Cafe.

The Solution of the Cafes of oblique fpherical Triangles, (Fig. 12 and 13.)

1

and the angle A oppofite to one of

pofite to
other.

the

The angle ACB.

and the angle A The other fide oppofite to one of

them.

Two fides AC, AB,

and the included angle A.

AB.

The other fide
BC.

Two fides AC, AB, Either of

and the included)

B.

B.

Solution.

As fige BC fine A :: fine A Cfme B. Note. When BC is lefs than AC, it cannot be determined whether B be acute or obtufe.

the As radius: cofine A: tangent AC : tanother angles, as gent AD. (by Theor. 1.) Whence BD is known; then as fine BD: fine AD :: tangent A tangent B. (by Theor. 4.) As radius: cofine AB:: tangent A: cotanThe other angle gent ACD. (by Theor. 4.) Whence BCD is also known; then as fine ACD : fine BCD:: cofine A: cofine B. (by Theor. 3.) As radius: cofine AC :: tangent A: cotangent AC D. (by Theor. 4.) Whence BCD is also known; then as cofine BCD cofine ACD :: tangent AC tangent BC. (by Theor. 1.)

Two angles A. Either of the
ACB, and the fide other fides, as
AC betwixt them BC.

Two angles A, B, and the fide AC

oppofite to one

of them.

Two angles A, B,

The fide BC,

oppofite to the As fine B; fine AC :: fine A ; fine BC.

other.

and the fide ACThe fide AB be-
oppofite to one
twixt them.
of them.

Two angles A, B,

and the fide AC The other angle
oppofite to one ACB.

of them.

All the three fides

AB, AC, and BC. Any angle, as A.

All the three angles

A, B, and ACB. Any fide, as AC.

VOL. 11.

t

The following propofitions concerning fpherical triangles, will render them more intelligible.

1. A fpherical triangle is either equilateral, ifofcelar, or fcalene, according as it has the three angles all equal, or two of them equal, or all three unequal; and vice versa.

2. The greatest fide is always oppofite the greateft angle, and the smallest fide oppofite the smallest angle.

3. Any two fides, taken together, are greater than the third.

4. If the three angles of a fpherical triangle be all acute, or all right, or all obtufe, angles, the three fides will be accordingly all less than 90 degrees, or equal to 90 degrees, or greater than 90 degrees; and vice verfa.

5. If from the three angles A, B, C, (fig. 14) of a sphe rical triangle, A, B, C, as poles, there be defcribed upon the furface of the sphere, three arches of a great circle, DE, DF, and FE, forming by their interfections another spherical triangle D E F; each fide of the new triangle will be the supplement of the angle at its pole; and each angle of the fame triangle will be the fupplement of the fide oppofite to it in the triangle ABC.

6. In any triangle A B C, or A b C, right angled at A, fig. 15,) the angles at the hypothenufe are always of the fame kind as their oppofite fides. And the hypothenuse is greater or less than a quadrant, according as the fides, including the right angle, are of different kinds, or of the fame kind; that is to fay, according as these same sides are either both obtufe, or both acute, or as one is obtufe, and the other acute, and vice verfa. Firft, the fides, including the right angle, are always of the fame kind as their oppofite angles. Secondly, the fides, including the right angle, will be of the fame or different kinds, according as the hypothenufe is lefs or more than 90 degrees; but one of them at least will be of go degrees, if the hypothenufe be fo.

CHAP.

CHAP. XIII.

OF GEOGRAPHY.

SECT. I.

GEOGRAPHICAL DEFINITIONS.

1. GEOGRAPHY is the knowledge of the earth, or a defcription of the terrestrial globe, particularly of the furface and known habitable parts thereof, with all its different divifions,

2. The earth is a globular body, furrounded with an atmosphere of air, by which all terreftrial bodies are confined to its furface, being attracted thereto by the laws of gravity.

That the earth is of a globular form, has been demonftrated by a number of experiments, particularly by obfervations of the eclipfes of the moon; in which it appears, that the fhadow of the earth always appears circular, whichever way it is projected. Alfo, by the observation of ships at fea, which, after their departure from any coaft, gradually disappear to an observer on land, from the bottom upwards; that is, the first part which disappears from the fight, is the keel, or lower part of the fhip; then those parts which are higher up, and so on; the top of the maft being the part

that is laft feen: this is owing to the convexity of the waters, which have the fame globular figure as the earth *.

3. The earth also has a diurnal motion on its own axis, performing one revolution in 24 hours; thereby occasioning the changes of the day and night, as will be feen in Aftronomy.

4. The circumference of the globe is fuppofed to be divided into 360 parts, called degrees, and each degree fubdivided into 60 minutes, and each minute into 60 seconds, &c. Every degree contains 60 geographic miles; confequently, the circumference of the globe is 21,600 such miles; and the diameter 6900 miles. But as 60 geographical miles are above 69 British measure, the circuit of the globe is therefore 24,840 English miles, and the diameter almost a third, or 7900 in round numbers.

5. The globe of the earth confists of land and water: the proportion of the land to the water is not accurately known, but it is generally believed to be near one third.

6. The waters are divided into three oceans, (befides the Imaller feas:) viz. The Atlantic, Pacific, and the Indian ocean. 1. The Atlantic or Western ocean divides Europe from America, and is 3000 miles wide. 2. The Pacific ocean divides America from Afia, and New Holland, and is 10,000 miles wide. 3. The Indian ocean lies between the Eaft In'dies and Africa, and is 3000 miles wide. The parts or

* The philofophers of the laft age differed greatly in their defcriptions of the true fpherical figure of the earth, of which there were two general opinions. The one maintained that the earth was a prolate fpheroid of this number was Caffini. The other party maintained it to be an oblate fpheroid: of this Sir Ifaac Newton was the chief, who infifted that the polar was shorter than the equatorial diameter by 36 miles. A party of philofophers from France was fent by the king of that country, to measure a degree on the polar circle, and alfo on the equator; the refult of their experiment turned out exactly in favour of Sir Ifaac Newton's theory. But this inequality of the shape of the earth, as well as the inequalities occafioned by the mountains, &c. make no fenfible difference in the form of the earth. 4

branches

branches of these oceans, called feas, a

as the Mediterranean

fea, &c. receive their names generally from the countries they border upon.

7. A bay, or gulf, is a part of the fea, almoft furrounded by land; as the gulf of Mexico, the bay of Biscay, &c.

8. A ftrait is a narrow paffage out of one fea into ans other; as the ftrait of Gibraltar.

9. A lake is a water surrounded by land; as the lake of Geneva.

10. Rivers are streams of water, iffuing from fprings in high grounds, and falling into the sea, or other rivers; and are wider near their mouths than towards their heads or fprings: they are described in maps by black lines.

II. The land is divided into two great continents: viz. The eastern and western continents, befides iflands. The eastern continent is fubdivided into three parts: viz. Eu rope, which is the north-west part; Afia, the north-eaft; and Africa, the fouth. The western continent confists of America only, divided into North and South America.

12. An island is a piece of land entirely furrounded by

water.

13. A peninsula is a country, or piece of land, furrounded by water, except on one fide, where it joins to fome other land.

14. An ifthmus is a narrow neck of land which joins a peninsula to some other country; as the ifthmus of Suez, which joins Africa to Afia; the ifthmus of Darien, which joins North and South America.

15. A cape, called fometimes a promontory, or head-land, is a point of land extending fome way into the fea.

16. The furface of the earth is fuppofed to be divided by feveral imaginary circles, for the better determining the fituation and boundaries of the feveral countries and parts of the world, of which the most confiderable circles are the following:-1. The equator, called alfo the equinoctial line, which divides the globe of the earth into two equal parts, or hemispheres,