4138 
 30.4302 
 30. 4467 
 30.4631 
 30.4796 
 30.4969 
 30.5123 
 30.6287 
 30.5450 
 30.6614 
 30.5778 
 30.5941 
 30.6105 
 30.6268 
 30. 6431 
 30.6694 
 30.6757 
 30.6920 
 30. 7083 
 30.7246 
 30.7409 
 30.7671 
 30.7734 
 30.7896 
 30.8058 
 30.8221 
 30.8383 
 30.8645 
 
 9.6727 
 9.6763 
 9.6799 
 9.6834 
 9.6870 
 9.6905 
 9. 6941 
 9. 6976 
 9.7012 
 9.7047 
 9.7082 
 9.7118 
 9.7153 
 9.7188 
 9.7224 
 9.7259 
 9.7294 
 9.7329 
 9.7364 
 9.7400 
 9.7435 
 9.7470 
 9. 7505 
 9.7540 
 9.7575 
 9.7610 
 9.7645 
 9.7680 
 9.7715 
 9.7750 
 9.7785 
 9.7819 
 9.7854 
 9.7889 
 9.7924 
 9.7959 
 9.7993 
 9.8028 
 9.8063 
 9.8097 
 9.8132 
 9.8167 
 9.8201 
 9.8236 
 9. 8270 
 9.8305 
 9.8339 
 9.8374 
 
 953 
 
 .908209 
 910116 
 912025 
 913936 
 915849 
 917764 
 919681 
 921600 
 923521 
 925444 
 927369 
 929296 
 931225 
 933156 
 936089 
 937024 
 938961 
 140900 
 942841 
 944784 
 946729 
 948676 
 960625 
 962576 
 954529 
 956484 
 958441 
 960400 
 962361 
 964324 
 966289 
 968266 
 970225 
 972196 
 974169 
 976144 
 978121 
 9801CO 
 982081 
 984064 
 986049 
 988036 
 990025 
 992016 
 994009 
 996004 
 998001 
 
 lOOOOOO 
 
 865523177 
 868260664 
 870983875 
 873722816 
 876467493 
 879217912 
 881974079 
 884736000 
 887503681 
 890277128 
 893056347 
 896841344 
 898632125 
 901428696 
 904231063 
 907039232 
 909853209 
 912673000 
 915498611 
 918330048 
 921167317 
 924010424 
 926859375 
 929714176 
 932574833 
 936441352 
 938313739 
 941192000 
 944076141 
 946966168 
 949862087 
 952763904 
 965671621 
 96858SiSB 
 961504803 
 964430272 
 967361669 
 970029000 
 973242271 
 976191488 
 979146657 
 982107784 
 986074875 
 988047936 
 991026973 
 994011992 
 997002999 
 100000000 
 
 30.8707 
 30.8869 
 30.9031 
 30.9192 
 30.9354 
 30.9516 
 30.9677 
 30.9839 
 31. 
 
 31.0161 
 31.0322 
 31.0483 
 31.0644 
 31.0805 
 31.0966 
 31.1127 
 31.1288 
 31.1448 
 31.1609 
 31.1769 
 31.1929 
 31.2090 
 ,31.2250 
 '31.2410 
 31.2670 
 31.2730 
 31.2890 
 31.3060 
 31.3209 
 31.3369 
 31.3528 
 31.3688 
 31.3847 
 31.4006 
 31.4166 
 . 31.4325 
 31.4484 
 31.4643 
 31.4802 
 31.4960 
 31.5119 
 31.5278 
 31.5436 
 31.6595 
 31.5753 
 31.5911 
 31.6070 
 31.6228 
 
 9.8408 
 
 906 
 
 954 
 
 9.8443 
 
 907 
 
 966 
 
 9.8477 
 
 908 
 
 966 
 
 937 
 
 9.8511 
 
 
 9.8546 
 
 910 
 
 958 
 
 9.8580 
 
 
 959 
 
 9.8614 
 
 912 
 
 960 
 
 9.8648 
 
 913 
 
 961 ...„ 
 
 962 
 
 9.8683 
 
 
 9.8717 
 
 915 
 
 963 
 
 9.8751 
 
 
 964 
 
 9.8785 
 
 917 
 
 965 
 
 9.8819 
 
 918 
 
 966 
 
 9.8854 
 
 919 
 
 967 
 
 9.8888 
 
 920 
 
 968 
 
 9.8922 
 
 921 
 
 969 
 
 970 
 
 9.8956 
 
 
 9.8990 
 
 923 
 
 971 
 
 9.9024 
 
 
 972 
 
 9.9068 
 
 925 
 
 973 
 
 9.9092 
 
 926 
 
 974 
 
 9.9126 
 
 
 976 
 
 9.9160 
 
 928 
 
 976 
 
 9.9194 
 
 
 977 
 
 9.9227 
 
 
 978 
 
 9.9261 
 
 931 
 
 979 
 
 9.9295 
 
 
 980 
 
 9.9329 
 
 933 
 
 981 
 
 9.9363 
 
 
 982 
 
 9.9396 
 
 935 
 
 983... 
 
 9.9430 
 
 936 . ... 
 
 984 
 
 9.9464 
 
 937 
 
 985 
 
 9. 9497 
 
 938 
 
 986 
 
 9.9531 
 
 
 987 
 
 9.956S 
 
 940 
 
 988 
 
 9. 9598 
 
 
 989 
 
 9.9632 
 
 042 
 
 990 
 
 9.9666 
 
 943 
 
 991 
 
 9.9699 
 
 
 992 
 
 9. 9733 
 
 945 
 
 993 
 
 9.9766 
 
 
 994 
 
 9.9800 
 
 947 
 
 996 
 
 9.9833 
 
 948 
 
 996 
 
 9.9866 
 
 
 997 
 
 9.9900 
 
 950 
 
 998 
 
 9.9933 
 
 951 
 
 999 
 
 9 9967 
 
 952 
 
 1000 
 
 10. 
 
 
 
 
 213. To find the square root of a decimal fraction or mixed number 
 from the foregoing table, multiply by 100 or by 10,000 and find the 
 product in the cokimn of squares. The corresponding number in 
 the first colunan, with the decimal point one or two places to the left 
 is the desired root. 
 
 For the cube root of a similar number, multiply by 1,000 or by 
 1,000,000, and find the nearest number in column of cubes. The 
 corresponding nurnber in the first column, with the decimal point 
 one or two places to the left, is the desired root. 
 
 Examples ; Required the square root of 5.246. 
 
 Multiply by 100- the result is 524, which found in column of squares 
 is opposite 23 in the column of numbers. Movtae the decimal point 
 one place to the left to correspond with the multiplication by 100, 
 gives 2.3 for the desired square root, to the first place of decimals 
 and hence approximate only. Second: Multiply by 10,000; the 
 result is 52,460, which found in the column of squares is opposite 
 
TOPOGEAPHY, MAP READING, AND EECONNAISSANCE. 137 
 
 ?n ^+S tu ?^''°^'' Of , numbers Moving the decimal point two places 
 ?e 9Q i-l 907espond to the multipUcation bv 10,000, the result 
 IS 2.29, which IS the desu-ed root to the second place of decimals. 
 
 Eequured the cube root of 5.246. Multiply by 1,000, giving 5,246, 
 which found m the column of cubes is opposite the number 17 in 
 the first column. Moving the decmaal point one place to the left 
 to correspond to the multipUcation by 1,000 gives 1.7, which is the 
 iT,fnl^"^® ™.°i^° °^® decimal place. Igain, multiplying by 
 1,000,000 gives 5,246,000, which found in the column of cubes 13 
 opposite the number 174 in the column of numbers. Moving the 
 decimal point two places to the. left to correspond with the multi- 
 plication by 1,000,000, gives the number 1.74, which is the desired 
 cube root correct to two places of decimals. 
 
 To find the square root or cube root of a number greater than 1,000, 
 find the nearest number in the column of squares or cubes and take 
 the corresponding number in the first column, which will be correct 
 for the number of figures it contains. 
 
 For the fourth root, take the square root of the square root. For 
 the sixth root, the square root of the cube root, or the cube root of 
 the square root. Higher roots, the indices of which can be factored 
 in 3's and 2's, may be taken in the same way. 
 
 CnJCULAR FUNCTIONS. 
 
 214. Those most jused are shown graphically in fig. 68. They bear 
 a definite relation to the radius of a circle in which they are drawn. 
 When the radius is unity, functions are called natural, as natural 
 sine, natural tangent, etc. Their values are given in Table XVI for 
 each 10' of arc. The tabulated values are ratios of the several func- 
 tions to the radius and if any length, expressed in any unit, considered 
 as a rad ius, be multiplied by a tabular number, the result wiU be the cor- 
 responding function of the circle of the given radius. The table gives 
 values from to 90°. For greater angles, use the following relations: 
 Subtract the given angle from 180° or 360°, or subtract 180° from the 
 angle, as may be required, to leave a remainder of 90° or less. Take 
 out the required function of the remainder, which is also that of the 
 given angle. 
 
 Interpolation for values not in the table may be done approxi- 
 mately Dy taking the proportional amount of the difference between 
 two consecutive values. Thus, for the sine of 28° 43' take the sine 
 of 28° 40' plus -^ of the difference between sine 28° 40' and sine 
 28° 50'. 
 
138 TOPOGBAPHY, MAP READING, AND RECONNAISSANCE. 
 
 Table YII.— Natural sines and tangents to a radius 1. 
 
 Arc. 
 
 Sine. 
 
 Tang. 
 
 Cotang. 
 
 Cosine. 
 
 
 ° / 
 
 
 
 JT 
 
 
 o / 
 
 .00 
 
 .0000000 
 
 .000000 
 
 Infinite. 
 
 1.0000000 
 
 90 00 
 
 10 
 
 .0029089 
 
 .002908 
 
 343. 7737 
 
 .9999958 
 
 50 
 
 20 
 
 . 0058177 
 
 .005817 
 
 171. 8854 
 
 . 9999831 
 
 40 
 
 30 
 
 .0087265 
 
 .008726 
 
 114.S886 
 
 .9999619 
 
 30 
 
 40 
 
 .0116353 
 
 .011636 
 
 85.93979 
 
 .9999323 
 
 20 
 
 50 
 
 .0145439 
 
 .014545 
 
 68.75008 
 
 .9998942 
 
 10 
 
 1 00 
 
 .0174524 
 
 .017455 
 
 S7. 28996 
 
 .9998477 
 
 89 00 
 
 10 
 
 .0203608 
 
 .020365 
 
 49. 10388 
 
 .9997927 
 
 50 
 
 20 
 
 .0232690 
 
 .023275 
 
 42. 96407 
 
 . 9997292 
 
 40 
 
 30 
 
 .0261769 
 
 .026185 
 
 38. 18845 
 
 .9996573 
 
 30 
 
 40 
 
 .0290847 
 
 .029097 
 
 34.36777 
 
 .9995770 
 
 20 
 
 50 
 
 .0319922 
 
 .032008 
 
 31.24157 
 
 .9994881 
 
 10 
 
 2 00 
 
 .0348995 
 
 .034920 
 
 28.63625 
 
 .9993908 
 
 88 00 
 
 10- 
 
 .0378065 
 
 .037833 
 
 26r43160 
 
 .9992851 
 
 60 
 
 20 
 
 .0407131 
 
 .040746 
 
 24. 54175 
 
 . 9991709 
 
 40 
 
 30 
 
 .0436194 
 
 .043660 
 
 22.90376 
 
 .9990482 
 
 30 
 
 40 
 
 .0465253 
 
 .046575 
 
 21. 47040 
 
 .9989171 
 
 20 
 
 50 
 
 .0494308 
 
 .049491 
 
 20.2ft5.'i5 
 
 .9987775 
 
 10 
 
 3 CO 
 
 .0523360 
 
 .052407 
 
 iq. 08113 
 
 .9986295 
 
 87 00 
 
 10 
 
 .0552406 
 
 .055325 
 
 18.07497 
 
 ,9984731 
 
 60 
 
 20 
 
 .0581448 
 
 .058243 
 
 17. 16933 
 
 .9983082 
 
 40 
 
 30 
 
 . 0610485 
 
 .061162 
 
 16.34985 
 
 .9981348 
 
 30 
 
 40 
 
 .0639617 
 
 .064082 
 
 15.60478 
 
 .9979530 
 
 20 
 
 SO 
 
 .0668544 
 
 .067004 
 
 14.924« 
 
 .9977627 
 
 10 
 
 4 00 
 
 .0697565 
 
 .069926 
 
 14.30066 
 
 .9975641 
 
 86 00 
 
 10 
 
 ;0726S80 
 
 .072860 
 
 13.72673 
 
 .9973569 
 
 50 
 
 20, 
 
 .0755589 
 
 .075775 
 
 13: 19688 
 
 .9971413 
 
 40 
 
 30 
 
 .0784691 
 
 .078701 
 
 12. 70620 
 
 .9969173 
 
 30 
 
 40 
 
 .0813587 
 
 .081629 
 
 12.25050 
 
 .9966849 
 
 20 
 
 SO 
 
 .0842576 
 
 .084558 
 
 11.82616 
 
 .9964440 
 
 10 
 
 5 00 
 
 .0871557 
 
 .087488 
 
 11.43006 
 
 .9961947 
 
 85 00 
 
 10 
 
 .0900532 
 
 .090420 
 
 11.05943 
 
 .9959370" 
 
 50 
 
 20 
 
 .0929499 
 
 .093354 
 
 10. 71191 
 
 .0056708 
 
 40 
 
 30 
 
 .0958458 
 
 .096289 
 
 10.38539 
 
 .9953962 
 
 30 
 
 40 
 
 .0987408 
 
 .099225 
 
 10.07803 
 
 .9961132 
 
 20 
 
 50 
 
 .1016351 
 
 .102164 
 
 9.788173 
 
 .9948217 
 
 10 
 
 6 00 
 
 .1045285 
 
 .105104 
 
 9.514364 
 
 .9945219 
 
 84 00 
 
 10 
 
 .1074210 
 
 .108046 
 
 9.265303 
 
 .9942136 
 
 50 
 
 20 
 
 .1103128 
 
 .110989 
 
 9.009826 
 
 .9938969 
 
 40 
 
 30 
 
 .1132032 
 
 .113935 
 
 8.776887 
 
 .9935719 
 
 30 
 
 40 
 
 .1160929 
 
 .116883 
 
 8.555546 
 
 .9932384 
 
 20 
 
 SO 
 
 .1189816 
 
 .119832 
 
 8.344955 
 
 .9928965 
 
 10 
 
 7 00 
 
 .1218693 
 
 .122784 
 
 8. 144346 
 
 .9925462 
 
 83 00 
 
 10 
 
 .1247560 
 
 .125738 
 
 7. 953022 
 
 .9921874 
 
 SO 
 
 20 
 
 . 1276416 
 
 . 128694 
 
 7.770350 
 
 .9918204 
 
 40 
 
 30 
 
 . 1305262 
 
 .131652 
 
 7.595754 
 
 .9914449 
 
 30 
 
 40 
 
 .1334096 
 
 . 134612 
 
 7.428706 
 
 .9910610 
 
 20 
 
 50 
 
 .1362919 
 
 .137575 
 
 7.268725 
 
 .9906687 
 
 10 
 
 
 Cosine. 
 
 Cotang. 
 
 Tang. 
 
 Sine. 
 
 Arc. 
 
TOPOGBAPHT, MAP BEADING, AND EECONNAISSANCE. 
 Table VII. — Natural sines and t(mgem,ts — Cointiiiued, 
 
 139 
 
 Arc. 
 
 Slna. 
 
 Tang. 
 
 Cotang. 
 
 Cosine. 
 
 
 / 
 
 8 00 
 
 .1391731 
 
 . 140540 
 
 7.115369 
 
 ^ .9802681 
 
 o t 
 
 82 00 
 
 10 
 
 . 14I20S31 
 
 .143608 
 
 6-968233 
 
 .9898590 
 
 50 
 
 20 
 
 .1478094 
 
 .148478 
 
 & 826943 
 
 .9894416 
 
 40 
 
 30 
 
 . Iffl451 
 
 6'. 691 166 
 
 .9890159 
 
 30 
 
 40 
 
 .1306867 
 
 .,1524ffl 
 
 6.560553 
 
 .9885817 
 
 20 
 
 50 
 
 .1536607 
 
 .I554()4 
 
 6. 434842 
 
 .9881382 ' 
 
 10 
 
 9 00 
 
 . 1564345 
 
 .158384 
 
 6.313751 
 
 .9876883 
 
 81 00 
 
 10 
 
 .1593^9 
 
 .161367 
 
 8.1»a)27 
 
 .9872291 
 
 60 
 
 20 
 
 .IffiI779 
 
 .164353 
 
 & 084438 
 
 .9867615 
 
 40 
 
 30 
 
 . 1650476 
 
 .167342 
 
 6. 976764 
 
 .9862856 " 
 .9858013 
 
 3,0 
 
 40 
 
 .1679159 
 
 .170334 
 
 5.870904 
 
 io 
 
 60 
 
 .1707828 
 
 .17^39 
 
 5.769368 
 
 .9853087 
 
 10 
 
 10 00 
 
 . 1736482 
 
 .176327 
 
 5.671281 
 
 .9848078 
 
 80 00 
 
 10 
 
 .17ffil21 
 
 .17K27 
 
 5.576378 
 
 .9842985 
 
 60 
 
 20 
 
 . 1793746 
 
 .182331 
 
 5.484505 
 
 .9837808 
 
 40 
 
 30 
 
 .1822365 
 
 .185339 
 
 5.396517 
 
 .9832549 
 
 30 
 
 40 
 
 .1850949 
 
 .188349 
 
 5.309279 
 
 .9827206 
 
 20 
 
 50 
 
 .1879528 
 
 .191363 
 
 5.226664 
 
 .9821781 
 
 10 
 
 11 00 
 
 .1908090 
 
 .194380 
 
 S. 144554 
 
 .9816272 
 
 79 00 
 
 10 
 
 .19^636 
 
 .197400 
 
 5.065836 
 
 .9810680 
 
 50 
 
 20 
 
 .I9«i5I66 
 
 .200424 
 
 4.988402 
 
 .9805005 
 
 40 
 
 30 
 
 .1«9^79 
 
 .203452 
 
 4.915157 
 
 .9799247 
 
 30 
 
 40 
 
 .2022176 
 
 .206483 
 
 4.843004 
 
 .9793406 
 
 20 
 
 SO 
 
 .2050655 
 
 .200.518 
 
 4.772856 
 
 .9787483 
 
 10 
 
 12 00 
 
 .2079117 
 
 .212656 
 
 4.704630 
 
 .9781476 
 
 78 00 
 
 10 
 
 .2107561 
 
 .215698 
 
 4.^8245 
 
 .9775386 
 
 50 
 
 20 
 
 .2135988 
 
 .218644 
 
 4.573628 
 
 .9769215 
 
 40 
 
 30 
 
 .2164396 
 
 ^221694 
 
 4.510708 
 
 .9762960 
 
 30 
 
 40 
 
 .2192786 
 
 .224748 
 
 4.449418 
 
 .W56823 
 
 20 
 
 50 
 
 .2^1168 
 
 .227806 
 
 4.389694 
 
 .9750203 
 
 10 
 
 13 00 
 
 .2249511 
 
 .230868 
 
 4.3ai475 
 
 .9743701 
 
 77 00 
 
 10 
 
 .2277844 
 
 . 233834 
 
 4.274706 
 
 .9737116 
 
 50 
 
 20 
 
 .2306159 
 
 .237004 
 
 4.219331 
 
 .9730449 
 
 40 
 
 30 
 
 .2334464 
 
 .-man 
 
 4.165299 
 
 .9723899 
 
 30 
 
 40 
 
 .2362729 
 
 .243157 
 
 4. 112561 
 
 .971^67 
 
 20 
 
 50 
 
 .2390984 
 
 .2462^1 
 
 4.061070 
 
 .^09953 
 
 10 
 
 14 00 
 
 .2419219 
 
 .249328 
 
 4.010780 
 
 .9702957 
 
 76 00 
 
 10 
 
 .2447433 
 
 .25^20 
 
 3.961661 
 
 .9685879 
 
 50 
 
 ; 20 
 
 .2475627 
 
 .255516 
 
 3.913642 
 
 .9688719 
 
 40 
 
 30 
 
 .2503800 
 
 .258617 
 
 3. 866713 
 
 .9681476 
 
 30 
 
 40 
 
 .2531962 
 
 .261723 
 
 3.820828 
 
 .9674152 
 
 20 
 
 1 50 
 
 .2660082 
 
 .264833 
 
 3.775961 
 
 .9666746 
 
 10 
 
 15 QO 
 
 .2588190 
 
 .267949 
 
 3.732060 
 
 .9659268 
 
 75 00 
 
 10 
 
 .2616277 
 
 .271869 
 
 3.689092 
 
 .9661689 
 
 50 
 
 : 20 
 
 .2644342 
 
 .274194 
 
 3.647046 
 
 .9644037 
 
 40 
 
 30 
 
 .2672384 
 
 .277324 
 
 3.605883 
 
 .9636306 
 
 30 
 
 40 
 
 .2700403 
 
 .280469 
 
 3.565574 
 
 .9628490 
 
 20 
 
 60 
 
 .2728400 
 
 .283599 
 
 3.526093 
 
 .9620594 
 
 10 
 
 
 Cosine. 
 
 Cotang. 
 
 Tang. 
 
 Sine. 
 
 Arc. 
 
140 TOPOGEAPHY, MAP KEAIUNG, AND EECONNAISSANOE. 
 
 Table VII. — Natural sines and tangents — Continued. 
 
 Arc. 
 
 Sine. 
 
 Tang. 
 
 Cotang. 
 
 Cosine. 
 
 
 16 00 
 
 .2756374 
 
 .268745 
 
 3.487414 
 
 .9612617 
 
 74 00 
 
 10 
 
 .2784324 
 
 .289896 
 
 3.449512 
 
 . 9604558 
 
 SO 
 
 20 
 
 . 2812251 
 
 .293052 
 
 3.412362 
 
 .9596418 
 
 40 
 
 30 
 
 .2840153 
 
 .296213 
 
 3.375943 
 
 .9588197 
 
 30 
 
 40 
 
 .2888032 
 
 .299380 
 
 3.340232 
 
 .9579895 
 
 20 
 
 50 
 
 .2895887 
 
 . 302552 
 
 3.305209 
 
 .9571512 
 
 10 
 
 17 00 
 
 .2923717 
 
 .305730 
 
 3.270852 
 
 .9563048 
 
 73 00 
 
 10 
 
 .2951522 
 
 .308914. 
 
 3. 237143 
 
 .9554502 
 
 50 
 
 20 
 
 .2979303 
 
 .312103 
 
 3.204063 
 
 .9545876 
 
 40 
 
 30 
 
 .3007058 
 
 .315298 
 
 3. 171594 
 
 .9537170 
 
 30 
 
 40 
 
 .3034788 
 
 .318499 
 
 3. 139719 
 
 .9528382 
 
 20 
 
 50 
 
 .3062492 
 
 .321706 
 
 -3.108421 
 
 .9519514 
 
 10 
 
 18 00 
 
 .3090170 
 
 .324919 
 
 3.077683 
 
 .9510565 
 
 72 00 
 
 10 
 
 .3117822 
 
 . 328138 
 
 3. 047491 
 
 . 9501536 
 
 50 
 
 20 
 
 .3145448 
 
 .331363 
 
 3.017830 
 
 .9492426 
 
 40 
 
 30 
 
 .3173047 
 
 . 334595 
 
 2. 988685 
 
 . 9483237 
 
 30 
 
 40 
 
 .3200619 
 
 .337833 
 
 2.960042 
 
 .9473966 
 
 20 
 
 60 
 
 .3228164 
 
 .341077 
 
 2.931888 
 
 .9464616 
 
 10 
 
 19 00 
 
 .3255682 
 
 .344327 
 
 2.904210 
 
 .9455186 
 
 71 00 
 
 10 
 
 .3283172 
 
 .347584 
 
 2. 876997 
 
 .9445675 
 
 50 
 
 20 
 
 .3310634 
 
 .350848 
 
 2. 850234 
 
 .9436085 
 
 40 
 
 30 
 
 .3338069 
 
 .354118 
 
 2. 823912 
 
 .9426416 
 
 30 
 
 40 
 
 .3365475 
 
 .357395 
 
 2. 798019 
 
 .9416665 
 
 20 
 
 50 
 
 .3392852 
 
 .360679 
 
 2.772544 
 
 .9406835 
 
 10 
 
 20 00 
 
 .3420201 
 
 .363970 
 
 2.747477 
 
 .9396926 
 
 70 CO 
 
 10 
 
 .3447521 
 
 .367268 
 
 2. 722807 
 
 . 9386938 
 
 50 
 
 20 
 
 .3474812 
 
 .370572 
 
 2. 698525 
 
 .9376869 
 
 40 
 
 30 
 
 .3502074 
 
 .373884 
 
 2.674621 
 
 .9366722 
 
 30 
 
 40 
 
 .3529306 
 
 .377203 
 
 2. 651086 
 
 .9356495 
 
 20 
 
 50 
 
 .3556508 
 
 .380530 
 
 2.627912 
 
 .9346189 
 
 10 
 
 21 00 
 
 .3583679 
 
 .383864 
 
 2. 605089 
 
 .9336804 
 
 69 00 
 
 10 
 
 .3610821 
 
 .387205 
 
 2. 582609 
 
 .9325340 
 
 60 
 
 20 
 
 .3637932 
 
 .390554 
 
 2.560464 
 
 .9314797 
 
 40 
 
 30 
 
 .3665012 
 
 .393910 
 
 2.538647 
 
 .9304176 
 
 30 
 
 40 
 
 .3692061 
 
 .397274 
 
 2.517150 
 
 .9293475 
 
 20 
 
 50 
 
 .3719079 
 
 .400646 
 
 2.495966 
 
 .9282696 
 
 10 
 
 • 22 00 
 
 .3746066 
 
 .404026 
 
 2.475086 
 
 .9271839 
 
 68 00 
 
 10 
 
 .3773021 
 
 .407413 
 
 2.454506 
 
 . 9260902 
 
 60 
 
 20 
 
 .3799944 
 
 .410809 
 
 2.434217 
 
 .9249888 
 
 40 
 
 30 
 
 .3826834 
 
 .414213 
 
 2.414213 
 
 .9238795 
 
 30 
 
 40 
 
 . 3853693 
 
 .417625 
 
 2. 394488 
 
 .9227624 
 
 20 
 
 50 
 
 .3880518 
 
 .421046 
 
 2.375037 
 
 .9216375 
 
 10 
 
 .23 00 
 
 .3907311, 
 
 .424474 
 
 2.355852 
 
 .9205049 
 
 67 00 
 
 10 
 
 .3934071 
 
 .427912 
 
 2.336928 
 
 .9193644 
 
 50 
 
 20 
 
 .3960798 
 
 .431357 
 
 2.318260 
 
 .9182161 
 
 40 
 
 30 
 
 .3987491 
 
 .434812 
 
 2. 299842 
 
 .9170601 
 
 30 
 
 40 
 
 .4014150 
 
 .438275 
 
 2.281669 
 
 . 9168963 
 
 20 
 
 50 
 
 .4040775 
 
 .441747 
 
 2.263735 
 
 .9147247 
 
 10 
 
 
 Cosine. 
 
 Cotang. 
 
 Tang. 
 
 Sine. 
 
 Arc. 
 
TOPOGEAPHY, MAP EEADING, AND BECONNAISSANCE. 141 
 
 Table VII. — Natural sines and tangents — Continued. 
 
 Arc. 
 
 Sine. 
 
 Tang. 
 
 Cotang. 
 
 Cosine. 
 
 
 o / 
 
 24 00 
 
 .4067366 
 
 .445228 
 
 2.246036 
 
 .9135455 
 
 66 00 
 
 10 
 
 .4093923 
 
 .448718 
 
 2. 228567 
 
 .9123584 
 
 50 
 
 20 
 
 .4120445 
 
 .452217 
 
 2.211323 
 
 .9111637 
 
 40 
 
 30 
 
 .4146932 
 
 .455726 
 
 2. 194299 
 
 .9099613 
 
 30 
 
 '40 
 
 .4173385 
 
 .459243 
 
 2. 177492 
 
 .9087511 
 
 20 
 
 SO 
 
 .4199801 
 
 .462771 
 
 2. 160895 
 
 .9075333 
 
 10 
 
 25 00 
 
 .4226183 
 
 .466307 
 
 2. 144506 
 
 .9063078 
 
 65 00 
 
 10 
 
 .4252528 
 
 .469863 
 
 2. 128321 
 
 .9050746 
 
 50 
 
 20 
 
 .4278838 
 
 .473409 
 
 2. 112334 
 
 .9038338 
 
 40 
 
 30 
 
 .4305111 
 
 .476975 
 
 2.096543 
 
 .9025853 
 
 30 
 
 40 
 
 .4331348 
 
 . 480551 
 
 2.080943 
 
 .9013292 
 
 20 
 
 60 
 
 .4367548 
 
 .484136 
 
 2.065531 
 
 .9000654 
 
 10 
 
 26 00 
 
 .4383711 
 
 .487732 
 
 2.060303 
 
 .8987940 
 
 64 00 
 
 10 
 
 .4409838 
 
 . 491338 
 
 2. 035256 
 
 .8975151 
 
 50 
 
 20 
 
 .4435927 
 
 .494954 
 
 2. 020386 
 
 .8962285 
 
 40 
 
 30 
 
 .4461978 
 
 .498681 
 
 2.005689 
 
 .8949344 
 
 , 30 
 
 40 
 
 .4487992 
 
 . 602218 
 
 1. 991163 
 
 .8936326 
 
 20 
 
 50 
 
 .4513967 
 
 .506866 
 
 1. 976805 
 
 . 8923234 
 
 10 
 
 27 00 
 
 .4539905 
 
 .509525 
 
 1.962610 
 
 .8910065 
 
 63 00 
 
 10 
 
 . 4565804 
 
 . 513195 
 
 1. 948577 
 
 .8896822 
 
 50 
 
 20 
 
 .4591666 
 
 .516875 
 
 1. 934702 
 
 .8883503 
 
 40 
 
 30 
 
 . 4617486 
 
 .520567 
 
 1. 920982 
 
 .8870108 
 
 30 
 
 40 
 
 .4643269 
 
 .524269 
 
 1. 907414 
 
 . 8866639 
 
 20 
 
 50 
 
 .4669012 
 
 .527983 
 
 1.893997 
 
 .8843095 
 
 10 
 
 28 CO 
 
 .4694716 
 
 .531709 
 
 1.880726 
 
 .8829476 
 
 62 00 
 
 10 
 
 .4720380 
 
 .535446 
 
 1. 867600 
 
 .8815782 
 
 60 
 
 20 
 
 .4746004 
 
 .639195 
 
 1. 854615 
 
 .8802014 
 
 40 
 
 30 
 
 .4771588. 
 
 .542955 
 
 1.841770 
 
 .8788171 
 
 30 
 
 40 
 
 . 4797131 
 
 .546728 
 
 1.829062 
 
 .8774254 
 
 20 
 
 SO 
 
 .4822634 
 
 .550612 
 
 1. 816489 
 
 . 8760263 
 
 10 
 
 29 00 
 
 .4848096 
 
 .554309 
 
 1. 804047 
 
 . 8746197 
 
 61 00 
 
 10 
 
 .4873517 
 
 .658117 
 
 1. 791736 
 
 .8732058 
 
 50 
 
 20 
 
 .4898897 
 
 . 561939 
 
 1. 779552 
 
 . 8717844 
 
 40 
 
 30 
 
 . 4924236 
 
 .565772 
 
 1.767494 
 
 .8703557 
 
 30 
 
 40 
 
 .4949632 
 
 .569619 
 
 1.765659 
 
 .8689196 
 
 20 
 
 SO 
 
 .4974787 
 
 .573478 
 
 1.743745 
 
 .8674762 
 
 10 
 
 30 00 
 
 .5000000 
 
 .577360 
 
 1.732050 
 
 .8660254 
 
 60 00 
 
 10 
 
 . 6025170 
 
 .681235 
 
 1.720473 
 
 .8645673 
 
 50 
 
 20 
 
 . 5050298 
 
 .685133 
 
 1. 709011 
 
 .86.31019 
 
 40 
 
 30 
 
 .5075384 
 
 . 589045 
 
 1.697663 
 
 .8616292 
 
 30 
 
 40 
 
 .5100426 
 
 .592969 
 
 1.686426 
 
 . 8601491 
 
 20, 
 
 50 
 
 .5125425 
 
 .596908 
 
 1.675298 
 
 .8586619 
 
 10 
 
 31 00 • 
 
 .5150381 
 
 .600860 
 
 ■ 1.664279 
 
 ,8571673 
 
 59 00 
 
 10 
 
 , 5175293 
 
 .604826 
 
 1.663366 
 
 .8556655 
 
 SO 
 
 20 
 
 .5200161 
 
 .608806 
 
 1.642567 
 
 .8541664 
 
 40 
 
 30 
 
 .5224986 
 
 .612800 
 
 1.631851 
 
 .8526402 
 
 30 
 
 40 
 
 .5249766 
 
 .616809 
 
 1.621246 
 
 ,8511167 
 
 20 
 
 SO 
 
 .5274502 
 
 .620832 
 
 1.610741 
 
 .8495860 
 
 10 
 
 
 Cosine. 
 
 Cotang. 
 
 Tang. 
 
 Sine. 
 
 Arc. 
 
142 T0PGGE4.PHY, MAP EEADINO, AND RECONNAISSANCE. 
 
 Table VII. — Natural sines and tangents — Continued. 
 
 Arc. 
 
 Sine. 
 
 Tang. 
 
 Cotang. 
 
 Cosine. 
 
 
 32 00 
 
 .5299193 
 
 .624869 
 
 1.600334 
 
 .8480481 
 
 o r 
 
 58 00 
 
 10 
 
 .532J839 
 
 .628921 
 
 1.690023 
 
 .8465030 
 
 50 
 
 20 
 
 .5348440 
 
 .63^88 
 
 1.579807 
 
 .8449508 
 
 40 
 
 30 
 
 .5372996 
 
 .637070 
 
 1.569685 
 
 .8433914 
 
 30 
 
 40 
 
 .5397507 
 
 .641167 
 
 1.569656 
 
 .8418249 
 
 20 
 
 50 
 
 .5421971 
 
 .645279 
 
 1.549715 
 
 .8402513 
 
 10 
 
 33 00 
 
 .5446390 
 
 .649407 
 
 1.539865 
 
 .8386706 
 
 57 00 
 
 10 
 
 .5470763 
 
 .653651 
 
 1.530102 
 
 .8370827 
 
 50 
 
 20 
 
 .5495090 
 
 .657710 
 
 1.520426 
 
 . 8354878 
 
 40 
 
 30 
 
 .5519370 
 
 .661885 
 
 1.510835 
 
 .8338858 
 
 30 
 
 40 
 
 .5543603 
 
 .666076 
 
 1.501328 
 
 .8322768 
 
 20 
 
 50 
 
 .6567790 
 
 -670284 
 
 1.491903 
 
 .8306607 
 
 10 
 
 34 00 
 
 .5591929 
 
 .674508 
 
 1.482661 
 
 .8290376 
 
 56 00 
 
 10 
 
 .5616021 
 
 .678749 
 
 1.473298 
 
 .82T4074 
 
 50 
 
 20 
 
 .5640066 
 
 .683006 
 
 1-464114 
 
 .8257703 
 
 40 
 
 30 
 
 .5664062 
 
 .687281 
 
 1.456009 
 
 .8241262 
 
 30 
 
 40 
 
 .5688011 
 
 .691572 
 
 1.446980 
 
 .8224751 
 
 20 
 
 50 
 
 .5711912 
 
 .695881 
 
 1.437026 
 
 .8208170 
 
 10 
 
 35 00 
 
 .5735764 
 
 .700207 
 
 i. 428148 
 
 .8191520 
 
 55 00 
 
 10 
 
 .6759568 
 
 .704561 
 
 1.419342 
 
 .8174801 
 
 60 
 
 20 
 
 .5783323 
 
 .708913 
 
 1.410609 
 
 .8158013 
 
 40 
 
 30 
 
 .5807030 
 
 .713293 
 
 1.401948 
 
 .8141155 
 
 30 
 
 40 
 
 .5830687 
 
 .717691 
 
 1.393357 
 
 .8124229 
 
 20 
 
 50 
 
 .5854294 
 
 .722107 
 
 1.384836 
 
 .8107234 
 
 10 
 
 36 00 
 
 .5877863 
 
 .726542 
 
 1.376381 
 
 .8090170 
 
 54 00 
 
 10 
 
 .5901361 
 
 .730996 
 
 1.367996 
 
 .8073038 
 
 50 
 
 20 
 
 .5924819 
 
 .735469 
 
 1.369676 
 
 .8065837 
 
 40 
 
 30 
 
 .5948228 
 
 .759961 
 
 1.351422 
 
 .8038569 
 
 30 
 
 40 
 
 .5971586 
 
 .744472 
 
 1.343233 
 
 .8021232 
 
 20 
 
 60 
 
 .5994893 
 
 .749003 
 
 1.335107 
 
 .8003827 
 
 10 
 
 37 00 
 
 .6018160 
 
 .753554 
 
 1.327044 
 
 .7986355 
 
 53 00 
 
 10 
 
 .6041356 
 
 .768124 
 
 1.319044 
 
 .7968815 
 
 60 
 
 20 
 
 .«)64511 
 
 .782715 
 
 1.311104 
 
 .7951208 
 
 40 
 
 30 
 
 .6087614 
 
 .767327 
 
 1.303225 
 
 .7933S33 
 
 30 
 
 40 
 
 .6110666 
 
 .771958 
 
 1.295405 
 
 .7916792 
 
 20 
 
 50 
 
 .6133666 
 
 .776611 
 
 1.287644 
 
 .7897983 
 
 10 
 
 38 00 
 
 .6166615 
 
 .781285 
 
 1.279941 
 
 .7880108 
 
 52 00 
 
 10 
 
 .6179511 
 
 .786980 
 
 1.272295 
 
 .7862165 
 
 50 
 
 20 
 
 .6202365 
 
 .790697 
 
 1.264706 
 
 .7844157 
 
 40 
 
 30 
 
 .6225146 
 
 .795435 
 
 1.267172 
 
 .7826082 
 
 30 
 
 40 
 
 .6247885 
 
 .800196 
 
 1.249693 
 
 .7807940 
 
 20 
 
 50 
 
 .6270671 
 
 .804979 
 
 1.242268 
 
 .7789733 
 
 10 
 
 39 00 
 
 .6293204 
 
 .809784 
 
 1.234897 
 
 .7771460 
 
 51 00 
 
 10 
 
 .6315784 
 
 .814611 
 
 1.227578 
 
 .77^121 
 
 60 
 
 20 
 
 .6338310 
 
 .819462 
 
 1.220312 
 
 .T?34716 
 
 40 
 
 30 
 
 .6360782 
 
 .824336 
 
 r. 213097 
 
 .7716246 
 .76977{0 
 
 30 
 
 40 
 
 .6383201 
 
 .829233 
 
 1.206932 
 
 20 
 
 50 
 
 .6405566 
 
 .834154 
 
 1.198818 
 
 .7679110 
 
 10 
 
 
 Cosine. 
 
 Cotang. 
 
 Tang. 
 
 Sine. 
 
 Arc. 
 
TOPOGEAPHT, MAP EEADING, AND IffiCONNAISSANCE. 
 Table VII. — Natural sines and tangents — Continued. 
 
 143 
 
 Arc. 
 
 Sine. 
 
 Tang. 
 
 Go tang. 
 
 Cosine. 
 
 
 40 00 
 
 . 6427876 
 
 .839099 
 
 1. 191753 
 
 .7660444 
 
 O f 
 
 50 00 
 
 10 
 
 . 6450132 
 
 .844068 
 
 1. 184737 
 
 .7641714 
 
 50 
 
 20 
 
 .6472334 
 
 . 849062 
 
 1. 177769 
 
 . 7622919 
 
 40 
 
 30 
 
 .6494480 
 
 .854a80 
 
 1. 170849 
 
 . 7604060 
 
 30 
 
 40 
 
 .6516572 
 
 .859124 
 
 1. 163976 
 
 . 7585136 
 
 20 
 
 50 
 
 .6538609 
 
 .864192 
 
 1.157149 
 
 . 7666148 
 
 10 
 
 41 00 
 
 .6560590 
 
 .869286 
 
 1.150368 
 
 . 7547096 
 
 49 00 
 
 10 
 
 .6582516 
 
 .874406 
 
 1.143632 
 
 .7627980 
 
 50 
 
 20 
 
 .6004386 
 
 .879552 
 
 1. 136941 
 
 . 7508800 
 
 40 
 
 30 
 
 .6626200 
 
 .884725 
 
 1. 130294 
 
 . 7489557 
 
 30 
 
 40 
 
 . 6647959 
 
 . 889924 
 
 1. 123690 
 
 . 7470251 
 
 20 
 
 50 
 
 .6669661 
 
 .895150 
 
 1. 117130 
 
 .7450881 
 
 10 
 
 42 00 
 
 . 6691306 
 
 .900404 
 
 1. 110612 
 
 .7431448 
 
 48 00- 
 
 10 
 
 .6712895 
 
 .905685 
 
 1. 104136 
 
 .7411953 
 
 60 
 
 20 
 
 . 6734427 
 
 . 910994 
 
 1. 097702 
 
 .7392394 
 
 40 
 
 30 
 
 .6755902 
 
 . 916331 
 
 1.091308 
 
 .7372773 
 
 30 
 
 40 
 
 .6777320 
 
 . 921696 
 
 1.084955 
 
 . 7353090 
 
 20 
 
 SO 
 
 .6798681 
 
 .927091 
 
 1.078642 
 
 . 7333345 
 
 10 
 
 43 00 
 
 .6819984 
 
 .932515 
 
 1.072368 
 
 . 7313537 
 
 47 00 
 
 10 
 
 . 6841229 
 
 .937963 
 
 1.066134 
 
 . 7293668 
 
 50 
 
 20 
 
 . 6862416 
 
 .943451 
 
 1.059938 
 
 . 7273736 
 
 40 
 
 30 
 
 .6883546 
 
 .948964 
 
 1.053780 
 
 . 7253744 
 
 30 
 
 40 
 
 .6904617 
 
 .954508 
 
 1.047659 
 
 .7233690 
 
 20 
 
 50 
 
 .6925630 
 
 .960082 
 
 1.041576 
 
 . 7213574 
 
 10 
 
 44 00 
 
 .6946584 
 
 . 965688 
 
 1. 035530 
 
 . 7193398 
 
 46 00 
 
 10 
 
 .6967479 
 
 . 971326 
 
 1.029520 
 
 . 7173161 
 
 60 
 
 20 
 
 .6988315 
 
 . 976995 
 
 1.023546 
 
 .7152863 
 
 40 
 
 30 
 
 . 7009093 
 
 .982697 
 
 1.017607 
 
 . 7132504 
 
 30 
 
 40 
 
 .7029811 
 
 .988431 
 
 1.011703 
 
 . 7112086 
 
 20 
 
 50 
 
 .7050469 
 
 .994199 
 
 1. 005834 
 
 .7091607 
 
 10 
 
 45 00 
 
 . 7071068 
 
 1.000000 
 
 1.000000 
 
 . 7071068 
 
 45 00 
 
 
 Cosine. 
 
 Cotang. 
 
 Tang. 
 
 Sine. 
 
 Arc. 
 
 PEOPEHTIES OF CIRCLES. 
 
 216. The ratio of the diameter to the circumference is represented 
 in mathematics by w, called Pi. Its value can not be exactly ex- 
 pressed. To 5 decimal places it is 3.14159, which equals ^ nearly. 
 Log. IT equals 0.4971499. 
 
 Diam. X3.14159 = circ. 
 
 Diam. X 0.886277 =side of square of equal area,, 
 
 Diam. X 0.7071 =side of inscribed square. 
 
 lir'D^ = 0.7854 X I> = area of the circle. 
 
 IT r^ = 3.1416 Xr^=area of the circle. 
 
 The length of an arc of n° =m X 0.017453. 
 
 Example: If the radius is 642 feet, the length of an arc of 18*' 20' 
 = 18°.33 X 542 X 0.017453 = 165.5 feet. 
 
 PROPERTIES OF SOME PLANE FIGURES. 
 
 216. Triangles are classed as equilateral when the three sides are 
 of equal length; isoceles, when two sides only are equal; acute- 
 angled, when each ef its angles is less than 90°; obtuse-angled, when 
 one ajigle is greater than 90°. 
 
 The sum of the angles of any triangle is 180°. The sides are 
 directly proportional to the sines of the opposite angles, the great- 
 est and least sides opposite the greatest and least angles. 
 
144 TOPOGBAPHY, MAP BEADING, AND BEO,ONNAISSANOE. 
 
 Formulas for the solution of plane triangles. (Fig. 69.) 
 Given two sides, as a and h, and an angle opposite to one of them, 
 as B. 
 
 Given 2 angles, as A and B, and the included side c, the most com- 
 mon case. 
 
 /y ■.o/^o / A , T,\ csiD..A , a sin. 5 
 
 O=180°-(A+B); a=^^-^; h-^^- 
 
 Given 2 sides at a and i, and the included angle 0. 
 
 180°- O^A+B 
 
 A 7? f r^ (A + B) 
 
 Tan ^-S ia-h)—,^ 
 
 2 
 
 a + b 
 
 ^- 2 + 2 '-^~ 2 ~ 2 ' ^- sin. A- 
 
 Given the 3 sides — 
 a + h + c 
 
 _ A_ KS-b) {S-a) . 
 -S, sm. ^=y j3 , 
 
 . B 
 
 sm. 2 
 
 l(S-a) (S-c) . C liS-a) (S-h) 
 
 =V ai — ' ^m. 2=y ^ — . 
 
 For every right-angled triangle the sine of the right angle is 1, 
 and the following relations result: The side opposite the right angle 
 is called the hypothenuse. (Fig. 69.) 
 
 Hypothenuse = a = c-^sin. C=cXsec. B=-p r^. 
 
 = &Xsec. O^-y/b' + c'; 
 
 b = ax sin. B = aX cos. 0==cX cotang. C= c X tang. S; 
 
 c = ax sin. C= a X cos. B = bx tang. C; 
 
 b c 
 
 sia. £=-cos. C; sin. C=- = cos. 5; 
 
 a a 
 
 b c 
 
 tang. B= — = cotang. 0; tang. C=-r- = cotang. 5. 
 
 The area of a triangle equals any side multiplied by }4 the perpen- 
 dicular distance from that side to the opposite angle. If the perpen- 
 dicular from the angle does not intersect the opposite side, prolong 
 the side, but do not include the prolongation in its length for.com- 
 puting the area. All triangles which have a common side and* their 
 opposite angles in a straight line parallel to the common side are 
 equal in area. '" 
 
TOPOGBAPHY, MAP KEADING, AND BEOONNAlSSANCE. 145 
 
 A line bisecting one angle divides the opposite side into parts pro- 
 portional to the adjacent sides. In fig. 70, ah bisects the angle at 
 a aiid ()c:ac::M:ad. 
 
 _ Lines drawn from each angle to the middle of the opposite side 
 mtersect m a common point, which is the center of gravity of the 
 triangle. The shorter part of each line is ^ the longer (fig. 71). 
 
 A line joining the middle points of two sides is parallel to the third 
 side and H its length. In fig. 71 the line ef, joiaing the middle points 
 of ab and he, is parallel to ac, and }4 its length. Lmes joiniag ea and 
 fg would be parallel to ab and he, and half their length, respectively. 
 
 Similar triangles are those which have the same angles and differ 
 only in Imigth of sides. The ratio between correspondmg sides of all 
 similar triangles is the same, since it is the ratio of the same fmiction 
 of th4 same angles. Hence, if two sides of a triangle and one of the 
 corresponding sides of a sinular triangle are known, the other corre- 
 sponding side may be determined. The simplest test of similar tri- 
 angles is that their corresponding sides are parallel or peraendicular. 
 The principle of similar triangles is of great utility in field geometry. 
 
 The side of a square equals the diameter of an inscribed circle; or 
 the diagram of a circumscribed circle X 0.7071. The diagonal of a 
 square equals one side X 1.4142. 
 
 The area of a trapezoid, fig. 72, equals }4 the sum of the parallel 
 sides ah and cd midtiplied by the distance between them, ef. 
 
 The area of a trapezium — ^no two sides paraUel^figure 73, equals 
 14 the diagonal ac multiplied by the sum of the perpendiculars, 
 bf and de. 
 
 The side of a hexagon equals the radius of a circumscribed circle. 
 The area equals the square of 1 side X 2.598. The side of an octagon 
 equals the radius of a circumscribed circle X 0.7633. The area 
 equals the square of one side X 4.8289. 
 
 To draw an octagon in a square (fig. 74). — From each corner, with a 
 radius equal to }4 the diagonal, describe arcs as shown. Join the 
 poiat at which they cutHhe sides. If a square stick be scribed at a 
 aistance from each corner equal to 0.3, the side of the square and the 
 corners chamfered to the marks, the resulting section will be nearly 
 a true octagon. 
 
 GEOMETRICAL CONSTRUCTIONS. 
 
 217. To divide a straight line into any number of equal parts: 
 Frbm one end of the line draw another, making any convenient angle 
 with it, as 10° or 20°. On this auxiliary line lay off any assumed 
 distance as many times as the number of equal parts desired. Join 
 the last point so determined with the end of the first line. Through 
 each of the points marked on the auxiliary line draw a line parallel 
 to the line joming the ends. These lines will divide the given Ime into 
 the desired number of equal parts. ,■ ,, , 
 
 'To draw a perpendicular from a given point on a line: Mark 2 
 points equidistant from the given point, fig. 75, and with them as 
 centers and a radius greater than their distance from the given point 
 describe arcs on each side of the line. Connect one intersection with 
 the given point by a straight line, which is the perpendicular required. 
 As a check on accuracy, note whether the Ime passes through the 
 other intersection. 
 68740°— 18 10 
 
146 TOPOGEAPHY, MAP SEADISfG, AND BECONNAISSANCE. 
 
 If the given point is at one end of th.e line, from a convenient point 
 c outside the une describe a semicircle passing through the given 
 
 Eoint and cutting the line agaia as at i, fig. 78. Draw a straight 
 ne ic through the center to the arc on the other side, as at d. The 
 line da is the perpendicular required. 
 
 From a given point to let fall a perpendicular to a given line : From 
 the given point, fig. 77, describe an arc cutting the hne twice. With 
 these two poiats proceed as in meeting a perpendicular at a given 
 poiat, fig. 75, or bisect the portion of the liue between the inter- 
 sections, as at d, and draw the line ad, which is the perpendicular 
 required. 
 
TOPOGEAPHY, MAP READING, AND EEOONNAISSANCE. 147 
 
 bv^'>'' wr«! It^'h'^^^^^ *^°"g^ ^ g^^«^ P«^*»= Join tJie points 
 dLflif «^' t^ ^^ ^'^' ^g- ^^' ^d construct a bisecting pL)en. 
 S^i?ed Srcle P^endiculaps intersect at the center^^the 
 
 tri«J,ll^?r.if^^ ^"^ ^^""^ f^^"" ^ ^^« ^q^^^es of similar Hnes; similar 
 frZ?^™ !i^''^''^i°^ corresponding sides, or of perper/diculars 
 irom corresponding angles to opposite sides, etc. 
 
 bquares are to each other as the squares of the sides or diagonals. 
 r.r.7.fl^ regular polygons are to each other as the squares of besides 
 or ot the radu of mscribed or circumscribed circles 
 
 ■.1,^.^^° f ^^® *i° ®^* "'^^^^ ^^ *^^ squares of diameters, or radii, or 
 cnords oi equal arcs. 
 
 SPHERES AND CUBES. 
 
 _ 219. The surface of a sphere =4 irr» = 12.5664 r='=3 1416 d2 = 3183 
 circ. squared = 4X area of a great circle = diam. X circ. = the curved 
 surface ot cu-cumscribed cyhndcEr. 
 
 The surfaces of two spheres are to each other as the squares of 
 correspondmg hnes. 
 
 The volume of a sphere =| t 1^=4.1888 r« = 0.5236 d* = 0.01689 
 circ.3 = j diam.Xarea of great circle =§ vol. of circumscribed cyl- 
 mder = 0.5326 vol. of circumscribed cube. 
 
 ■ Tie volumes of spheres and cubes are to each other as the cubes 
 ot their correspondmg lines, or the squares of corresponding surfaces; 
 for a sphere, the radius, the diameter, the area of^a great circle, or 
 of any cffcle subtraiding equal angles at the center; for a cube, an 
 edge, a diagonal of a side, or a diagonal of the cube. The diagonal 
 of a cube=the edgeX 1.7321. 
 
 GRAVITATION. ' 
 
 220. The earth's attraction is measured by the increase in the 
 velocity of a falling body which that attraction produces in a second 
 of time. This quantity is represented by g, and its value at the sur- 
 face of the earth on the equator is 32.092 feet. This means that if 
 any body is falling freely its velocity at the ead of any second of time 
 is 32.092 feet per second greater than at the beginning of that second. 
 If the body starts from rest, its velocity at the beginning is 0, and at 
 the end of the first second is g. The velocity of any fafling body at 
 any instant equals g multiplied by the number of seconds the body 
 has been falling. This relation is strictly true for a vacuum only 
 but for moderate heights is nearly correct in air. ' 
 
 The value of g varies slightly with the latitude as shown in the 
 following table: 
 
 221. 
 
 Values of g 
 
 Table XVII. 
 at surface of earth in different latitudes : 
 
 
 Latitede. 
 
 Value oig 
 in feet. 
 
 T,aHtude. 
 
 Value of ff 
 in feet. 
 
 
 32.09 
 32. n 
 32.13 
 32.15 
 32.17 
 
 62° ly... 
 
 32 19 
 
 20* 40' 
 
 60° :: 
 
 
 SO" 
 
 89°15' 
 
 32 23 
 
 
 90° pole 
 
 
 45° 
 
 
 
 
 
148 TOPOGRAPHY, MAP EEAMNG, AND EEOONNAISSANCE. 
 
 222. The value of g varies also with the distance from the center 
 of the earth; or the distance above or below the surface, diminishing 
 ba. both cases. This diminution is approximately 0.016 foot for 
 each mile above the surface and 0.008 foot for each mile below. 
 
 223. ITie fundamental law of motion of faUing bodies is '\^=2g'h, 
 in which ■u = the velocity at any point in feet per second; Ti, the dis- 
 tance througlx which the body has fallen from rest to the given instant. 
 
 As v=gt, 'h, = \gf = lW^. These relations are strictly true only in 
 vacuo, but for small, smooth, dense objects are approximately cor- 
 rect for motion in air up to 5 seconds. 
 
 CENTRIFUGAL FORCE. 
 
 224. If w = the weight of a revolving body and n the number of 
 revolutions per minute, r = the radius of revolution or the distance 
 of center of gravity of the body from center of motion, and c = the 
 centrifugal force or puU on the radius in pounds, then c = O.OOQZ^wrin? 
 
 Equivalents of measure. 
 
 ' LENGTHS. 
 
 1 meter, m=10 decimeters, dm=100 centimeters, cm=1000 millimeters, mm. 
 1 meter, m=0.1 decameter, dkm=0. 01 hectometer, hm=O.0Ol kilometer, km. 
 1 meter, m=39.37 inches, U. S. Standard— 39.370113 inches, British standard. 
 1 millimeter, mm= 1000 microns, ii= 0.03937 inch= 39.37 mils. 
 
 Meters, 
 
 Inches, 
 in. 
 
 Feet, 
 It. 
 
 Yard, 
 yd. 
 
 Eods, 
 
 r. - 
 
 Chains, 
 ch. 
 
 MUes, TT. S. 
 
 Kilo- 
 meters, 
 - km. 
 
 Statute. 
 
 Nautical. 
 
 1 
 0.02540 
 0.30480 
 0. 91440 
 5.02921 
 20. 1168 
 1609.35 
 1853.25 
 1000 
 
 39.37 
 1 
 
 12 
 
 36 
 
 198 
 
 792 
 
 63360 
 
 72962.5 
 
 39370 
 
 3.28083 
 
 0.08333 
 
 1 
 
 3 ' 
 
 16.5 
 
 66 
 
 5280 • 
 
 6080.20 
 
 3280.83 
 
 1. 09361 
 
 0. 02778 
 
 0.33333 
 
 1 
 
 5.5 
 
 22 
 
 1760 
 
 2026.73 
 
 1093.61 
 
 0.19884 
 0. S6051 
 0. 06061 
 0. 18182 
 1 
 4- 
 
 320 
 36^497 
 198.838 
 
 0.04971 
 0.S1263 
 0.01515 
 0.04545 
 ' 0.25 
 1 
 
 80 
 92.1243 
 49. 7096 
 
 0.^6214 
 0. 81578 
 0.J1894 
 C.J5682 
 0. g3125 
 0.01250 
 
 1 
 1. 15165 
 0.62137 
 
 0. S6396 
 0. S1371 
 0. S1645 
 0. g4934 
 0. S2714 
 0.01085 
 0.86839 
 
 1 
 0.53959 
 
 0.001 
 0.S2540 
 0.g3048 
 0. S9144 
 0. S5029 
 0.02012 
 1.60935 
 1.85325 
 1 
 
 1 yard, XT. S.=-1.0000029 yards British. 1 yard British- 0.9999971 yard U. S. 
 
 1 chain, Gunter's= lOO links. llink= 7.92 inches. 
 
 1 cable length, V. S.= 120 fathoms=960 spans=720 Ieet= 219.457 meters. 
 , 1 league, U. S.=3 statute miles=24 furlongs. 
 
 1 international geographical mile=s A ° at equator= 7422 m= 4.611808 U. S. statute miles. 
 
 1 international nautical mile= ,>„ ° at meridian— 1852 m= 0.999326 U.S. nautical miles. 
 
 1 U. S. nautical mile= ,'0° of circumference of sphere whose surface equals that of the earth= 6080.27 feet— 
 1.1S15S statute miles= 1853.27 meters. 
 
 1 British nautical mile= 6080.00 foet= 1.16152 statute miles= 1853.19 meters. 
 
 SURFACES AND AEEAS. 
 
 1 sq. meter, m2=100 sq. decbneters, dm!= 10000 sq. centimeters, cm." 
 1 sq. meter, m2=0.01 are, a= 0.0001 hectare, ha. 
 1 sq. millimeter, mm2=0.01 om2=0.00165_sq. inch= 1217.36 circ. mils. 
 1 are, a=l sq. decameter, dkm= 0.0247104 acre. 
 
 Square 
 
 meters, 
 
 m«. 
 
 Square 
 
 inches, 
 
 sq;in. 
 
 Square 
 feet, 
 sq.ft. 
 
 Square 
 yards, 
 sq. yd:. 
 
 Square 
 rods, 
 sq. r. 
 
 Acres, 
 
 A. 
 
 Hectares, 
 ha. 
 
 Square 
 miles, 
 statute. 
 
 Square 
 kilo- 
 meters, 
 Irmi, 
 
 1 
 
 1660.00 
 
 10.7639 
 
 1. 19599 
 
 0.03954 
 
 0.J2471 
 
 0.00001 
 
 0. 13861 
 
 0.S1 
 
 0. 16452 
 
 1 
 
 0. S6944 
 
 0. g7716 
 
 0. S2561 
 
 0. S1594 
 
 0. g6452 
 
 0. g2491 
 
 0.S6452 
 
 0.09290 
 
 144 
 
 1 
 
 0. 11111 
 
 0. g3673 
 
 0. 52296 
 
 0.S9290 
 
 0. 53587 
 
 0.59290 
 
 0. 83613 
 
 1296 
 
 9 
 
 1 
 
 0.03306 
 
 0g2066 
 
 0.J8361 
 
 0. S3228 
 
 0. S8361 
 
 25.2930 
 
 39204 
 
 272.25 
 
 30.25 
 
 1 
 
 0.00625 
 
 0. g2629 
 
 0.g9766 
 
 0.S2629 
 
 4046. 87 
 
 6272640 
 
 43560 
 
 4840 
 
 160 
 
 1 
 
 0. 40469 
 
 0. gl563 
 
 0.g4047 
 
 10000 
 
 15499969 
 
 107639 
 
 11959. 9 
 
 395.366 
 
 2.«104 
 
 1 
 
 0. g3861 
 
 0.01 
 
 2689999 
 
 
 27878400 
 
 3097600 
 
 102400 
 
 640 
 
 269.000 
 
 1 
 
 2.69000 
 
 1000000 
 
 
 10763867 
 
 1195985 
 
 39536. 6 
 
 247.104 
 
 100 
 
 0.38610 
 
 1 
 
 1 sq. rod, sq. pole, or sq. perch=625 sq. links^yj^ acre. 
 
 1 sq. Cham, (}unter's= 16 sq. rods= A acre. 
 
 1 acre= 4 sq. roods= 160 sq. rods. Square of 1 acre= 208.7103 feet square. 
 
 Notations a, 5, J, etc., indicate that the %, %, %, etc., are to be replaced by 2, 3, 4, etc., ciphers. 
 
 Example: 1 sq. rod= 0.89766= 0.000009766 sq. imles. 
 
TOPOGEAPHY, MAP EEABING, AND EECONNAISSANCE. 
 
 149 
 
 Equivalents of measure. 
 "volume and capacity. » 
 
 l"pu;nieter,-m'=l,HX)ecu. decimeteridm'= 1,000,000 cu. centimeters,. cm«. 
 
 1 Uter, 1= 10 deciliters, dl= 100 centiliters, cl= 1,000 milliliters, ml= 1,000 cu. centimeters, cm', or co. 
 
 1 liter, 1= 0.1 decaliter, dlil= 0.01 hectoliter, hl= 1 6a. decimeter, dm*. 
 
 Cubic 
 deci- 
 
 
 
 
 U. S. quarts. 
 
 U. S. gallons. 
 
 
 Cubic 
 inches. 
 
 Cubic 
 feet, 
 cu. ft. 
 
 Cubic 
 yards, 
 cu. yd. 
 
 
 
 
 
 U.S. 
 bushels. 
 
 
 
 
 
 dm', 1'. 
 
 cu. in. 
 
 Liquid, 
 
 Dry, 
 
 Liquid, 
 
 Dry, 
 
 bu. 
 
 
 
 
 
 l.qt. 
 
 d. qt. 
 
 1. gal.- 
 
 d. gal. 
 
 
 1 
 
 61.0234 
 
 0.03631 
 
 0.31308 
 
 1.05638 
 
 0.90808 
 
 0.26417 
 
 0.22702 
 
 0.02838 
 
 0.01639 
 
 1 
 
 0. I57S7 
 
 0. S2143 
 0.03704 
 
 0.01732 
 
 0.01488 
 
 - 0. ;4329 
 
 0.53720 
 
 0. S4660 
 
 28.3170 
 
 1728 
 
 1 
 
 29.9221 
 
 25.7140 
 
 7.48055 
 
 6.42851 
 
 0. 80356 
 
 764.559 
 
 46656 
 
 27 
 
 1 
 
 807.896 
 
 694.279 
 
 201.974 
 
 173.570 
 
 21.6962 
 
 0.94636 
 
 57.75 
 
 0.03342 
 
 0. S1238 
 
 1 
 
 0.85937 
 
 0.25 
 
 0.21484 
 
 0.02686 
 
 1. 10123 
 
 67.2006 
 
 0.03889 
 
 0. ;1440 
 
 1. 16365 
 
 1 
 
 0.29091 
 
 0.25 
 
 0.03125 
 
 3.78543 
 
 231 
 
 0.13368 
 
 0. J4951 
 
 4 
 
 3.43747 
 
 1 
 
 0.85937 
 
 0. 10742 
 
 4.40492 
 
 268.803 
 
 0. 15556 
 
 0. 55761 
 
 4. 65460 
 
 4 
 
 1.16365 
 
 1 
 
 0.125 
 
 35.2393 
 
 2150.42 
 
 1.24446 
 
 0.04609 
 
 37.2368 
 
 32 
 
 9.30920 
 
 8 
 
 1 
 
 V. S. dry measure: 1 bushel=4 peoks=8 gallons=32 quarts= 64 pints. 
 
 U. S. liquid measure: 1 gaIlon=4 quarts=8 pints=32 gills=128 fluid ounces. 
 
 U. S. apoth. measure: In. ounce, i S=8 fl. <hams, f3 = 480 minims, ni=29.574 cu. cm'. 
 
 British Imperial gallon dry and liquid measure= 1.03202 U, S. dry gal.= 1.20001 U. S. liquid gal. 
 
 British Imperial gaUon= 277.410 cu. in.= 4545.9631 cm'. 
 
 Weight of water at maximum density, 4° C, 45° lat., and sea level. 
 
 1 cu. ft.=62.4283 lbs. av.= 28.3170 kg. 1 cu. in.= 0.57804 oz. av.= 16.3872 g. 
 
 1 gal., U. S. liquid- 8.34545 lbs.=3.7S543 kg. 
 
 1 gal., British lmperial= 10.0221 lbs.= 4.5459631 kg. 
 
 MASSES AND WEIGHTS. 
 
 1 gram, g=10 decigrams, dg=100 centigrams, cg= 1,000 milligrams, mg. 
 1 gram, g=0.1 decagram, dkg—O.Ol hectogram, hg=0.001 kilogram, kg. 
 
 1 kilogram, kg=l cu. decimeter of water or liter, 4° C, 45° lat., and sea level= 15432.35639 grains, U. S. 
 and British standard. 
 
 Kilo- 
 grams, 
 kg. 
 
 Grains, 
 gr. 
 
 Ounces. 
 
 Pounds. 
 
 Tons. 
 
 Troy, 
 oz. t. 
 
 Avoir., 
 oz. av. 
 
 Troy, 
 lb. t. 
 
 Avoir., 
 lb. av. 
 
 Net, 
 
 short, 
 
 2,000 lbs. 
 
 Gross, 
 2,240 fts. 
 
 Metric, 
 1,000 kg. 
 
 1 
 0.J6480 
 0.03110 
 0.02835 
 0.37324 
 0.45359 
 907.185 
 1016.05 
 1000 
 
 15432^4 
 1 
 
 480 
 437.5 
 
 6760 
 
 7000 
 14000000 
 15680000 
 15432356 
 
 32.1607 
 0.;2083 
 
 1 
 0.91146 
 
 12 
 14.5833 
 29166. 7 
 32666.7 
 32150.7 
 
 35.2740 
 0. S2286 
 1.09714 
 
 1 
 13.1657 
 
 16 
 32000 
 35840 
 35274.0 
 
 2.67923 
 0. gl736 
 0.08333 
 0.07595 
 
 1 
 1.21528 
 2430.56 
 2722.22 
 2679.23 
 
 2.20462 
 
 0;sl429 
 
 0.06857 
 
 0.06250 
 
 0.82286 
 
 1 
 
 2000 
 
 2240 
 
 2204.62 
 
 0. S1102 
 0. J7143 
 0.S3429 
 0.S3125 
 0.S1114 
 0.00050 
 1 
 1.12 
 1.10231 
 
 0. 19842 
 0. S6378 
 0. S3061 
 0.;2790 
 0.53674 
 0. S4464 
 0.89286 
 
 1 
 0.98421 
 
 0.001 
 0.56480 
 0.{3110 
 0.S2835 
 0. S3732 
 0. g4536 
 0.90719 
 1.01605 
 1 
 
 1 ounce aToir.=16 drams, avoir. 1 ounce troy= 20 pennyweight, dwt. 
 
 1 ounce apoth., 3=8 drams, 8=2i scrulpes, 9=480 grams, gr=31.103S g. 
 
 1 hundredweight— 1/20 Jong ton= 4 quarters— 8 stone= 112 lbs.= 60.8024 Kg. 
 
 Notations S, 8, S, etc., indicate that the S, J, }, etc., are to be replaced by 2, 3, 4, etc., ciphCTS. 
 
 Sxample: 1 graiB=0;2083=0.002083 oz. t. 1 grain=0.S6480=0.00006480 kg. 
 
150 TOP06EAPHY, MAP EEADIKG, AND EECONNAISSANCE. 
 
 Equivalents of measure. 
 
 JORCES OR WEIGHTS PER UNITS OF LENGTH, LINEAR WEIGHTS. 
 
 1 dyne per centimeter=0.00101979 g/cm =.0.000183719 poundal/iJU 
 1 gram pet centimeter^ 980.5966 dyiies/cmsO.180154 poundaUin. 
 1 poundal per inch = 5443.11 dynes/cm=S.55081g/cms=0.03IO8Sponnd/In. 
 
 Grams 
 
 
 
 
 
 Kilo- 
 
 Net tons, 
 
 2,000 
 pounds, 
 per mile. 
 
 Gross 
 
 Metric 
 
 per 
 
 Grains 
 
 Pounds 
 
 Pounds 
 
 Pounds 
 
 grams 
 
 tons, 
 2,240 
 
 ton^, 
 
 centi- 
 
 per inob. 
 
 per inch. 
 
 per loot. 
 
 per yard. 
 
 per 
 
 1,000 kg. 
 
 meter, 
 Efcm. 
 
 gr./in. 
 
 Ib./in. 
 
 Ib./ft. 
 
 Ib./yd. 
 
 meter, 
 
 kg/a. 
 
 pounds, 
 per mile. 
 
 per kilo- 
 meter. 
 
 1 
 
 39.1983 
 
 0.55600 
 
 0.06720 
 
 0.20159 
 
 0.10 
 
 0.17740 
 
 0.15839 
 
 0.10 
 
 0.02551 
 
 1 
 
 0.S1429 
 
 (>.*I714 
 
 0.S51© 
 
 0.gZ551 
 
 0.i4526 
 
 0. 54041 
 
 0.52551 
 
 178.579 
 
 7080 
 
 1 
 
 12 
 
 36 
 
 17.8679 
 
 31.6800 
 
 28.2857 
 
 17.8579 
 
 14.8816 
 
 583.333 
 
 &. 08333 
 
 1 
 
 3 
 
 1.48816 
 
 2.94000 
 
 2.35714 
 
 1.^8816 
 
 4.960S4 
 
 194. 4«4 
 
 0.02778, 
 
 0.33333 
 
 i 
 
 0.49605 
 
 0.88000 
 
 0.78571 
 
 0.49605 
 
 M 
 
 391.9^ 
 
 Q. 05600 
 
 0.67197 
 
 2.0-1591 
 
 1 
 
 1.77400 
 
 1.59393 
 
 I 
 
 5.63698 
 
 220.960 
 
 0. 03157 
 
 0.37879 
 
 1.13636 
 
 0.56370 
 
 1 
 
 0v^286 
 
 0.56370 
 
 6.31342 
 
 247.475 
 
 0.03635 
 
 0.42424 
 
 1.27273 
 
 0.63134 
 
 1.12 
 
 1 
 
 0.^134 
 
 10 
 
 391.983 
 
 0. 05600 
 
 0.67197 
 
 2.01591 
 
 1 
 
 1.77400 
 
 1.58393 
 
 1 
 
 FORCES OR WEIGHTS PEE UNITS OF AREA, PBESSUEE. 
 
 1 dyne per sq. centimeter=O.001<a979 g/cm' =0.000466646 poundals/in^. 
 1 gram per sq.eentimeter=98Q.5966 <iyiies/em^=0:4S7592 poundab/in'. 
 1 poundal per sq. inch =2142.95 dynes/cm»=2.18536 g/cms=0.(»10832 poand/ina. 
 
 Kilo- 
 grams 
 
 per 
 square 
 centi- 
 meter, 
 kg/cm.! 
 
 Pounds 
 
 per- 
 
 square 
 
 inch, 
 
 lb./in.s. 
 
 Pounds 
 
 per 
 
 square 
 
 foot, 
 
 Ib./lt.!. 
 
 Net tons, 
 
 2,000 
 pounds 
 
 per 
 square 
 
 Atmos- 
 pheres, 
 standard, 
 760 mm. 
 
 Colmnns ot mereury, 
 Hg. 13.59593 sp. g. 
 
 Columns of water, 
 max. d«nsity 4° C. 
 
 MUli- 
 metses. 
 
 iDGhes. 
 
 Meters. 
 
 Feet. 
 
 1 
 0.07031 
 0. SJ882 
 0.97648 
 1.03329 
 0.31360 
 O.U3453 
 
 0.10 
 0.03048 
 
 14.2234 
 
 1 
 0.3^4 
 13.8889 
 14.6969 
 0.01934 
 0.49119 
 1.42234 
 0.43353 
 
 2048.17 
 
 144 
 , 1 
 
 2000 
 2116.35 
 2.78468 
 70.7310 
 204.817 
 82.4283 
 
 1.02408 
 0.07200 
 0.00050 
 
 1 
 1.05818 
 1. 31392 
 Ot 63537 
 0. 10241 
 0.03121 
 
 0.96778 
 0:06804 
 0.34725 
 0.94502 
 
 1 
 0. 31316 
 0.03342 
 0.09678 
 0.02950 
 
 735.514 
 51.7116 
 0.359U 
 
 718.219 
 760 
 1 
 25.4001 
 73.5614 
 22.4185 
 
 28.9572 
 2.08588 
 0.01414 
 28.2762 
 29.9212 
 0.03937 
 
 1 
 2. 89572 
 0.88262 
 
 10 
 6.70307 
 0. 34882 
 9.76482 
 10.3329 
 0. 01360 
 0.34634 
 
 1 
 0.30480 
 
 32.8033 
 2.30666 
 0.01602 
 32.0367 
 3&9006 
 0.04461 
 1.13299 
 3.28083 
 1 
 
 FORCES OK WEIGHTS PER UNITS OF VOLUME, DENSITY. 
 
 1 dyne per cu. oentimeter=O.00101979 gram/om» =0.00118528 poundj|ls/in3. 
 1 gram per cu. centimeter= 980.5066 dynes/cm» = 1.162283 poundals/in». 
 1 pooni&l per CO. inch =843.683 dynes/cm» =0.880378 g/cm»-0i)310832 pound/in». 
 
 Grams ■ 
 per 
 cubic 
 centi- 
 meter, 
 g/cma. 
 
 Founds 
 
 per 
 
 cubic 
 
 inch, 
 
 lb./in.3. 
 
 Pounds 
 
 per 
 
 cubic 
 
 loot, 
 
 lb./K.». 
 
 Pounds 
 
 per 
 
 cubic 
 
 ; l^.^d'.». 
 
 Kilo- 
 grams 
 per 
 cubic 
 meter, 
 kg/ms. 
 
 Pounds 
 
 per 
 
 bushel, 
 
 U.S. 
 
 Pounds 
 gallon, 
 
 Pounds 
 
 gaUon, 
 HqnM,, 
 
 xi.s. 
 
 KBo- 
 grams 
 
 per 
 hecto- 
 
 Fiter, 
 kg/hi. 
 
 1 
 27.6797 
 0.01602 
 0. 55933 
 
 0.001 
 0.01287 
 0.10297 
 0.11983 
 
 0.01 
 
 0.03813 
 
 1 
 0. 55787 
 0. S2143 
 0. S3613 
 0. 54650 
 0. 33720 
 0.34329 
 0. 13613 
 
 62.4283 
 1728 
 1 
 0.03704 
 0.06243 
 0.80356 
 6.42851 
 7.48062 
 0.62428 
 
 1685.56 
 46656 
 27 
 1 
 1.68556 
 21.6962 
 173.570 
 201.974 
 16.8557 
 
 1000 
 27679.7 
 16.0184 
 0.59327 
 
 1 
 
 12.8718 
 
 102.974 
 
 119.826 
 
 10 
 
 77.6893 
 2150.42 
 1.24446 
 0.04609 
 0.07769 
 
 1 
 
 8 
 9.30020 
 0.77689 
 
 9.71116 
 268.803 
 0. 16556 
 0. 35762 
 0. 39711 
 0.125 
 
 1 
 1.16365 
 0.09711 
 
 8.34545 
 
 231 
 0.13368 
 0. 34951 
 0. 38345 
 0.10742 
 0.85937 
 
 1 
 0.08345 
 
 100 
 2767.97 
 1.60184 
 6.05933 
 
 0.10 
 
 1.28718 
 
 10.2974 
 
 11.9826 
 
 1 
 
 Notations 3, 8, S, etc., indicate that the 3, 5, ; 
 Example: 1 kg/ma=O.S3613=0.000036131b./in«. 
 
 , etc., are to be replaced by 2, 3, i, etc., ciphers. 
 
TOPOGEAPHY, MAP EEADING, AND EECONNAISSANCE. 
 
 151 
 
 Eqwivalents of ■measure. 
 ENERGY, ■VVOEK, HEAT. 
 
 1 dyn6-oentlmeter=l erg=a 0.00101979 gram-centimeter- 0.S737612 toot-pound. 
 1 gram-oentimeter= 980.5966 ergs=0.s7233 foot-pound. 
 1 ioot-pound= 13557300 ergs= 13825.5 gram-oentimeters. 
 
 Kilogram- 
 meters, 
 kg-m. 
 
 Foot- 
 pounds, 
 ft.-lbs. 
 
 Horsepower-hour. 
 
 Poncelet- 
 hours, 
 100 
 
 kg-m-h. 
 
 Kilowatt- 
 hours, 
 kw-h. 
 
 Joules, 
 
 10? ergs, 
 
 3-s. 
 
 Thermal units. 
 
 U.S., 
 h. p.-h. 
 
 Metric 
 75 kg-m-h. 
 
 B.T.tJ., 
 b. t. u. 
 
 Calorie, 
 kg-oal. 
 
 1 
 0. 13826 
 273745 
 270000 
 360000 
 367123 
 0. 10198 
 107.577 
 426.900 
 
 7.2.1300 
 
 1 
 ■1980000 
 1952910 
 2603880 
 2655403 
 0. 73761 
 778. 104 
 3087. 77 
 
 0. S3653 
 0. S5051 
 
 0.98632 
 1.31509 
 1.34111 
 0.33725 
 0.^3930 
 0. 31559 
 
 0. '3704 
 0. |5121 
 1.01387 
 
 1 
 1.33333 
 1.35972 
 0. S3777 
 0. 33984 
 0. 31581 
 
 0. 12778 
 0. S3840 
 0.76040 
 0.75 
 1 
 1.01979 
 0. 32833 
 0-32988 
 0. 31186 
 
 * 
 0. '2724 
 0. S3766 
 0. 74565 
 0.73545 
 0.98060 
 
 1 
 0. 32778 
 0. S2930 
 0. 31163 
 
 9.80597 
 1.35573 
 2684340 
 2647610 
 3530147 
 3600000 
 
 1 
 1054.90 
 4186. 17 
 
 0. 39296 
 0.31285 
 2544.65 
 2509.83 
 3346.44 
 3412.66 
 0. 39480 
 ~ 1 
 3.96832- 
 
 0.32342 
 0. 33239 
 641.240 
 632.467 
 843.289 
 859.975 
 0. 32389 
 0.25200 
 1 
 
 POWER, BATE OP ENERGY AND HEAT. 
 
 1 erg per sec.=l dyne<!m/seo.= 0.00101979 gram-cra/sec.=0.;737612 foot-pounds/sec.' 
 1 gram-centimeter per second= 980.5966 ergs/seo.= 0.S7238 foot-pounds/seo. 
 1 foot-pound per seoond= 13567300 ergs/sec.= 13825.5 gram-om/seo. 
 
 Kilogram- 
 meters 
 per 
 
 S^(Hld, 
 
 tg-m/s. 
 
 Foot- 
 pounds 
 
 per 
 second, 
 ft.-lbs./s. 
 
 Horsepower. 
 
 Fonceletj 
 100 ■ 
 
 kg--myij. 
 
 Kilowatt, 
 kw. 
 
 Watts, 
 lO'ags^. 
 
 Thermal units 
 per sec. 
 
 tJ-. S.,5S0 
 ft.-lTJs./s. 
 
 Metric, 
 
 B. T. U. 
 
 btu/s. 
 
 Calorie, 
 kg-cal/s. 
 
 1 
 0.13826 
 76.0404 
 
 75 
 .100 
 101.979 
 0.10198 
 107.577 
 426.900 
 
 7.23300 
 1 
 550 
 542.475 
 723.300 
 737.612 
 0. 73761 
 778. 104 
 3087. 77 
 
 0.01315 
 0. 31818 
 
 0.98632 
 1.31509 
 1.34111 
 0. 31341 
 1.41474 
 5. 61412 
 
 0.01335 
 .0. 11843 
 1.01387 
 
 1 
 1.33333 
 1.35972 
 0. 31360 
 1.43436 
 5.69200 
 
 0.01 
 0-31383 
 0.76B40 
 
 0.75 
 1 
 1.01979 
 0. 31020 
 1.07577 
 4.26900 
 
 0. 39806 
 0. 3 1356 
 a 74565 
 0-73545 
 0.98060 
 1 
 0.001 
 1.05490 
 4. 18617 
 
 9. 80597 
 1.35573 
 745. 650 
 735.448 
 980.697 
 1000 
 1 
 1054.90 
 4186.17 
 
 0- 39206 
 0. 11285 
 0. 70685 
 0.69718 
 0.92957 
 0.94796 
 0:39480 
 
 3.96832 
 
 0.32342 
 0.33237 
 0. 17812 
 0.17.569 
 0.23425 
 0.23888 
 0.32389 
 0.25200 
 1 
 
 VELOCITIES AND ACCELERATIONS. 
 
 1 kine= 1 centimeter per seoond=0.0328083 foot per second. 
 
 1 radian per second= 57.2958 degrees per seo.= 0.1591.55 revolutions per sec. 
 
 1 gravity— 980.5966 centimeters per sec. per sec.= 32. 1717 feet per see. per sec. 
 
 Meters 
 
 per 
 second, 
 
 m/s. 
 
 Feet per 
 
 second, 
 
 ft./s. 
 
 Miles 
 
 per hour, 
 
 M/i. 
 
 Knots 
 
 per hour, 
 
 U.S. 
 
 Kilo- 
 meters 
 hour, 
 km/h. 
 
 Meter 
 
 -per 
 
 sec/sec, 
 
 m/s!". 
 
 Feet per 
 see/seo 
 ft./sa- 
 
 Miles 
 
 per 
 
 hour/sec, 
 
 MA-s. 
 
 KUo- 
 
 meter per 
 
 hour/sec, 
 
 km/h-s. 
 
 1 
 0.30480 
 0.44704 
 0.51479 
 0.27778 
 
 3.28083 
 
 1 
 1.46667 
 1.68894 
 0.91134 
 
 2.23693 
 0.68182 
 
 1. 15155 
 0.62137 
 
 1.94254 
 0.,ra209 
 0.86839 
 
 1 
 0. 53959 
 
 3.6 
 
 1.09728 
 
 1.60835 
 
 I.8S325 
 
 1 
 
 ■ 1 
 
 0.30480 
 0.44704 
 0.27778' 
 
 3.28083 
 
 1 
 1.46667 
 0.91134 
 
 2.23693 
 0.68182 
 
 1 
 0. 62137 
 
 3.6 
 
 1.09728 
 
 1.60935 
 
 1 
 
 Notations 3, S. %, eto-j indicate that the *, 3, S, etc., a 
 Example: 1 calorie- 0.51163= 0.001163 kilowatt-hours. 
 
 J, etc., are to be replaced by 2, 3, 4, etc., ciphers. 
 
 o 
 
ADDITIONAL COPIES 
 
 OP Tma PUBLICATION MAY BE PBOCTOED FBOU 
 
 THE SIJPEKDITENDENT OP DOCUMENTS 
 
 GOVEENMENT PEINTINO OFHCE 
 
 WASHINGTON, D. 0. 
 
 AT 
 
 20 CENTS PER COPY