be taken as an unit, and if Q

180 be

put for the figure BDEC,

A ABC = LA + 2B+ 2C – 180;
..A ADE - A ABC=Q= LB + 2C-180;

:: IQ = Į (B + C) – 90;
.. tan įQ=cotį (B+C)
cos (AC+AB)

. tan į A (Art. 244.)
cos Ž (AC AB;
sin (BD+CE)

. tanA (Introd. 13.) sin I (BD-CE) If, then, B be put for the base, and H, H', for the two sides that are at right angles to it,

sin } (H+H') tan Q

tan 5 B. cos į (H-H')

(272.) ScHolium. The problem, which has been solved in Art. 271, may often be advantageously employed in determining the number of square miles contained in any large tract of the Earth's surface. For, when the latitudes, and the longitudes, of several points, in the boundary of that region, are known, the whole space may be very conveniently divided into quadrilateral spherical figures, having, each of them, two sides perpendicular to the base ; and the calculation may then be carried on with great facility.

Thus, let BFGC be the space to be computed ; let the points B, F be in the circle of latitude ABD; and

the points C, G in the circle of latitude ACE; let DE be the equator, and A its pole. The angle A, or the arch DE, measures the difference of the longitudes of B and C, or of F and G; also BD, FD, CE, and GE, are the latitudes of the points B, F, C, and G, respectively: wherefore, by means of the equation, found in Art. 271, the spaces BDEC and FDEG may be computed; and their difference, BFGC, is the space required.

Again, let ABCDEQFGH be the tract to be mea

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sured ; let it lie on both sides of the equator AQ; and let the latitudes, and the longitudes, of the several points A, B, C, D, E, F, G, H, be given: Then, if circles of latitude, Bb, Cc, Dd, &c. be supposed to be drawn through those several given points, and if the points themselves be supposed to be joined, by arches of great circles, the whole space will be divided into four rightangled spherical triangles, and five such quadrilateral figures, as are described in Art. 271 : its superficial content may, therefore, readily be found, by means of that and of the preceding Articles.

PROP. V. (273.) Problem. To bisect an isosceles quadrantal triangle, by an arch of a great circle, drawn through a given point, in one of its equal sides.

Let ADE. be an isosceles quadrantal triangle, and B a given point in AD, one of its equal sides: It is required to bisect the triangle ADE, by an arch of a great circle, drawn through B.


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Let the arch BC be supposed to bisect the given triangle; and, the same notation being employed, as in Art. 271, let x be put for the arch CE: Then, (Art. 271.)

sin (H+x) tan , B
cos į (H-X)

tan į B

b b being put for

tanĮ B

tan, B
sin į H cos { x + cos H sin Ž x

cos į H cos į x + sin H sin į x ъ bo (Introd. 27, 28.)

tan H + tan į x
1+tan į H tan Ź x


(Introd. 17.)
.:: b.tan H + b tan { x=1+tan ļH tan į x;

1- b tan Į H
... tan


b-tan 5 H whence, the value of x, or CE, becomes known; and the arch of a great circle, which joins B, C, bisects the given isosceles quadrantal triangle ADE.

* See the Figure in Art. 271.



Spherical Trigonometry.




(274.) In attaining a knowledge of the theory of Spherical Trigonometry, no greater exercise of the memory is required, than that which is necessarily implied in the mental process, of following the connexion of the chain of proofs. When, however, the study of Trigonometry is finished, and occasion is, afterwards, found for a practical application of its theorems, it is plain, that, unless the theorems themselves can be recollected, reference must continually be made, to the books in which they are exhibited.

The objections which may be urged, against the latter mode of supplying an immediate want of such theorems, are very obvious. It is not only toublesome, but, under some circumstances, it


become impracticable ; and it always consumes much more time than does the mere act of recollection.

On the other hand, although the memory, in some individuals, appears to be naturally strong, and although it is, perhaps, of all our faculties the most manageable, and the most susceptible of improvement, yet there are very few persons, who would chuse to burden it with the details of Spherical Trigonometry. It becomes, therefore, necessary to enquire, whether there are any methods, by which the memory may well be relieved from such a weight of matter, uninteresting in itself, and chiefly valuable on account of the purposes to which it may be applied.

(275.) Now, there are, in reality, three ways, by which this kind of relief may be afforded.

The first consists merely in directing the attention to those particular theorems, which are of the greatest importance, and of the most frequent use; thus abridging the quantity, without altering the form, of what is to be remembered.

The second mode of relief is the invention of general rules; which, although they are comprised in few words, and are, therefore, easily gotten by heart, do nevertheless,

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