Surveying (Fr. survoir, to overlook), the art of measuring portions of the surface of the earth, either for the purpose of calculating the contents of areas, of laying out tracts of required extent, of establishing roads, or of preparing maps. The ancient science of geometry grew out of the practice of surveying, and now embodies the mathematical princi-. ples upon which the work is conducted. This science was cultivated by the Egyptians at a very early period, and many of the old Greek writers ascribe its origin to changes which annually took place from the inundation of the Nile, and to the consequent necessity of adjusting the claims of each person respecting the limits of lands. The progress of the art of surveying to its higher application in determining the figure of the earth has been traced in the article Earth; and the operations in trigonometrical surveys upon a grand scale are described under Coast Survey. - The systems of surveying may be classed according to its special objects; as land surveying, for determining the contents of areas, or dividing tracts into lots of smaller dimensions; topographical surveying, which includes the measurement of horizontal lines and angles, and the variations of level, so that the superficial inequalities may be graphically represented; hydro-graphical or maritime surveying, the object of which is the determination of the positions of channels, shoals, rocks, and the shore line; and mining surveying, for fixing the positions of the underground works in mines, so that these can be correctly mapped.

Surveys extending over large territories involve the consideration of the curvature of the earth and the use of spherical trigonometry, and are called geodetic in contradistinction from ordinary surveying over more limited areas, which may with sufficient accuracy be conducted without reference to the figure of the earth, and which may be termed plane surveying. (See Geodesy.) These systems all involve the same principles of measuring lines and angles between definite points upon the area included in the survey, and reproducing these upon paper, reduced to a convenient scale. Calculating the content of the area is commonly the conclusion of the work of land surveying. Tracts of any shape or size may be accurately surveyed, if tolerably level and clear, with no other instrument than the surveyor's chain (see Gunter, Edmund); and for this may bo substituted a measuring tape, a measured rope, or leather driving reins. This is done by measuring all the sides of the tract, and then diagonals from one corner to another, so selected as to divide the tract into triangles as nearly equilateral as possible. The number of diagonals will be two less than the number of sides.

In using the chain it is to be kept as nearly horizontal as possible, or if the measurement is made on a slope the variation from the horizontal is to be determined and duly allowed. In case the corners are not visible from each other, intermediate points may be adopted and used for the terminations of lines from corners, the object being in every case to divide the tract into triangles of which the sides are all measured. Proof lines measured from a corner of each triangle to the opposite side serve to rectify the other measures of the triangle, and if perpendicular to the side afford a convenient means of calculating upon the ground the area of the triangle. Perpendiculars to any line are readily laid out with a chain, as carpenters and masons draw right angles by what they call the G, 8, and 10 rule, the popular application of the principle of the square of the hypothenuse being equal to the sura of the squares of the two other sides. The method is to measure from the point where the perpendicular meets the line, either along this line or along the perpendicular, a distance equal to six units of any kind, and then upon the other of these lines a distance of eight units. The two lines are perpendicular to each other when the two termini are just ten units apart.

Convenient distances for this measurement might be 3, 4, and 5 rods or chains, or any similar multiples of these numbers, as 21, 28, and 35. Other trigonometrical methods readily suggest themselves. A number of convenient instruments of simple form, known as the surveyor's cross, are in use for setting out perpendiculars by lines of sight, crossing each other at right angles; and a temporary substitute for them is easily mado by sticking a pin in each corner of a square piece of board, and sighting across these in the direction of the line and at right angles to it. Angles in the field are determined by a chain, by measuring a "tie line" from a measured point on one side to another measured point on the other side. By this means the boundaries of a tract may be determined when it cannot be-conveniently measured off in triangles. A great variety of expedients are adopted for overcoming natural obstacles and determining the extent and shape of inaccessible objects, systems of triangles being in such cases formed outside of and around such objects. Crooked lines are determined by means of perpendicular offsets measured from different points along a straight lino run as nearly coincident to the crooked line as may be.

In all the methods of surveying, the measurements, together with various incidental observations, are recorded, after some established system, in what are called field notes, and from these the results of the survey are afterward plotted" to a convenient scale. - A more common system of surveying is that in which instruments for taking angles are employed in connection with the chain. A graduated horizontal circle, with a straight edge called an alidade turning upon its central point, which may be conveniently sighted along, furnishes the means of ascertaining the angular distance of two lines, the instrument being set at their intersection, and the alidade pointed in the direction of one and then of the other. This involves the principle of the engineer's transit, or of the theodolite. (See Theodolite.) With these instruments angles can be determined with great accuracy, especially when the observations are repeated by reversing the instrument and taking the mean, each including the reading of both verniers.

With the transit and the chain for measuring distances, a tract of almost any dimensions is accurately surveyed by measuring the angles at its corners, and the correctness of the work is proved when the sum of all the interior angles is found equal to the product of two right angles, or 180°, by the number of sides of the tract less two; or if the instrument be used by the method called traversing, or "surveying by the back angle " (which consists in noting the angle which each successive line makes, not with the preceding line, but with the first line observed, which is hence called the meridian of the survey), then the reading, on getting round to the last station, and looking back to the first line, should be 3G0°, or 0°. A compass and chain may be employed in filling up the interior details of a large survey with the transit; and the compass may be used for determining the magnetic bearing of one of the lines, unless this be astronomically ascertained by observations of the north star or of the shadows before and after noon. The compass is the instrument in most common use in ordinarv surveying.

The magnetic nee-dle, wherever the instrument is set, establishes the meridian line, and from this, the sights of the instrument being turned to any other line, the angle of divergence is read on the graduated circle around the compass box. This instrument has been described under its own name; also the more perfect instrument, in which its inaccuracies are obviated, under the head of Compass, Solar. - The details of surveys are variously modified according to the extent of the area, character of the ground, etc. With the transit or compass, the boundary lines may be all followed out, the angles they make with each other determined, and their lengths measured by the chain; the points of crossing of roads, brooks, fences, etc, measured, and the bearings of these objects taken; and increased accuracy may be given to the work by running diagonal or proof lines, as in chain surveying. Additional checks are furnished by taking at each station the bearings of some marked objects, which when the work is plotted should severally fall at the points of intersection of the lines directed toward these objects from the several stations.

Sometimes a tract may be surveyed from a measured base line, either a line within or without it, or one of the boundary lines, by placing the compass successively at each end of this line and taking the bearings of each corner; or without a compass the work may be very conveniently performed with approximate correctness by the plane table method, provided no angles are taken less than 30° nor larger than 150°. A drawing board covered with paper is set up at one end of a measured base line, and a ruler furnished with upright sights at each end, exactly over the drawing edge, is set with this edge against a fine needle stuck up in the board, and is then directed successively toward the corners of the tract to be surveyed and any other prominent objects, toward which from the needle lines are to be drawn on the paper. One of these lines should also be in the direction of the measured line. The instrument is then taken to the other end of the measured line, the needle is removed along the last line named on the board a distance corresponding, according to the scale adopted, to that of the measured line on the ground, and the board is so placed as to make the line toward the former station correct.

The ruler is then again pointed to the same objects, and lines are drawn toward each from the new position of the needle. Their intersections with the former lines designate the places of these objects on the plane. The plane table is used in various other ways, as by moving it from one corner to the next, and placing it at each so that the last line drawn coincides with that in the ground. From any central point also radiating lines may be measured to the corners, and the distances measured and marked off according to the proper scale. - Rivers, brooks, and roads are surveyed by measuring a succession of lines following their general course, and taking offsets from the sides of the line. Streets are followed in a similar manner. Distances are sometimes measured upon roads, where expedition is more important than extreme accuracy, by various substitutes for the chain, some of which, as the odometer and pedometer, have been noticed under the former head. One may soon accustom himself to pace in straight lines, and with steps of uniform lengths, the most exact method being to regulate the natural step, rather than to try to attain one of any determinate length. The usual average step of a man is that of the English military pace, 2½ ft.

The French geographical engineers accustom themselves to take regular steps of 8/10 of a metre, or 2 ft. 7½ in. - The field work being completed, the figure of the tract surveyed is reproduced upon a diminished scale by what is termed plotting; and from this plot the contents are ascertained by a series of mathematical calculations applied successively to the several divisions, or by the method of calculation of latitudes and departures, for which a table of natural sines is required, unless " traverse tables" giving the latitude and departure for any bearing, as furnished in some books on surveying, are at hand. An approximate estimate of the number of acres included in the survey is sometimes made by drawing the plan upon sheet lead of uniform thickness, or upon Bristol board or heavy paper, cutting out the piece on the boundary lines, and weighing it in a delicate balance. The weight may then be compared with that of a similar piece that exactly comprises a definite number of acres, laid out upon the same scale. - The extensive territories of the United States are surveyed upon a peculiar system, planned with reference to the division of the lands into squares of uniform size, so arranged that any tract of 160 acres, or a " quarter section," may have its distinct designation and be readily found upon the map or recognized upon the ground by the marks left by the surveyors.

Each great survey is based upon a meridian line run due N. and S. by astronomical measurements the whole extent of the survey in these directions; and upon a "standard parallel " or base line running E. and W., similarly established with great accuracy. Parallels to these lines are run every 6 m., usually with the solar compass corrected by frequent celestial observations; and thus, as nearly as the figure of the earth admits, the surface is divided into squares of 6 m. N. and S. and the same E. and W., each one containing 36 sq. m. or sections, into which the territory is further divided by meridians and parallels run at every mile; while the half mile being marked on these lines by setting what is called a quarter post, the points are established for the subdivision into quarter sections. The squares of 36 sq. m. are termed townships, often contracted to "towns;" and each line of them E. and W. is numbered either N. or S. from the base line, and each line of them N. and S. is termed a range and is numbered E. or W. from the meridian. The N. and S. lines bordering the townships are known as range lines, and the E. and W. as township lines.

Each survey is designated by the meridian on which it is based, and of these principal meridians there are 6 designated by numbers and 18 by special names. The following table, compiled from information furnished from the general land office in August, 1875, gives the designation of the meridians, their longitude W. from Greenwich, and the N. latitude or other description of the principal base lines:

 DESIGNATION OF PRINCIPAL MERIDIAN. Longitude W. fr' m Greenwich. Principal base lines, N. lat. First, boundary bet. Ohio and Indiana.. 84° 51' Second, through Indiana ............. 86° 28' ............... Third,through Illinois 89° 10' 30" Fourth.N.from mouth of Illinois river___ 90° 29' 56" 38° 58' 12" Fifth. N. from mouth of Arkansas river.. 90° 5S' Mouth of St. Francis R. Sixth............... 97° 22' 40° Michigan ............... 84° 19' 9" 7 m. X. of Detroit. Tallahassee.......... 84° 18' Tallahassee, F1a. St, stephen ................. 8S° 2' 31° Huntsville........... 86° 31' N. boundary of Alabamn. Choctaw ............ 89º 10' 30" 29 m. S. of Jackson, Miss. Washington......... 91° 5' 31° St. Helena, S. from base................. 90° 11' 31° Louisiana........... 92° 20' 31° New Mexico......... 106° 52' 9" 34° 19' Great Salt Lake..... 111" 53' 47" 40° 46' 4" Bolse ........... 116° 20' 43° 26' Mt. Diablo.......... 121° 54' 37° 53' San Bernardino ......... 1160 56' 34° 6' Humboldt ............ 124° 11' 40° 25' 30" Willamette.......... 122" 44' 45° 30' Montana............ 111° 3S' 45° 46' 27" Gila and Salt river.... 112° 15'46" 33° 22' 57" Indian ..................... 97° 15' 56" 34° 31'

The 36 sections of each township are numbered in order, beginning with the N. E. corner and thence proceeding along the N. side of the township to section 6 in the N. W. corner; section 7 begins the next line of sections S., the numbers running E. to 12, and then beginning the third line with 13 and running W. to 18, and so on, bringing No. 36 in the S. E. corner of the township. The quarter sections are designated by their position as N. E., N. W., S. E., and S. W. Fractional sections of irregular shapes are admitted on the borders of lakes, rivers, etc. With these explanations any tract may be readily pointed out upon the government maps from its abbreviated description, or any locality in the wildest territory may be correctlv defined; thus the S. W. qr. sect. 13, T. 66 N., R. 34 W., meridian Michigan, is traced directly to an old mining location near the N. E. extremity of Isle Royale, Lake Superior. The law which established this system, while it required that the N. and S. lines should be true meridians, also required that the townships should be six miles square.

To satisfy both of these conditions is physically impossible, for the figure of the earth causes the meridians to converge toward the pole, thus making the N. line of each township shorter than its S. line; an inequality which becomes more and more marked the higher the latitude of the surveys. Provision is consequently made for correcting the errors thus caused, by establishing what are called correction lines, which are parallels bounding a line of townships on the north when lying N. of the principal base, or the S. line of townships when lying S. of the principal base, from which the surveys as they are continued are laid out anew, the range lines again starting at correct distances from the principal meridian, In Michigan these correction lines are repeated at the end of every tenth township, but in Oregon they have been repeated with every fifth township. The instructions to the surveyors have been that each range of townships should be made as much over 6 m. in width on each base and correction lino as it will fall short of the same width where it closes on to the next correction line N.; and it is further provided that in all cases where the exterior lines of the townships shall exceed or shall not extend 6 m., the excess or deficiency shall be specially noted and added to or deducted from the western or northern sections or half sections in such township, according as the error may be in running the lines from E. to W. or from S. to N. In order to throw the excesses or deficiencies on the N. and on the W. sides of the township, it is necessary to survey the section lines from S. to N. on a true meridian, leaving the result in the 1ST. line of the township to be governed by the convexity of the earth and the convergency of the meridians.

Navigable rivers, lakes, and islands are "meandered" or surveyed by the compass and chain along the banks. - The insus struments employed on these surveys, besides the solar compass, are a surveying chain 33 ft. long of 50 links, and another of smaller wire as a standard to be used for correcting the former, as often at least as every other day; also 11 tally pins made of steel, telescope, targets, tape measure, and tools for marking the lines upon trees or stones. In surveying through woods, trees intercepted by the line are marked with two chops or notches, one on each side; these are called sight or line trees. Other trees near by not touched by the line are blazed on two sides, quartering toward the line; but if at some distance from the line, the two blazes should be near together, on the side facing the line. These are generally found to be permanent marks, not only recognizable for many years, but carrying with them their own age by the rings of growth around the blaze, which may at any subsequent time be cut out and counted as years; and the same are recognized in courts of law as evidence of the date of the survey. They cannot be obliterated by cutting down the trees or otherwise without leaving evidence of the act.

Corners are marked upon trees if found at the right spots, or else upon posts set in the ground, and sometimes a monument of stones is used for a township corner and a single stone for section corners; mounds of earth are made where there are no stones nor timber. At the corners the four adjacent sections are designated by distinct marks cut into a tree, one in each section. These trees facing the corner are plainly marked with the letters B. T. (bearing tree) cut into the wood. Notches cut upon the corner posts or trees indicate the number of miles to the outlines of the township, or, if on the boundaries of the township, to the township corners. - A useful text book is the " Treatise on Land Surveying" of W. M. Gillespie (new ed., New York, 1S75).