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Triangles contain mix parts; vix, three angles, and three sides, and any one of the one and two of the other being given, the other three may be found either by projection, or by logarithms.

This art is called trigonometry: and by nicans of it are performed most problems in astronomy, geography, navigation, and our. veying

It is founded on the great principle that all triangles which have equal angles, have all their salles in equal proportion. This is the foundation of tables for calculating triangles.

059. In every triangle, the three angles to gether coutain 180 degrees; and as a right angle is 90 degrees, the other two angles are, of course, equal to 90 degrees; all triangles may be reduced to right-angled triangles,

Tables, then, are calculated tiom the proportions of triangles; whose base or hypoThrnwese is 1,000,000,000, for every degree and minute of the acute angles,

Penoe, if the base of a triangle be 67 yards, and the angle 30 degrees, I can, in a moment, ascertain the length of the other sides, by making a rule of proportion from the tables.

Obs.--In these tables, it should be understood, that the Aypothenuse corresponds to the radius of 1,000,000,000, that the base e orresponds to the co-sine; and the perperdionlar to the sine. Or, when the buse is deemed the redius, the perpendicular is the tangent, and the hypothenuse the secant.

The elements of trigonometrical tables may, in a mo. ment, be understood by attending to the following ildenam;

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D A is the diameter: 0 Q in the rudiunt RF in the #me O kis te cardue, Oli C Athe radh** A the tong ui kia the sc***,

Tablee, then, are calculated for these several times, every degree and minute of the quadrant from A tuta and as the sides of all triangles, which have equal angles. are in exact proportion, it is evident, that we have only to adapt the palevedy calculated proportions de other triangles; and the latter way be ealeulated by the simple rule of properties

500, Superficial contenta are ascertained by multiplying the length by the breadth; and solid contents, by multiplying the length breadth, and depth, together,

Irregular superticial figures are to be reduced to regular ones; and in solids, or casks, roues, wo, a mean or average height or breadth is as vertained.

Lines are in the proportion to each other's respective lengths : superficies in the proportion of their warm and solids of their ewbie

ádl, Every diameter of a virole is to its cir ounitorogee, as I to 8,14130.

The superticies of every eirele ia to the square of its diameter, as 11 to 14, or as 0.7304 te i,

The contents of every sphere is to the cube of its diameter, as 0.5236 to 1.

Every square foot contains 144 square inches.
Every cubic foot 17:28 solid inches.

282 cubic inches are a gallon of ale ; and 231, of wine.

562. The length of a pendulum vibrating seconds at London, is 398 inches.

The English yard is 36 inches; the mile 1760 yards; and a degree of the earth's surface, 69.! miles nearly,

The French metre is the 10 milliouth of the distance from the equator to the nowh pole; and is 39,371 inches English.

The English acre is 4,840 square yards; and 640 acres are a square mile.

The surveyor's chain is 100 links, 22 yards, or 4 poles; and 10-square chains are an acre.

Oos---As the preceding nombers are the foundation of all calculations relative to quantity, and are frequently called into use in real life, every young person should . be expert in the recollection and use of them.

563. The tables in which all the proportions of triangles are cæleulated, which have 1,000,000,000 for the radus

are calleet tables of sines and tangents, and are to be found in various books of mathematics.

The numbers are reduced to logarithms for : greater ease in making the proportionis ; addition, in working logarithms, being a substitute for multiplication, and subtraction for division, 80 that the process is finished in a momenty

664. Trigonometry also caleulates the sides

of triangles, whose sides are parts of the circles of the cuttle and heavens: henee, it is bigbly useful to the astronomer and navigator. 11 e. ables is to enleulate the heights of buildings und mountains, and the distance of celestial bodies.

The projection of spherical triungles's part of the earth or heavens, and of maps on a lot tulur principle, in one of the most beautiful branches of practical geometry and astronomy

608. Logarithms are mimber in arithmeticul progression; which, met with others in gros metrical progression, express their ratios or proportions to one another, tin in the two following series, viz., Logarithms, 0. 1. 2. 3. 4, 6; 0. Arith. Prog, Numbers, 1. 2. 4. 9.16.32.04. Geom, I'rog.

600. It is the peculiar and useful properly of Loxarithms, that for every addition and subtraction of one series, there corresponds to it in the other, a multiplication and division of the number to which they belong.

Thus, by wding 2 und 4 in the logarithmic series you have 0, which is the logarithm of the number in the lower series 04, the productor 4 times 10; and the contrary for division.

By dividing a logarithm, you extract the rout ofiti number: No 0, the logarithm of 64, mi vided by %, given 3, the logarithm of 8; 'which in the square root of 64 ; or divide ( by 3, gives 2, the logarithm of 4, the cube root of 04; and so of others.

Obs. llaving, therefore, completed a table of logas

rithons for all large numbers, the tedions labour of mul. tiplication, division, and extraction of roots, is saved by the addition, subtraction, and division of logarithms.

567. Perspective is that part of the mathematies, which gives rules for delineating objects on a plain superficies, just as they would appear to the sight, if seen through a transparent plane, a pane of glass, or window.

lu the representation of solid bodies, buildings, &c., there are three divisions :di 1. Ichnography, which shews the plap or ground-work of the building. 1, 2, Orthography, which exhibits the front or parts in direct view.,

3. Scenography, which is the perspective view of the whole building, fronts, sides, and height.--See Drawing, &e.

Obs. 1.---Sciagrapby, or dialling, is the art of making dials on all kinds of planes ; as horizontal, erect, or declining, or erect and reclining. The 'hour-liner, the height of the stile or gnomon ahove the plane, the di tance of the substile from the meridian, and the differe ence of longitude, are all calculated by spherical trigonometry, 9-19 a work like the present, correct general views are all that can reasonably be expected ; and the details of the common sciences of reading and writing, grainmar, arithmetic, and bookkeeping, are supposed to be acquired in the routine of school-business, from such comhow books as Blair'. English Grammar Jovce's Arithm netici

, Morrison's Book-Keeping, and Nicholson's Pen Pular Elements of Mathematics.

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