Popular Science Monthly/Volume 30/December 1886/Measuring the Earth's Surface
GEODETICAL science—that is, the particular branch of human investigation which is devoted to ascertain what are the exact form and dimensions of the earth—has not been slow to follow the general progress. The advance made in this branch of studies since it was first proved that the earth's form was spheric, and since Galileo uttered his historical "Eppur si muove," has been parallel to the advance made in all other branches of scientific knowledge and methods of investigation.
The first notions concerning the form of the earth were that its form was that of a tablet, ending abruptly at its extremities into what would be considered the abyss, which could not be reached by man. The idea that the earth was nothing but a plane was abandoned before the beginning of the Christian era. The earlier attempts at calculating the size of the globe were based on astronomical observations. It would be difficult to-day to say within what degree of accuracy the figures then obtained could have been relied upon, as the units of measurement used by those pioneers have been lost and could not be compared with the units now in use.
One of the earlier attempts at obtaining the actual length of the earth's meridian by direct measurement of a portion of the same was made in the sixteenth century by a French doctor. The means employed, although very ingenious, would be considered perfectly clumsy and inadequate by the modern scientist. There was in this early measurement no attempt at mathematical precision as understood in the present century, and, considering the simplicity of the method employed by the doctor, it is only to be wondered that no greater error was obtained in its final result. The measurement consisted simply in driving from Paris to Amiens, and counting the revolutions of the wheels of the carriage, and from the number of revolutions of the wheels obtain the distance between the two cities, which could serve as a basis for calculating the length of the meridian. Of course, this calculation could not by any means be considered accurate, but, taking into account the means employed, the result obtained has been subsequently found to be wonderfully precise. The most curious thing about it is, that what would now be considered grave errors and inexactitudes were so distributed that they almost compensated each other, and the dimensions then obtained show only slight differences with the dimensions given by the most recent measurements. Thus chance (and no better name could be found) permitted of the same results, with only a small final error, being obtained with that crude method, that are now obtained with the most precise instruments and with the most complicated calculations,
Geodetical triangulation is, like many of the other branches of scientific applications, essentially a child of the modern era. It is not older than the seventeenth century; the first application of geodetical triangulation to the measurement of an arc of the earth's meridian having been made in Holland at the beginning of that century. It was followed by similar measurements in England and in France, but in all these measurements the arc measured was never greater than two degrees, and the importance of such measurements on the question of the length of the earth's meridian could therefore not be considered very great. This fact was so keenly felt that the work done in France was extended both in a southward and in a northward direction at the beginning of the eighteenth century, and the distance between Dunkerque and Perpignan, the northern and southern extremities of France, was obtained by triangulation.
What is geodetical triangulation? If two sides and one angle, or one side and two angles, or three sides of a triangle, are known, the remaining parts of the triangle can be calculated by means of well-known formulas. It is on this property of triangles that geodetical or trigonometrical triangulation is based. Supposing the exact distance between two cities situated from one hundred to five hundred miles from each other has to be measured, it is not necessary to tramp the whole distance with a surveyor's chain or other measuring instrument. Such measurement would be too tedious, besides being incorrect, and could not be made in a straight line, even supposing that the ground between the two cities were all level, and that no obstacles intervened to render such straight-line measurement altogether impossible. But this difficulty can be obviated and the exact distance ascertained by means of triangulation. A number of intermediate points are taken, situated so that each three of them form a triangle in which the angles are not too small to be measured. The two ends of the line whose length has to be calculated are also used as points. A series of triangles is thus obtained, the sides of which are of course imaginary, between the points chosen. These points are called stations. The whole system of stations, and of the imaginary lines between these, is what is known as a triangular or trigonometrical net, because when drawn on paper all the lines between the various stations form a sort of net. If the actual distance between two of the stations of the net is known, and if the angles between any two lines of the net are measured by means of special instruments, all the distances between the various stations can be calculated, and thus the distances between any two stations, whether terminal or intermediate, can be ascertained.
However simple this work may seem in appearance, the difficulties to be encountered in its execution, and the probabilities of errors to be avoided, are so many that special scientific skill and thorough ability and training are required in those who have to undertake the practical execution of the work. Like much other scientific work, it has to be a work of love rather than a matter of duty on the part of the executors on whose observations the accuracy of the result necessarily depends. Dangerous ascents and solitary life on the top of high mountains, with no other society than that of the few assistants who accompany him, are common occurrences for the geodete. Not less dangerous to him is the ignorance and greed of the mountaineers, who, seeing his bright, well-kept instruments, imagine that they are made of gold, and often do not stop at any means to get possession of what they consider will make their fortune.
The progress made by mathematical science during the seventeenth and eighteenth centuries, and the great controversy raging then concerning the exact form of the earth, resulted in a serious attempt being made to measure arcs of the meridian at different places on the surface of the globe, and as much as possible near the central parallel, the equator, and the extreme parallel that could be reached, the polar circle. This work was undertaken by the French Academy of Sciences, and two expeditions were fitted out to undertake such measurements, one in Sweden and the other in Peru. The execution of the work was very accurate, considering the difficulties under which it was undertaken. Their purpose was to obtain the exact length of a degree at those different latitudes, and from these lengths the exact form of the earth. The results of the work done by the two expeditions were made known about the year 1750, and showed that the length of a degree near the equator was shorter than that of a degree in a northern latitude, the difference, expressed roughly and in a popular manner, being a little less than one per cent. This confirmed the theory which had been previously proposed, that the earth was depressed near the pole, so that, although this theory had been already advanced before the end of the seventeenth century, it was not generally accepted until it was shown to be correct by actual measurement. The impetus given to geodetical measurements by the last-named expeditions and by the results obtained was so great that geodetical work began to be done in earnest. The English triangulation was begun before the close of the last century. In India a short arc measurement was also executed about the same time.
The outbreak of the French Revolution, and the new ideas which it gave rise to, were the direct cause of the most interesting scientific work done at the close of the last century. As they abolished the privileges of classes, the new ideas tended also to abolish the privileges of systems, and a new system of computation was tried to be introduced which would give uniformity in division. This division was the decimal instead of duodecimal or others which had been until then the prevailing ones. Thus the year was divided into twelve months, and the month into three weeks of ten days each, the tenth day being made a civil holiday; the remaining five days of the year not being distributed in the various months, as with the Gregorian Calendar, but being put together as a civic yearly period of festivity at the end of the year, which was made to begin with the September moon, on the twenty-second day of September. The same was done as regards the system and units of measurement, value, etc.; but, while the time-divisions were made on a rather arbitrary basis, and have, therefore, not been able to supersede the older and more natural divisions, the decimal system of measures, weights, and values which was then introduced, rested on a thoroughly scientific basis, and has therefore been able to withstand all attacks and to gain introduction into the larger number of states into which Europe is divided. This system is the metric system.
Aside from the fact that the metric or decimal system permits of all sorts of calculations being made more quickly and easily, it possesses as its foundation a basis which is thoroughly scientific. The other systems of measurement are based on a more or less arbitrary standard, which may be indestructible, but which is liable to alter like anything made by the hands of mortal. But the basis on which the metric system rests can be supposed to be unalterable, being the earth itself as measured by the length of its meridian. The standard of unit in the metric system is the metre, the length of which is the 10000000 part of the quadrant, or the 40000000 part of the whole meridian. In order, therefore, to obtain the exact length of the metre, it was necessary to measure an arc of meridian of sufficient length to guarantee the exact calculation of the whole meridian. This arc, which had a length of nearly ten degrees, was measured between Dunkerque and Barcelona, the most prominent mathematicians of the time in France being intrusted with the execution. Although the greatest possible accuracy obtainable at that time was secured, the method of execution was not so perfect as those now in use. The metallic thermometer, invented by Borda, and which is described further on in the base measurement, was then used for the first time, but, instead of iron and zinc, the metals used were platinum and copper.
This arc was later on extended from Barcelona southward as far as the Balearic Isles by Arago and other French observers, who ran their net southward through Spain, and measured some very large triangles between the continent and the islands.
In all European countries geodetical measurements were made during the first half of the present century. To combine all these different lines measured by uniting them by means of special chains of triangles, and so obtain series of uninterrupted observations over a comparatively large area, was the work undertaken by Bessel in Prussia. This work possesses a high scientific and historical value on account of the thoroughness with which it was executed, and because the methods of execution then applied have become standard and have been accepted and imitated in modern geodesy. The geographical position of Prussia rendered the triangulation there of special importance as a means of connection between the different lines measured in the countries north and south, east and west of Prussia. This work was begun in 1831, and a connection was made between the French and Russian triangulations, and between the Danish and the South German nets.
Continental Europe has always taken the first place as the home of science; and scientific work, of whatever kind, can not fail to be duly appreciated there and to obtain that encouragement it necessarily needs. But Europe is politically so divided that the various states, however populous and powerful, are comparatively small in extent. For the accurate solution of geodetical problems vast areas and long distances are necessary, and these none of the European countries possess except Russia with its Asiatic dependencies. It has therefore been found necessary, in the interest of science, that the various countries combine together so as to have reliable observations extending over a vast area of territory which could be put together and aid in the solution of the problem that most interests the modern geodete—the exact form and size of the earth.
Although one by one the triangulations of each continental country were connected with those of the countries immediately surrounding it, there was no uniformity in the whole work until the proposal was made that all the countries combine together and act on a uniform scientific plan.
In the United States no geodetical work was done before the year 1831. The only arc measurement executed previous to that date was made by Mason and Dixon in the eighteenth century. It is a part of the line now separating the States of Maryland and Delaware, the direction of which is almost exactly north to south. The length of the arc was not over one and one half degree, and was measured directly with long wooden rods.
More recently the United States Coast and Geodetic Survey have been actively engaged in completing the measurement of an arc north to south and another east to west, the work done so far on the first bringing its length to twenty-two degrees, while the parallel arc across the continent will have when completed a length of forty-nine degrees. The advantages which these measurements in the United States have as scientific results are their great length, and, being executed by one authority, their uniformity in the methods of execution.
Although the principle is always the same, the methods of execution in geodetical work may show slight variations from one country to another. The following description of the field-work that has to be done is therefore of a general character, being intended to give the reader who is not familiar with mathematical studies, and with methods of measurement, an insight into the thoroughness with which geodetical work has to be executed, and the minuteness with which all the details of the work have to be carefully considered.
If the length of one side of a triangle is known, the length of the other two sides can be calculated, provided that at least two of the angles of the triangle can be measured by direct or indirect observation. It is therefore indispensable that a straight line be carefully measured, and the length of this line can be used as a basis by means of which all the distances between the various points or stations of the net can be calculated. This straight line that is actually measured in the field is used as the side of one of the triangles, and the other two sides are calculated with the help of the angles which can be measured by means of angular observations. The two sides of the first triangle, the lengths of which have been calculated, form with other imaginary lines other triangles, which may be designated as triangles numbers 2, 3, 4, etc. Each triangle has one side in common with the following triangle. Triangle number 1, for instance, may have one side in common with number 2, and one with number 3. If, therefore, the lengths of the three sides of triangle number 1 are known, these, together with the observations of the angles of number 2 and number 3, permit of the lengths of the sides of triangles number 2 and number 3 to be calculated, and so on.
The straight line that has to be actually measured in the field is known as a geodetical base. The accuracy necessary in the measurement of a geodetical base leaves all ordinary methods of which surveyors dispose altogether out of the question as too incorrect; a system of measurement has therefore to be applied which permits of the measurement being executed in a line scrupulously straight, of all variations in temperature which can affect the length of the measuring-rods being carefully noted and kept account of, and of the rods themselves being kept in a perfectly horizontal position. The measuring-rods are themselves very delicate and costly instruments. They consist of a prism of iron or steel, four metres long, on the upper surface of which another rod of metallic zinc rests, the zinc rod being somewhat shorter than the other, both being so placed on their supports as to prevent their bending and allowing them free expansion. The coefficient of expansion of zinc is much greater than that of iron, therefore the expansion or contraction of the zinc rod is much greater than that of the other. Changes in temperature in the two metals can thus be easily ascertained by actual measurement of the distance which separates the end of the zinc rod from a given point, marked on the iron rod. Four or six such rods are used for measuring a geodetical base. The rods are each in a long, wooden case, provided with micrometric arrangements for placing the rods in a straight line, raising or lowering the ends so as to have them perfectly horizontal; with spirit-levels, glasses, microscopes, etc. When the rods have been carefully placed in line, the distance between the end of one rod and the following has to be ascertained, and also, for getting at the actual length of each rod, the difference in the length of the iron and zinc rod in each of the cases. A small space is always left between two rods, which are not made to touch each other, in order to avoid sudden and too sharp contact. This intervening space, as well as the varying distance between the ends of the zinc and iron rods, is measured by means of small pieces of crystal a few inches long which have the form of half-prisms, being larger at one end and growing gradually smaller toward the other end. In fact, two of the four sides lengthwise have the form of a trapeze, while the other two are rectangles. One of these rectangular faces is divided to scale, and the observer has only to insert this piece of crystal between the two ends of the rods, without forcing it in, and to call the scale. The same is done for noting the variation in temperature by measuring the space between the ends of the iron and zinc rods in each case. In order to leave no cause for error, the two ends of each rod are cut to sharp edges and made one vertical and the other horizontal, and the horizontal end of one rod faces the vertical end of the other. The same is done for the zinc and iron. When one of the cases is moved forward and placed in position ahead of the others, all the measurements between any two rods are repeated in order to detect whether any of the other rods has moved while the other was being placed in position. Spirit-levels are on each case, and their variation is carefully measured with microscopes. All the data are registered, and have afterward to be carefully gone over, and all compensations for error, temperature, etc., duly allowed. This apparatus is known as Bessel's base-measuring apparatus, the chief feature of which is the metallic thermometer with iron and zinc.
The length of a geodetical base varies according to the area to be covered by the net and to the possibility of finding a good ground for laying such a base. From five to ten miles would be considered a good geodetical base, one longer than the higher figure not being advisable on account of the possible increase in the inevitable error of such measurement. Even bases of about two miles have been measured, but these could not be used for very large triangulations, and are more intended as a check on other measurement and as a means of compensating errors. The ground has to be carefully leveled before the base can be measured. The two end-points of the base are marked by stone monuments which can be seen at a distance, and the whole length is divided into so many sections, each of which constitutes a day's work, stone and metallic tablets being laid and the line marked in advance before the actual measurement can be undertaken.
It is not necessary that the base should be actually measured before the real triangulation work—that is, the measuring of the angles at each of the stations of the net—is begun. The work of measuring a base is necessarily very slow. All apparatuses have to be carefully tested before being actually used in the field. Each of the rods has to be subjected to a series of experiments at different temperatures, in order to determine the actual expansion of the iron by comparison with that of the zinc. The actual measuring in the field occupies one or two months' time, and may be longer if the weather is not favorable, the whole distance, divided into sections, each of which constitutes a day's work, being measured at least twice, once in each sense. Five or six experienced operators are required, besides a number of assistants to do the menial work—such as carrying the rods forward, etc. The calculations are, later on, done in the office, and are in themselves a very slow and exhaustive process. Taking all together, the time required is little short of one year, including the preparatory experiments and the calculations.
When the base-line has been established and measured, and its terminals have been so marked with permanent material as to be practically indestructible, the base has to be developed—that is, a complete set of observations has to be made for the purpose of connecting the base with the stations of the net. The base being, as a rule, much shorter than the sides of the triangles of the actual net, it can not be connected directly with these large sides, as the triangles thus formed would have very small angles. A special net of triangles, the sides of which grow larger by degrees until they reach the large sides of the actual net, is established. This small net is given the form of a polygon for the purpose of increasing the accuracy. Longer bases are sometimes divided into two halves, and, besides the two terminals, a central station is established in the middle of the base, and thus three base stations are obtained instead of two. All the angular observations at the base stations, and at those which may be called stations of development, are made in the same number and with the same accuracy as in those of the net proper. They are all considered as first-class stations.
All geodetical points at which angular observations are made can be divided into four classes. In the first class are included all the base stations, the developing stations, and the actual stations of the geodetical net. The second class includes those stations which are of secondary importance geodetically, and which do not belong to the net proper. The observations at these stations are not so exhaustive as in first-class stations, although they are used also for controlling the observations of the others. Third- and fourth-class stations have more importance as topographical points, as they are used by the topographical operators as starting-points when mapping out the country. For scientific purposes, only the data collected at first-class stations are used, all others being rejected.
For ordinary topographical purposes, the number of angular observations at each station is not so large as when these have to be used for scientific purposes.
Except for the measurement of bases, geodetical triangulation consists almost exclusively in angular observations. In fact, it can be called essentially a measurement by angles, the work to be done, and on which many years may be spent before even a small net can be called complete, being an uninterrupted series of measurements of angles. Very delicate instruments are used in these measurements. The best part of an observer's outfit consists of a good theodolite. Although simple in principle, the theodolite is a very complicated instrument, and a good deal of practice is necessary to enable an observer to become efficient in handling this delicate machine. It consists chiefly in a good field-glass, which can be turned in every direction, so as to enable the observer to see the exact spots in the distance which are the stations of the net, and of a circle on which a scale, carefully divided, enables the observer to read the angle between any two directions in which the glass may be pointed. The glass may be turned toward any point on the horizon, and also in an upward or downward direction. The scale on the circle is read by means of two microscopes, diametrically opposite to each other. Spirit-levels, micrometric arrangements, etc., are provided, and the observer has to be very careful about placing his instrument in the right position before he can actually begin operations. Although the stations are not at the same altitude, the angles between any two directions have to be so measured as if the stations were all on the same level, on a perfect plane. This plane is supposed to be vertical to the earth's radius which crosses that particular station—that is, a plane tangent to the earth's circumference and perfectly horizontal. The movement of the field-glass has, therefore, to be such that, although two stations may be situated lower or higher, the angle between the two directions can be read as if each of them had been raised or lowered vertically to the level of the station from which the observations are being made. The circle of the theodolite represents this horizontal plane; its center is supposed to be mathematically "the point," and the theodolite has, therefore, to be so placed that the center of this circle is in a perfect perpendicular to the point, while the surface of the circle itself is perfectly horizontal. The field-glass is situated on a support in the form of a double column, and the central axis of this support is vertical to the circle, and passes through its center. Delicate and exact mechanical arrangements permit of the glass being turned toward all points of the compass, and also of its being turned in an upward or downward direction; but each movement is either in a horizontal or a vertical direction to the circle. This enables the observer to obtain the angle desired—that is, the angle which any two directions would give if all the stations were at the same level.
The length of the sides of the triangles varies according to the facilities for extending a good net which the ground offers. From twelve miles upward is a suitable distance, the distance being in some cases only limited by the visual power of the glass.
The scale on the circle is divided into degrees, minutes, and possibly seconds, the latter and their fractions being read with the microscope.
The angles are measured as follows: When the theodolite is placed in its exact position, and the circle is perfectly horizontal, the glass is pointed successively at each of the surrounding stations, and for every direction the scale on the circle is read and noted on the field-book. Supposing the scale reads 56° 18' 12·075" when the glass is pointed toward one station, and 115° 56' 18·850" when pointed in another direction, the angle between the two directions is equal to the difference between the two readings, which is in this case 59° 38' 6·775". These readings are repeated several times, the circle being every time moved around its center in order not to have all the readings on the same divisions of the scale. When a complete set of observations has been made, another set of observations is begun with the field-glass inverted. The observer places the larger end of the glass toward him, and turns the glass upside down around the axis, so that by this movement the smaller end of the glass comes to stay near the observer. With the glass thus inverted observations are repeated. The moving around of the circle and inverting of the glass are intended to avoid any errors which might be caused by faulty construction of the instrument, as, however costly and delicate the instrument may be, human skill can not make it mathematically precise.
When the angles have been measured, all the calculations have to be made for each triangle and for each polygon separately. Neither the work of mensuration in the field nor the calculations are what would be called work done by steam or by electricity. The season for the field observations being necessarily limited, each observer can not cover more than two or three first-class stations each year, as much as two months being often spent for one station alone when the climatic conditions are unfavorable. Angular observations for scientific purposes can not be executed in all weathers and at all hours.
Geodetical triangulation can be more successfully executed in mountainous regions, where the peaks act as natural observatories, and nothing interferes to prevent distant points being seen with the glass. Before the measuring of the angles actually begins, the net has to be laid out, the stations have to be visited, the most suitable point to be chosen and marked with some permanent sign. Stone blocks are used for this purpose, in the center of which a square metallic tablet is laid. The intersection of the two diagonals of the metallic tablet is the geodetical point. A small pyramid is sometimes placed in lieu of the tablet. In order to see the exact spot at a distance, a pyramid of wood or other material is built over the same, and a metallic rod, similar to a lightning-rod, situated in an exact perpendicular to the "point," is placed on top of the pyramid. This rod acts for all practical purposes as the real spot when the observer is at a distance. For stations situated in the plains, church-steeples, towers, or the tops of high buildings are used, and a given spot on these is chosen as the geodetical point, care being taken to choose only such points as are likely to be permanent for future reference, and are not liable to get out of the perpendicular.
The carefully executed observations described in the foregoing have to be made so accurate in order to avoid errors, which, although they may be allowed up to a certain limit when a topographical survey is alone being made, can not be allowed when the triangulation has to be made use of for purposes of getting at the real dimensions of the globe. An error which may be neglected on an area of a few hundred square miles is not permissible, and would be too large if multiplied to the whole length of the earth's meridian.The accompanying illustration will serve to show what is a geodetical or trigonometrical net of triangles. It is not an imaginary net, but
one which has been actually laid out and measured. This net is a part of the work done by the United States Geodetic Survey, and is that portion of it which covers the whole State of New Jersey. It is copied from reports kindly furnished by Mr. G. H. Cook, State Geologist of New Jersey. It includes all the stations in New Jersey, and a few in the surrounding States, in all forty-seven stations. They are all first-class stations. To show how slow geodetical triangulation has necessarily to be, it may be stated that in the small State of New Jersey alone this work has been going on for nearly thirteen years, and is now nearly completed, only a few stations remaining to be covered at the beginning of the year 1886.
I may here state also that a few of the historical facts I have given are taken from "Elements of Geodesy," by J. Howard Gore, B. S., just published.
The idea of connecting the various measurements in the different European states was later on improved upon, and for the purpose of obtaining good, reliable data, collected on a system of uniformity sufficiently numerous and covering a large area of territory, all the states of Continental Europe have combined in the interest of science. If each country did its work separately, and the data obtained in one could not be compared with others, the observations made would have only a local value, and, being limited in extent, could not have that scientific weight which it is necessary they should have. All European countries have felt the necessity of having thorough topographical surveys made, so as to possess good, detailed maps of their territory. This work being considered necessary for military purposes, its execution has been undertaken by the military authorities. The triangulation work necessary for this purpose could, with little addition, be extended so as to connect the geodetical nets of the various countries and form a complete system of nets extending over the whole continent. An agreement was entered into, by the various states into which Europe is divided, that the geodetic data which were being collected and the observations that were being made should become common property, and that all the observations being made on a standard of uniformity agreed upon by all parties concerned, they should be used in common for the purpose of furthering the scientific problem and obtaining a series of nets, by means of which the exact distance between any two points on the European Continent could be easily calculated.
The common work—that is, the direction of the whole as an international undertaking, each country doing its own share within its own borders—has been confided to an international commission specially founded for the purpose, and which is known as the Commission Internationale pour la Mesure du Degré en Europe. All the states are represented in this commission, the representatives being mostly the heads of the geodetical department of each country, and some of the best-known astronomers. The best specialists of Europe, who have devoted their life to this branch of studies, belong to this commission.
The international agreement makes it possible to have uninterrupted chains of triangles across the whole continent, from north to south, and from east to west. It is, however, not necessary that the nets should extend over the whole area of each country. Neither the topographical necessities nor the scientific requirements make such a complete geodetical survey indispensable. Those countries possessing a comparatively larger territory would find a complete triangulation too costly and too slow. Each country has therefore laid its nets as it thought best. The Atlantic countries, France and Spain, have laid a series of parallel nets from north to south, and another from east to west, crossing each other almost at right angles. Supposing each of these nets to be drawn on the map, but instead of the various triangles a thick black line to be laid down as a sort of central line of each separate net, the whole systems would have the appearance of square grates or pigeon-holes. Germany and Austria have not observed the same rule; their lines are less regular in form, although just as convenient for the purpose. Italy and the smaller states have found it necessary to cover their whole territory, on account of their irregular geographical form, or their smallness. Sweden and Norway have run several nets, and also Russia; but the vast area to be covered leaves a thorough and systematic triangulation of the whole country out of the question.
The control which observations extending over such a vast area permit is very great. The possibilities of its being extended over a still wider field are only limited by political difficulties and by the great cost necessary for its execution in half-civilized countries. The result of the work undertaken by the International Commission can not fail to be of the highest scientific value, and the standing of the men who compose it is a guarantee that the greatest efficiency and thoroughness characterize the work done.
Of the calculations necessary, and which follow the field-work, the least said the better. To an outsider, one who is incapable of comprehending the scientific purpose of the same, they look very much like time wasted and which could have been better employed. It will suffice to say that all calculations are made twice and independently of each other. Each set of calculators do their work independently of the other, and only compare the final result. Months and months elapse before a partial result is reached, and before other and more complicated calculations can be begun. But the battle-field is one worthy of man; he has arrayed himself against figures, and, although slowly, he conquers them with the help of formulas, equations, and logarithms, all children of his fertile brain. The scientific result is obtained with scientific means. From beginning to end, geodetical triangulation is purely scientific; nothing is left to chance, and, although it can not by any means be mathematically precise, it comes as near the point of complete correctness as it is possible for any human thing to be.