23. From one of the angles of a rectangular meadow there are two straight foot paths, one leading to the opposite angle, and the other to a stile 110 yards distant from it; this distance, with the two paths, forms a triangle, of which the angles are as the numbers 2, 3, and 10; what sum will pay for the mowing, making, and carting of the said meadow at 27s. 6d. per acre? Ans. 71. Ss. 24d.

24. There are three seaport towns A, B, and C, B bears ESE, and C, E by N from A; a telegraph is erected, for the purpose of speedy communication with the metropolis, at 5 miles distance from each of the towns, and in the line AC; required the distance of B from A and C, and its bearings from the telegraph? Ans. from B to A 8.3147 miles, from B to C 5.5557 miles; and B bears S Eb S from the telegraph.

25. A flag-staff is placed on a castle wall 163 feet long, in such a situation that a line of 100 feet in length will reach from its top to one end of the wall, and a line of 89 feet from its top to the other; required the height of the flag-staff, and its distance from the extremities of the wall? Ans. height 47.7244 feet; distance from one extremity 87.8773 feet, from the other 75.1227 feet.

26. In the hedge of a circular inclosure 500 yards in diameter three trees A, B, and C were planted in such a manner, that if straight lines be drawn from each to the other two, the angle at A will be double the angle at B, and the angle at C double of A and B together; required the distance between every two of the trees? Ans. from A to B 433.013 yards, from B to C 321.394 yards, and from A to C 171.01 yards.

A's assumed distance :: 558 : A's real distance; whence also B's distance will be found; and the distance divided by the number of hours, will give the rate of sailing per hour.

To find the angles, see the note on prob. 20. To find the sides; First, with the radius 250 describe a circle, and from it cut off a segment containing an angle equal to the greatest angle of the proposed triangle (34.3.), draw straight lines from the extremities of this chord to the centre, and an isosceles triangle will be formed by these three lines, of which the vertical angle (at the centre) will be double the supplement of the said greatest angle (20 and 22. 3.), and the three angles of this isosceles triangle will be known (32. 1.). Secondly, find the base (Art. 67.) which will be the greatest side of the proposed triangle (19. 1.), whence the two remaining sides will likewise be found by Art. 67.

27. An English sloop of war having orders to survey an enemy's port, placed two boats A and B at 1100 fathoms distance apart, A being directly east from B: at the inner extremity of the harbour there is a spire visible from the boats, likewise a castle on one point of the entrance, and a light-house on the other; at A the castle bore SSW, the spire S W by S, and the light-house W SW. At B the castle bore S E, the spire south, and the light-house S by W; required the length and breadth of the harbour? Ans. length from middle of entrance 1056 fathoms; breadth of entrance 920.59 fathoms.

28. On the opposite sides of an impassible wood, two cities A and B are situated; C is a town visible from A and B, distant from the former 3 miles, and from the latter 2, and they make at C an angle of 28°; now, it is desirable to cut a passage from A to B, and an engineer undertakes to make one, 19 feet wide, at 7s. 6d. per square yard; the inhabitants of A agree to furnish of the expense, which they can accomplish, by every 7 persons paying 31 shillings; those of B can make up the remainder, by every six persons subscribing 33 shillings; required the number of inhabitants in A and B ? Ans. A 43626,* B 8839, to the nearest unit.

29. An isosceles triangle has each of the angles at the base double that at the vertex; now, if the vertical angle be bisected, and either of the angles at the base trisected, the segment of the trisecting line, intercepted between the opposite side and the bisecting line, will be three inches; required the sides of the triangle? Ans. each of the equal sides 13.8314 inches; the base 8.35371 inches.

30. In a circle, whose radius is 5, a triangle is inscribed, and the perpendiculars from the centre of the circle to the sides of the triangle are as 1, 3, and 4; required the sides and angles of the triangle?

31. The altitude of a balloon as seen from A was 47°, and its bearings SE; from B, which is 24 miles south of A, it bore NE b N; required the perpendicular height of the balloon, and its distance from B?

PART X.

THE CONIC SECTIONS.

HISTORICAL INTRODUCTION.

IF a solid be cut into two parts by a plane passing through it, the surface made in the solid by the cutting plane, is called A SECTION.

If a fixed point be taken above a plane, and one of the extremities of a straight line passing through it be made to describe a circle on the plane, then will the segments of this line by their revolution, describe two solids (one on each side of the fixed point) which are called OPPOSITE CONES".

A plane may be made to cut a cone five ways; first, by passing through the vertex and the base; secondly, by passing through the cone parallel to the base; thirdly, by passing through it parallel to its sides; fourthly, by passing through the side of the cone and the base, so as likewise to cut the opposite cone; and fifthly, so as to cut its opposite sides in unequal angles », or in a position not parallel to the base.

a If the segment of the generating line between the fixed point and the base be of a given length, the cone described by its motion will be A RIGHT CONE, having its axis perpendicular to the base; but if the length of the segment be variable in any given ratio, so as to become in one revolution a maximum and a minimum, the cone produced will be AN OBLIQUE CONE, and its axis will make an oblique angle with the base.

Of course a right cone is here understood; for if the cone be oblique, the base, which is a circle, will cut the opposite sides in unequal angles, and the segment made by cutting them in equal angles will evidently be an ellipse.

If the plane pass through the vertex and the base, the section is a triangle; if it be parallel to the base, the section is a circle; if parallel to the side of the cone, the section is called A PARABOLA; if the plane pass through the side and cut the opposite cone, the section is called AN HYPERBOLA; and if it cut the opposite sides of the cone at unequal angles, the section is called

AN ELLIPSE.

The triangle and circle pertain to common elementary Geometry, and are treated of in the Elements of Euclid; the parabola, the ellipse, and the hyperbola, are the three figures which are denominated THE CONIC SECTIONS.

There are three ways in which these curves may be conceived to arise, from each of which their properties may be satisfactorily determined; first, by the section of a cone by a plane, as above described, which is the genuine method of the ancients; secondly, by algebraic equations, wherein their chief proporties are exhibited, and from whence their other properties are easily deduced, according to the methods of Fermat, Des Cartes, Roberval, Schooten, Sir Isaac Newton, and others of the moderns; thirdly, these curves may be described on a plane by local motion, and their properties determined as in other plane figures from their definition, and the principles of their construction. This method is employed in the following pages.

WHEN, or from whom the ancient Greek geometricians first acquired a knowledge of the nature and properties of the cone and its sections, we are not fully informed, although there is every reason to suppose that the discovery owes its origin to that inventive genius, and indefatigable application to science, which distinguished that learned people above all the other nations of antiquity. Some

of the most remarkable properties of these curves were in all probability known to the Greeks as early as the fifth century before Christ, as the study of them appears to have been cultivated (perhaps not as a new subject) in the time of Plato, A. C. 390. We are indeed told, that until his time the conic sections were not introduced into Geometry, and to him the honour of incorporating them with that science is usually ascribed. We have nothing remaining of his expressly on the subject, the early history of which, in common with that of almost every other branch of science, is involved in impenetrable obscurity.

The first writer on this branch of Geometry, of whom we have any certain account, was Aristæus, the disciple and friend of Plato, A. C. 380. He wrote, a treatise consisting of five books, on the Conic Sections; but unfortunately this work, which is said to have been much valued by the ancients, has not descended to us. Menechmus, by means of the intersections of these curves (which appears to have been the earliest instance of the kind) shewed the method of finding two mean proportionals, and thence the duplication of the cube; others applied the same theory, with equal success, to the trisection of an angle; these curious and difficult problems were attempted by almost every geometrician of this period, but the solution (as we have remarked in another place) has never yet been effected by pure elementary Geometry. Archytas, Eudoxus, Philolaus, Denostratus, and many others, chiefly of the Platonic school, penetrated deeply into this branch, and carried it to an amazing extent; succeeding geometers enriched it by the addition of several other curves as the cycloid, cissoid, conchoid, quadratris, spiral, &c. the whole forming a branch of science justly considered by the ancients