school of mathematics which he founded, and those “Elements” which ever since have formed the foundation of every great mathematician's knowledge. Euclid was undoubtedly the father of “Geometry,” the Greek word signifying literally “land-surveying.” It may be worth adding that even King Ptolemy became one of Euclid's pupils, and apparently a singularly stupid student too, for the ordinarily-gifted prince reprimanded his tutor for his own inability to learn geometry more quickly than his common subjects | The professor of surveying was, however, no courtier, but courageously “snubbed ” his imperial pupil with the remark—which has ever since remained a household proverb throughout the civilized world,—“Sire, there is no royal road to learning.” Any one hearing this little anecdote of Euclid for the first time, might think it worth while to run through his Elements again, when he will find that they are in the main directly or indirectly devoted to the measurement of figured space. An intending surveyor must have a good practical knowledge of simple arithmetic, decimal fractions, geometry and geometrical drawing, and should understand logarithms and algebra. In all cases, whether an area be that of a field, parish, county, or country, the fundamental principle upon which the whole practice of land surveying is based is that of reducing that area to the dimensions of a triangle or triangles. As we usually associate the area of an enclosed space with “square measure,” it may be objected that the square rather than the triangle should have been adopted as the unit of division. The reason, however, for choosing the triangle, or, rather, the reason why it is the only practicable unit of superficial measurement, is the remarkable fact that it, alone of all geometrical figures, is the one which cannot change its form without changing the length of its enclosing boundaries. Provided the length of a triangle's sides be constant, its form is unalterable. This may easily be demonstrated by drawing a triangle, each of whose sides measures, say, six inches. Now, on trying to “plot" another triangle of a different form, but with sides also measuring six inches each, it will be found that an absolute impossibility is being attempted. On the other hand, if a square with a side of six inches be constructed, it may readily be seen that such a figure could be pulled —without altering the length of its sides—into an infinite number of forms, each more or less resembling the outline of the conventional diamond. In like manner, the line enclosing a circle may be compressed into countless and still more varied shapes. Two other remarkable properties of the triangle may also here be mentioned : in proportion to the length of its boundary, it is the least capacious of geometrical figures, and the sum of its interior angles is invariably equal to two right angles. It must not be assumed, however, that the adoption of the triangle ensures accurate surveying, or that the measurement of merely its three sides is sufficient in practice correctly to draw the figure and to calculate its area. For these latter purposes, a proof or tie line touching the triangle should be added, and it may be laid down and measured from any one of its angles to the opposite side, or from one side to either of the other sides. In computing the area of a plot of any size or shape, the boundaries, however irregular they may be, are reduced to a sufficient number of straight lines. Even the arc, or circumference of a circle, is assumed to be the boundary of a “polygon,” a figure enclosed by an infinite number of sides, each of which is a straight line. Though it may be objected that this arbitrary system of reduction is not absolutely accurate, it is demonstrable that, with reasonable care, any error so arising is so infinitesimal that in practice it may be entirely disregarded. Land is usually measured by means of the chain alone or in conjunction with angle-measuring instruments such as the theodolite, sextant and others. The theodolite is, however, unnecessary for ordinary field-work, and, in practice, it is mainly used for urban, maritime, mine, railway or forest surveying, the measurement of inaccessible distances and especially in the survey of large areas, such as counties or countries. In these cases a knowledge of trigonometry and logarithms is essential, forming an extension of surveying beyond the scope of these elementary notes. In the British dominions, “Gunter's chain,” so-called from having been devised by Dr. Gunter, an eminent mathematician, is used. Its length is the tenth of a furlong, and thus the eightieth of a mile, or four poles, or twenty-two yards, or sixty-six feet. This unit of lineal measurement was adopted by Gunter in order that a rectangular space, measuring one furlong, or ten chains, in length, by one chain in width, should have an area of exactly one imperial acre. As the chain is sixty-six feet long, but is divided into one hundred links, it follows that each link has a length of sixty-six hundredths of a foot or 7-92 inches. The “Field-book '' is a record made in the field of all measurements; other necessary memoranda, and sketches relating to the work being done by the surveyor. These entries are begun at the foot of the page, and are continued upwards towards the top, thus following upwards or onwards, as it were, in the direction of the line being measured. Each page is usually divided into three parts extending from the foot to the top. The middle division is the narrowest, being merely wide enough to contain the entries of distances as measured on the chain line—where, for instance, fences are crossed, ineasurements to objects on either side of the chain are made, and “stations,” where two or more lines join or intersect, are entered. In the wider column on each side of the chainage entries, notes of the perpendicular distances from the line of near objects necessary to be taken into account, and called “offsets,” sketches of buildings, the direction of crossing or adjacent fences, names and other descriptions are appended to the right or left accordingly as they take place to the right or the left of the chain. Instead of the middle column, it is perhaps preferable to have merely one red-ink line running up the middle of each page, on and across which the chainage distances are entered. This one line undoubtedly represents, better than the wider column, the actual line measured on the ground, and it enables fences and other features occurring thereon to be sketched in with less distortion, and consequently with more facility for future recognition and reference. In surveying a considerable area, such as a parish or large estate, the surveyor should commence by carefully perambulating the tract, as he may thereby prevent a faulty construction of main lines of measurement, and may save himself the setting out and chaining of several subsidiary lines. If he finds that the boundary encloses— as it often does—a space approximating in form more or less to that of the trapezium or other quadrilateral figure, he should adopt the following system of six fundamental lines. Of these, four should run as nearly as possible alongside the four boundaries of the area, and should be joined at their extremities. The other two should be diagonals connecting the four angles of the quadrilateral, and intersecting each other, as nearly as may be, at right angles, and in such a manner as to make their four sections approximate each other in length. One main triangle is thus described and “tied" on each side of one diagonal, and the accuracy of such two triangles may be effectively checked by comparing the sum of their areas with the collective area of the two main triangles described on the other diagonal, for as the two former triangles together occupy precisely the same space as the two latter triangles together, it is obvious that the area of the two former should coincide exactly with that of the two latter. From the above it will be seen that the imperial acre contains one hundred thousand square links. If, therefore, any area be found in square links, its acreage may be ascertained by dividing the number of the latter denomination by one hundred thousand, a process instantaneously performed by simply placing a decimal point so as to cut |