PROBLEMS, &c. FOR EXERCISE IN CONIC SECTIONS.

1. Demonstrate that if a cylinder be cut obliquely the section will be an ellipse.

2. Show how to draw a tangent to an ellipse whose foci are F, f, from a given point P, situated on or without the curve.

3. Show how to draw a tangent and a normal to a given parabola from a given point P, either in, or without the curve.

4. The diameters of an ellipse are 16 and 12. Required the parameter and the area, and the magnitude of the equal conjugate diameters.

5. The base and altitude of a parabola are 12 and 9. Required the parameter, and the ordinates corresponding to the abscissæ 2, 3, and 4.

6. In the actual formation of arches, the voussoirs or arch-stones are so cut as to have their faces always perpendicular to the respective points of the curve upon which they stand. By what constructions may this be effected for the parabola and the ellipse?

7. Construct accurately on paper, a parabola whose base shall be 12 and altitude 9.

8. A cone, the diameter of whose base is 10 inches, and whose altitude is 12, is cut obliquely by a plane, which enters at 3 inches from the vertex on one slant slide, and comes out at 3 inches from the base on the opposite slant side. Required the dimensions of the section?

9. Suppose the same cone to be cut by a plane parallel to one of the slant sides, entering the other slant side at 4 inches from the vertex, what will be the dimensions of the section?

10. Let any straight line EFR be drawn through F, the focus, of a parabola, and terminated by the curve in E and R; then it is to be demonstrated that EF. FR = ER. parameter, and that any two such chords have the same ratio as the rectangles of the segments into which they are divided at the focus. 11. All circles are similar figures: all parabolas are similar figures: and hyperbolas between the same asymptotes are similar figures.

12. Similar ellipses or similar hyperbolas, have their axes in the same ratio: and if two similar figures of either kind have either their centres or a focus, coin

cident in position, and their transverse axes also coincident in direction, all lines drawn through the common centre or common focus, will be divided by the curves in the same ratio.

13. To describe a polygon in a conic section similar to a given polygon in a given and similar conic section.

14. Let TP, tp be tangents to a parabola

whose axis is VM at the extremities P and p of

the right ordinates PM, pm: then

1. tan. TPM : tan. tpm ;; PM : pm, and

2. tan. TPM-tan. tpm= 2 tan. pPM. Required demonstrations of these two useful theorems in practical gunnery.

M

15. If the opposite sides of a hexagon inscribed in a conic section be produced till they meet, the three points of intersection will be in the same straight line: and if the opposite angles of a hexagon circumscribed about a conic section be joined by the diagonals, these three diagonals will intersect in the same point.

16. If any point in the plane of a given conic section be given, then a line can be found such that all lines drawn through the point will be harmonically divided by the curve and that line: and, conversely, if the line be given, the point can be found.

17. If the opposite sides of a quadrilateral inscribed in a conic section be produced to meet, and likewise the opposite sides of quadrilateral whose sides touch the curve at the angles of the inscribed one be also produced to meet: then the four points of intersection are in the same straight line; and the four diagonals of the two quadrilaterals intersect in the same point.

18. If about two points as poles two given angles revolve, so that one leg of one angle intersect one leg of the other angle in a straight line, then the other legs will always intersect in a conic section.

19. If three lines revolve about these given points so that two of their intersections are always in two given lines, then the third point of intersection will always be in a conic section.

20. If tangents be drawn to touch two of three similar, and similarly situated conic sections, then the three points of mutual intersection of those tangents will be in the same straight line.

ON GEODESIC OPERATIONS, AND GEOMORPHY, OR THE FIGURE OF THE EARTH.

SECTION I.

General Account of this Kind of Surveying.

ART. 1. In the treatise on Land Surveying in the first volume of this Course of Mathematics, the directions were restricted to the necessary operations for surveying fields, farms, lordships, or at most counties; these being the only operations in which the generality of persons, who practise this kind of measurement, are likely to be engaged but there are especial occasions when it is

:

requisite to apply the principles of plane and spherical geometry, and the practices of surveying, to much more extensive portions of the earth's surface; and when of course much care and judgment are called into exercise, both with regard to the direction of the practical operations, and the management of the computations. The extensive processes which we are now about to consider, and which are characterised by the terms Geodesic Operations and Trigonometrical Snrveying, are usually undertaken for the accomplishment of one of these three objects. 1. The finding the difference of longitude, between two moderately distant and noted meridians; as the meridians of the observatories at Greenwich and Oxford, or of those at Greenwich and Paris. 2. The accurate determination of the geographical positions of the principal places, whether on the coast or inland, in an island or kingdom; with a view to give greater accuracy to maps, and to accommodate the navigator with the actual position, as to latitude and longitude, of the principal promontories, havens, and ports. These have, till lately, been desiderata, even in this country: the position of some important points, as the Lizard, not being known within seven minutes of a degree; and, until the publication of the Board of Ordnance maps, the best county maps being so erroneous, as in some cases to exhibit blunders of three miles in distances of less than twenty. 3. The measurement of a degree in various situations; and thence the de rmination of the figure and magnitude of the earth.

When objects so important as these are to be attained, it is manifest that, in order to ensure the desirable degree of correctness in the results, the instruments employed, the operations performed, and the computations required, must each have the greatest possible degree of accuracy. Of these, the first depend on the artist; the second on the surveyor, or engineer, who conducts them; and the latter on the theorist and calculator: they are these last which will chiefly engage our attention in the present chapter.

2. In the determination of distances of many miles, whether for the survey of a kingdom, or for the measurement of a degree, the whole line intervening between two extreme points is not absolutely measured; for this, on account of the inequalities of the earth's surface, would be always very difficult, and often impossible. But, a line of a few miles in length is very carefully measured on some plain, heath, or marsh, which is so nearly level as to facilitate the measurement of an actually horizontal line; and this line being assumed as the base of the operations, a variety of hills and elevated spots are selected, at which signals can be placed, suitably distant and visible one from another: the straight lines joining these points constitute a double series of triangles, of which the assumed base forms the first side; the angles of these, that is, the angles made at each station or signal staff, by two other signal staffs, are carefully measured by a theodolite, which is carried successively from one station to another. In such a series of triangles, care being always taken that one side is common to two of them, all the angles are known from the observations at the several stations; and a side of one of them being given, namely, that of the base measured, the sides of all the rest, as well as the distance from the first angle of the first triangle, to any part of the last triangle, may be found by the rules of trigonometry. And so, again, the bearing of any one of the sides, with respect to the meridian, being determined by observation, the bearings of any of the rest, with respect to the same meridian, will be known by computation. In these operations, it is always advisable, when circumstances will admit of it, to measure another base (called a base of verification) at or near the ulterior extremity of the series: for the length of this base, computed as one of the sides of the chain of triangles, compared with its length determined by actual admeasurement, will be a test of the accuracy of all the operations made in the series between the two bases.

H

3. Now, in every series of triangles, where each angle is to be ascertained with the same instrument, they should, as nearly as circumstances will permit, be equilateral. For, if it were possible to choose the stations in such manner, that each angle should be exactly 60 degrees; then, the half number of triangles in the series, multiplied into the length of one side of either triangle, would, as in the annexed figure, give at once the total distance ; and then also, not only the sides of the scale or ladder, constituted by this series of triangles, would be perfectly parallel, but the diagonal steps, marking the progress from one extremity to the other, would be alternately parallel throughout the whole length. Here, too, the first side might be found by a base crossing it perpendicularly of about half its length, as at H; and the last side verified by another such base, R, at the opposite extremity. If the respective sides of the series of triangles were 12 or 18 miles, these bases might advantageously be between 6 and 7, or between 9 and 10 miles respectively; according to circumstances. It may also be remarked (and the reason of it will be seen in the next section), that whenever only two angles of a triangle can be actually observed, each of them should be as nearly as possible 45°, or the sum of them about 90°; for the less the third or computed angle differs from 90°, the less probability there will be of any considerable error. See prob. 1, sect. 2, of this chapter.

R

4. The student may obtain a general notion of the method employed in measuring an arc of the meridian, from the following brief sketch and introductory illustrations.

The earth, it is well known, is nearly spherical. It may be either an ellipsoid of revolution, that is, a body formed by the rotation of an ellipse, the ratio of whose axes is nearly that of equality, on one of those axes; or it may approach nearly to the form of such an ellipsoid, or spheroid, while its deviations from that form, though small relatively, may still be sufficiently great in themselves, to prevent its being called a spheroid with much more propriety than it is called a sphere. One of the methods made use of to determine this point, is by means of extensive Geodesic operations.

The earth, however, be its exact form what it may, is a planet, which not only revolves in an orbit, but turns upon an axis. Now, if we conceive a plane to pass through the axis of rotation of the earth, and through the zenith of any place on its surface, this plane, if prolonged to the limits of the apparent celestial sphere, would there trace the circumference of a great circle, which would be the meridian of that place. All the points of the earth's surface, which have their zenith in that circumference, will be under the same celestial meridian, and will form the corresponding terrestrial meridian. If the earth be an irregular spheroid, this meridian will be a curve of double curvature; but if the earth be a solid of revolution, the terrestrial meridian will be a plane curve.

5. If the earth were a sphere, then every point upon a terrestrial meridian would be at an equal distance from the centre, and of consequence every degree upon that meridian would be of equal length. But if the earth be an ellipsoid of revolution slightly flattened at its poles, and protuberant at the equator; then, as will be soon shown, the degrees of the terrestrial meridian, in receding from the equator towards the poles, will be increased in the duplicate ratio of the right sine of the latitude; and the ratio of the earth's axes, as well as their actual magnitude, may be ascertained by comparing the lengths of a degree on the meridian in different latitudes. Hence appears the great importance of measuring a degree.

6. Now, instead of actually tracing a meridian on the surface of the earth,a measure which is prevented by the interposition of mountains, woods, rivers, and seas,-a construction is employed which furnishes the same result. It consists in this.

Let ABCDEF, &c. be a series of triangles, carried on, as nearly as may be, in the direction of the meridian, according to the observations in art. 3. These

triangles are really spherical or spheroidal triangles; but as their curvature is extremely small, they are treated the same as rectilinear triangles, either by reducing them to the chords of the respective terrestrial arcs AC, AB, BC, &c. or by deducting a third of the excess, of the sum of the three angles of each triangle above two right angles, from each angle of that triangle, and working with the remainders, and the three sides, as the dimensions of a plane triangle; the proper reductions to the centre of the station, to the horizon, and to the level of the sea, having been previously made. These computations being made throughout the series, the sides of the successive triangles are contemplated as arcs of the terrestrial spheroid. Suppose that we know, by observation, and the computations which will be explained in this chapter, the azimuth, or the inclination of the side AC to the first portion AM of the measured meridian, and that we find, by trigonometry, the point M where that curve will cut the side BC. The points A, B, C, being in the same horizontal plane, the line AM will also be in that plane: but, because of the curvature of the earth, the prolongation MM', of that line, will be found above the plane of the second horizontal triangle BCD: if, therefore, without changing the angle CMM', the line MM' be brought down to coincide with the plane of this second triangle, by being turned about BC as an axis, the point M' will describe an arc of a circle, which will be so very small, that it may be regarded as a right line perpendicular to the plane BCD: whence it follows, that the operation is reduced to bending down the side MM' in the plane of the meridian, and calculating the distance AMM', to find the position of the point M'. By bending down thus in imagination, one after another, the parts of the meridian on the corresponding horizontal angles we may obtain, by the aid of the computation, the direction and the length of such meridian, from one extremity of the series of triangles, to the other.

A line traced in the manner we have now been describing, or deduced from trigonometrical measures, by the means we have indicated, is called a geodetic or geodesic line: it has the property of being the shortest which can be drawn between its two extremities on the surface of the earth; and it is therefore the proper itinerary measure of the distance between those two points. Speaking rigorously, this curve differs a little from the terrestrial meridian, when the earth is not a solid of revolution: yet, in the real state of things, the difference between the two curves is so extremely minute, that it may safely be disregarded. 7. If now we conceive a circle perpendicular to the celestial meridian, and passing through the vertical of the place of the observer, it will represent the prime vertical of that place. The series of all the points of the earth's surface which have their zenith in the circumference of this circle will form the perpen