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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.

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.)

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.. b. tan 1⁄2 H + b tan 1⁄2 x=1+tan 1⁄2 H tan 1⁄2 x ;

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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.

PART II.

THE ELEMENTS OF

Spherical Trigonometry.

SECTION V.

ON THE FORMATION OF A TECHNICAL MEMORY, FOR THE PURPOSES OF 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 may 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,

comprehend many particular cases, which it would be irksome to remember, as so many distinct propositions.

The third method is founded on the well-known principle of the association of ideas; and consists in the substitution of such mathematical forms as are easily retained; and which, being recollected, suggest to the mind more complicated expressions.

It is evident, that only the two last of these methods can, with propriety, be said to belong to the province of Mnemonics. We shall proceed, however, to a brief illustration of all of them.

(276.) The most extensively useful, of all spherical theorems, is that which foreign writers have called the Theorem of the Four Sines; according to which, the sines of the sides, are proportional to the sines of the opposite angles, of a spherical triangle: and it is that, of all others, which is the most easily retained.

Of next, if not of equal importance, are Naper's four Analogies, demonstrated in Arts. 244. 245. And it appears, from the table, exhibited in Art. 264, that in addition to those analogies, and the theorem of the four sines, only two other expressions, one for the sine of half an angle, the other for the sine of half a side, are wanted, for the ready solution of all the cases of obliqueangled spherical triangles; which cases necessarily include those, also, of right-angled spherical triangles :

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