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Ex. 5. Required the surface and the solidity of the greatest parallelopiped, whose length, breadth, and depth, together make 18?
Ex. 6. The surface of a square prism is 546: what is its solidity when a maximum ?
Ex. 7. The content of a cylinder is 169.645968: what, is its surface when a minimum ?
Ex. 8. The whole surface of a right cone is 201·061952: what is its solidity when a maximum ?
Ex. 9. The surface of a triangular pyramid is 43.30127: what is its capacity when a maximum ?
Ex. 10. The radius of a sphere is 10. Required the solidities of this sphere, of its circumscribed equilateral cone, and of its circumscribed cylinder.
Er. 11. The surface of a sphere is 28.274337, and of an irregular polyedron circumscribed about it 35: what are their respective solidities?
Ex. 12. The solidity of a sphere, equilateral cone, and Archimedean cylinder, are each 500: what are the surfaces and respective dimensions of each?
Ex. 13. If the surface of a sphere be represented by the number 4, the circumscribed cylinder's convex surface and whole surface will be 4 and 6, and the circumscribed equilateral cone's convex and whole surface, 6 and 9 respectively. Show how these numbers are deduced.
Er. 14. The solidity of a sphere, circumscribed cylinder, and circumscribed equilateral cone, are as the numbers 4, 6, and 9. Required the proof.
PLANE TRIGONOMETRY CONSIDERED ANALYTICALLY.
ART. 1. There are two methods which are adopted by mathematicians in investigating the theory of Trigonometry: the one Geometrical, the other Algebraical. In the former, the various relations of the sines, cosines, tangents, &c, of single or multiple arcs or angles, and those of the sides and angles of triangles, are deduced immmediately from the
figures to which the several enquiries are referred; each individual case requiring its own particular method, and resting on evidence peculiar to itself. In the latter, the nature and properties of the linear-angular quantities (sines, tangents, &c,) being first defined, some general relation of these quantities, or of them in connection with a triangle, is expressed by one or more algebraical equations; and then every other theorem or precept, of use in this branch of science, is developed by the simple reduction and transformation of the primitive equation. Thus, the rules for the three fundamental cases in Plane Trigonometry, which are deduced by three independent geometrical investigations, in the second volume of this Course of Mathematics, are obtained algebraically, by forming, between the three data and the three unknown quantities, three equations, and obtaining, in expressions of known terms, the value of each of the unknown quantities, the others being exterminated by the usual processes. Each of these general methods has its peculiar advantages. The geometrical method carries conviction at every step; and by keeping the objects of enquiry constantly before the eye of the student, serves admirably to guard him against the admission of error: the algebraical method, on the contrary, requiring little aid from first principles, but merely at the commencement of its career, is more properly mechanical than mental, and requires frequent checks to prevent any deviation from truth. The geometrical method is direct, and rapid, in producing the requisite conclusions at the outset of trigonometrical science; but slow and circuitous in arriving at those results which the modern state of the science requires while the algebraical method, though sometimes circuitous in the developement of the mere elementary theorems, is very rapid and fertile in producing those curious and interesting formula, which are wanted in the higher branches of pure analysis, and in mixed mathematics, especially in Physical Astronomy. This mode of developing the theory of Trigonometry is, consequently, well suited for the use of the more advanced student: and is therefore introduced here with as much brevity as is consistent with its nature and utility.
2. To save the trouble of turning very frequently to the 2d volume, a few of the principal definitions, there given, are here repeated, as follows:
The SINE of an arc is the perpendicular let fall from one of its extremities upon the diameter of the circle which passes through the other extremity.
The COSINE of an arc, is the sine of the complement of that arc, and is equal to the part of the radius comprised between the centre of the circle and the foot of the sine.
The TANGENT of an arc, is a line which touches the circle in one extremity of that arc, and is continued from thence till it meets a' line drawn from or through the centre and through the other extremity of the arc.
The SECANT of an arc, is the radius drawn through one of the extremities of that arc and prolonged till it meets the tangent drawn from the other extremity.
The VERSED SINE of an arc, is that part of the diameter of the circle which lies between the beginning of the arc and the foot of the sine.
The COTANGENT, COSECANT, and coverSED SINE of an arc, are the tangent, secant, and versed sine, of the complement of such arc.
3. Since arcs are proper and adequate measures of plane angles, (the ratio of any two plane angles being constantly equal to the ratio of the two arcs of any circle whose centre is the angular point, and which are intercepted by the lines whose inclinations form the angle), it is usual, and it is perfectly safe, to apply the above names without circumlocution as though they referred to the angles themselves; thus, when we speak of the sine, tangent, or secant, of an angle, we mean the sine, tangent, or secant, of the arc which measures that angle; the radius of the circle employed being known.
4. It has been shown in the 2d vol. (pa. 6), that the tangent is a fourth proportional to the cosine, sine, and radius; the secant, a third proportional to the cosine and radius; the cotangent, a fourth proportional to the sine, cosine, and radius; and the cosecant a third proportional to the sine and radius. Hence, making use of the obvious abbreviations, and converting the analogies into equations, we have
Or, assuming unity for the rad. of the circle, these will become
These preliminaries being borne in mind, the student may pursue his investigations.
5. Let ABC be any plane triangle, of which the side BC opposite the angle a is denoted by the small letter a, the side AC opposite the angle в by the small letter b, and the side AB opposite the angle c by
the small letter c, and CD perpendicular to AB: then is, c = a. cos B + b. cOS A.
For, since AC b, AD is the cosine of a to that radius ; consequently, supposing radius to be unity, we have AD b, COS. A. In like manner it is BD = α . cos. B. Therefore, AD † BD = AB = c = a. cos. B + b. cos. A. By pursuing similar reasoning with respect to the other two sides of the triangle, exactly analogous results will be obtained. Placed together, they will be as below:
6. Now, if from these equations it were required to find expressions for the angles of a plane triangle, when the sides are given; we have only to multiply the first of these equations by a, the second by b, the third by c, and to subtract each of the equations thus obtained from the sum of the other For thus we shall have
b2 + c2
a2 2bc. cos. A, whence cos. A =
7. More convenient expressions than these will be dedu, ced hereafter: but even these will, often be found very convenient, when the sides of triangles are expressed in integers, and tables of sines and tangents, as well as a table of squares, (like that in our first vol.) are at hand.
Suppose, for example, the sides of the triangle are a=320, b=562, c = 800, being the numbers given in prop. 4, pa. 161, of the Introduction to the Mathematical Tables: then we have
The remainder being log. cos. A, or of 18°20'
The remainder being log. cos. B, or of 33°35′ = 9·9207060 Then 180° (18° 20′ + 33° 35′) = 128° 5′ = c; where all the three triangles are determined in 7 lines.
8. If it were wished to get expressions for the sines, instead of the cosines, of the angles; it would merely be necessary to introduce into the preceding equations (marked II),
instead of cos. A, cos. B, &c, their equivalents cos. A=√(1— sin2. A), COS. B =√(1-sin2. B), &c. For then, after a little reduction, there would result,
Or, resolving the expression under the radical into its four constituent factors, substituting s for a+b+c, and reducing, the equations will become
These equations are moderately well suited for computation in their latter form; they are also perfectly symmetrical: and as indeed the quantities under the radical are identical, and are constituted of known terms, they may be represented by the same character; suppose K: then shall we have
Hence we may immediately deduce a very important theorem: for, the first of these equations, divided by the second,
Or, in words, the sides of plane triangles are proportional to the sines of their opposite angles. (See th. 1 Trig. vol. ii).
9. Before the remainder of the theorems, nécessary in the solution of plane triangles, are investigated, the fundamental proposition in the theory of sines, &c, must be deduced, and the method explained by which Tables of these quantities, confined within the limits of the quadrant, are made to extend to the whole circle, or to any number of quadrants whatever. In order to this, expressions must be first obtained for the sines, cosines, &c, of the sums and differences of any two arcs or angles. Now, it has been found (I) that a = b. cos. c + c COS. B. And the equations (IV) give Substituting these va
b = a
sin. c sin. A