miles; required the ratio of their surfaces, and also of their volume, supposing them both to be spherical. Ans. the surfaces are as 134 to 1 nearly; and the volumes as 49 to 1 nearly. 58. A rectangular cistern whose length, breadth, and depth, internally, were 3ft 10in, 2ft 1in, and 2ft 9in respectively, was rested on props at the corners, of 4in high; but by accident one of the props was knocked out of its place: how much less water would the cistern hold when it was brought with that corner to rest upon the ground; and how much less, still, when two adjacent props were removed, either those under the side or under the end? 59. Let the section of the breast-work be as in Ex. 4, p. 486, and EO the breadth of the ditch at top be 20ft; the slopes of the ditch unequal so that ER : RD :: 2 : 3 and SO: SP :: 2 : 4; what must be the depth of the ditch, so that the earth thrown out shall form a glacis whose height is 3ft and base OL is 14ft? Ans. 6.8ft. 60. If the area of the profile ABHC be 100ft; and BF = 1, FH = 6, BG = 10, GR 13, RD = 6, and ER = 3ft: what must be the breadth of the ditch so that its section EDPS shall be equal to the profile ABHC and OKL (the section of the glacis) together, when the slopes BH, KL are in the same plane, and the slopes ED, OP, are equal? Ans. 25.778 ft. 61. (1) The four sides of a trapezium are 64, 15, 12, and 9 respectively, the first two of these sides make a right angle: required the area of the quadrilateral. (2) When the same four sides form a quadrilateral inscriptible in a circle, find its area, angles, and diagonals. 62. Find the ratio of the surfaces of the torrid zone, the two temperate, and the two frigid zones, of the earth; supposing the two tropics to be 23° 28′ from the equator, and the two polar circles to be 23° 28' from their respective poles. 63. A cone, whose altitude is 63, and diameter of the base 32, is to be cut, by sections parallel to the base, into four portions of equal curved surface: required the respective distances from the vertex, measured on the slant side, at which the sections are to be made. 64. The solid content of a spherical shell, is equal to that of a conic frustrum, the areas of whose two ends are respectively equal to the exterior and interior curve surfaces of the shell, and whose height is equal to the shell's thickness. 65. A sphere is to any circumscribing polyhedron, as the surface of the sphere to the surface of the polyhedron. 66. The surface of a sphere is double the curve surface of an inscribed cylinder whose height and diameter of the base are equal. Also, the surface of a sphere is to the curve surface of an equilateral inscribed cone, as 8 to 3. 67. If a cone be cut by a vertical section, the segment of the base cut off by that section, is to the corresponding segment of the same surface, as the radius of the base to the slant side of the cone. 68. The volume of a regular octahedron inscribed in a sphere is to the cube of the radius as 4 to 3. 69. If a, ẞ, y, be the angles under which any three diameters of a sphere to radius a intersect, and σ = (a + B + y): show that the volume of the parallelopiped which is formed by tangent planes at the extremities of their diameter, is expressed by 4a3 {sin o sin (σ — a) sin (σ — ß) sin (o — y) } ̃ ̄1. 70. Find the volumes of the pyramids which envelope a triangular and a square pile of balls, respectively, the side of the lower course in each being n. 71. A cannon-ball whose radius is r, may be touched by four, or by eight, or by twenty shells of equal radii, R, and each of which touches three of the remain ing ones; or it may be touched by six equal shells of radii R1, each of which touches four of the remaining ones; or, again, it may be touched by twelve equal shells of radii R2. each of which touches five of the others: it is required to prove these contacts, and assign the several external radii R, R,, R, of these shells in terms of r. 72. If a rectangular pile of six inch balls of 150 in length and 40 in breadth be roofed over, the roof being in close contact with the balls: how much empty space would be enclosed? NOTES. NOTE I. Synthetic Division, p. 129. ... A different and much more simple investigation of this process has occurred to me since the sheet on this subject was printed off. It is as follows:To divide Ax” + Bæ”¬1 + Cœ”−2 + · by xTM + ɑ1×TM−1 + α ̧ï3¬2` + ... Assume the quotient to be Aw"-" + A1, A2 are unknown: then, since multiplication be made. ... n-m-l А11⁄2”—m−1 + A2x”—m−2 + ...; in which quotient × divisor dividend, let this .... +.... In this we have worked by detached coefficients as usual. The first part of the operation shows the manner in which the coefficients of the divisor, which are known, are combined with those of the quotient, which are unknown, in forming those of the dividend; and, conversely, the second operation shows the formation of the addends to coefficients of the dividend to form those of the quotient, to be precisely the same as before, except that all the signs are changed. Moreover, when A (which is the same in the dividend and quotient) is known, we can form the diagonal column composed of — a1A, — α‚Â‚ — α ̧Â‚ — α‚Â‚ . . . ; and thence we obtain A,, and, consequently, the next diagonal column a1A1, аA1, — αA, ; then, similarly for A, and the next diagonal column, as far as it is necessary to carry the process. The change of sign of the coefficients of the divisor, it is obvious, is a consequence of the converse nature of the operation of finding the coefficients of dividend from those of the quotient and divisor, to that of finding the quotient from the divisor and dividend. The entire result is so strictly in accordance with the prescribed rule (p. 128) that any further detail in addition to what has been given would be superfluous. NOTE II. Small arcs, p. 436. THE tabular sine and tangent of a very small angle, or vice versa, which cannot be obtained from the tables on account of the rapid variations which those functions undergo at that stage, is often required to be found with great accuracy. They are, however, found in the following manner, without much labour. π Now, as the length of an arc of 1o, or of is 01745329 180' term of the series for this arc has no effective figure within the first ten decimal places; and hence, à fortiori, the series for a smaller arc can have no effective figure within the first seven decimal places. The series will be, then, effectively reduced in this case to its first two terms, and we shall have But when it is very small, cos a varies very slowly, and may be taken, for this purpose, with sufficient accuracy from the tables: whence the value of sin æ can also be computed from tab sin x = log x + 10 + } (tab cos x — 10). π Let the arc a contain p seconds, or a = 180.60.60 .p": then log x = log p + log — - log (180.602) = log p - 5.3144251... Substituting this in the preceding general formula, we have tab sin x = log p + 10 — 5·3144251 + ¦ (tab cos x — = 2. To find tab tan x when x is very small. 10) By the preceding we have sin x = x 3√ cos x, and hence tan x = - tab cos x); and, proceeding ac tab cos x. 3. Given tab sin x or tab tan x to find x itself. Here, by merely reversing the former processes, we have log p = tab sin x + 5·3144251 + ac tab cos x — 10, from either of which, according to the data, the value of a is found *. The rules indicated by these formula were first given, in a verbal form, by Dr. Maskelyne, in the Introduction to Taylor's Tables. Many investigations of them have since been given; but the above, whilst they are the most generally adopted ones, are amongst the most simple. This method of investigation itself was originally given in Woodhouse's Trigonometry. NOTE III. p. 423. The notation for powers of trigonometrical functions. In comparing the notation for powers with that for inverse functions, it cannot have escaped the student's attention that the same notation is used in two different senses; and a degree of confusion might arise from it, if he were not apprised of the reason, and of the relations of the two things signified by it. The strictly correct notation for powers is (cos x)", (tan 0)", etc.: but the increased space required for writing it in this way, as well as the additional trouble, has caused it to be written cos"x, tanme, etc., by some authors; and cosa", tan 0m, etc., by others. Now, so long as no idea of the successive trigonometrical functions—such as the cosine of a cosine, or the trigonometrical functions of any power of an arc-was entertained, mathematicians very naturally abridged the notation as far as possible, so as not to create doubt in the mind as to the signification of the expression. The introduction of the notation for inverse functions has, however, interfered with the former of these notations; whilst the latter has never been extensively used: and, in strict accuracy, we should be compelled to adopt (cos x)", (tan 0)", etc., as our standard notation; and especially, should the inquiries of mathematicians ever lead to results involving the successive trigonometrical functions, direct and inverse, of any expressions for the arc, in the same manner that they have already led to the consideration of successive logarithmic functions (p. 262). Instances of this, however, are so rare, that it would be difficult to quote one in any subject of even a moderately elementary character. We have hence, for the present, adhered to the old notation: though it was necessary to point out the circumstance for the satisfaction of the inquiring student. It may be added, too, that sin1r does really, in reference to the thing signified, bear the same meaning, though founded on a different view of the subject, as sin r; and hence, as a fundamental principle, there is no real difference in this stage (the utmost extent to which the notations under the two aspects coalesce) of the inquiry, between the quantities signified. NOTE IV. The angular unit taken as the arc equal to radius, p. 422. THE angular unit employed by the Greeks and by all the moderns, till the period of the French Revolution, was the ninetieth part of the quadrant: but by the French, the quadrant was divided centesimally, the hundredth part being the unit; with, however, a general impression that these were more advantageously considered as minor divisions of the quadrant, taken as the standardunit. Amongst many distinguished men of science, however, the radius of the circle by which the angle was estimated, has been considered the most advantageous standard-unit: and in some important applications of trigonometry it is undoubtedly the case, though in reference to the ordinary ones for which tables have already been computed, it would be altogether useless till very extensive tables adapted to this division shall have been published. The main applications, indeed, of this unit, are to purposes for which, from particular circumstances, the existing tables are inadequate. Denote by a and a the length and the number of degrees of the arc subtending the angle A°; and let the radius r contain p degrees of that circle. Then p° : 180° ::r: rπ, and A° : 180° :: a : ræ. In reference to this mode of estimating angles, p = 57°2957795 ... is to be considered the unit: and the angle is then said to be estimated in circular measure. NOTE V, pp. 451-6. Certain conditions amongst the data of plane triangles. In some of the examples given for solution, the student will have discovered a certain degree of uncertainty in the results which he obtained. This arises from the great relative variations of the trigonometrical functions compared with those of the angles themselves, in certain parts of the tables. In actual trigonometrical observations, the classes of conditions which require these computations to be employed, will, where practicable, be avoided: but numerous instances arise in practice where they cannot be avoided, and hence to remove the resulting uncertainty, other methods of solution have been devised, one or two of which are given in this note, with one or two other particulars relating to the subject. 1. There are given a, b, C, where b is very small in comparison with a, to find the remaining parts of the triangle. and taking log, of both sides, and expanding, we have 2 loge c=2logę a— a or log, c = b2 = loge a cos C cos 2C a 2a2 cos 3C. 3a2 ..... |