sides in equal proportion. This is the foundation of tables for calculating triangles.

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

Tables, then, are calculated from the proportions of triangles; whose hypothenuse is 1,000,000,000, for every degree and minute of the acute angles.

Hence, if the base of a triangle be 67 yards, and the angle 36 degrees, I can ascertain the length of the other sides, by making a proportion from the tables.

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

The elements of trigonometrical tables may be understood by attending to the following diagram.

E

C F

DA is the diameter; C B the radius; B F is the sine; CF is the co-sine. Or, CA is the radius; A E is the tangent; CE is the secant.

Tables, then, are calculated for these several lines, to every degree and minute of the quadrant from A to G ; and as the sides of all triangles, which have equal angles, are in equal proportion, it is evident, that we have only to adapt these already calculated proportions to other triangles; and the latter may be calculated by the simple proportion.

550. Superficial contents are ascertained by multiplying the length by the breadth; and solid contents, by multiplying the superfices by the height.

Irregular superficial figures are to be reduced. to regular ones; and in solids, or casks, cones, &c., a mean or average-height or breadth is ascertained.

Lines are in the proportion to each other's respective lengths; superfices in the proportion of their squares; and solids of their cubes.

551. Every diameter of a circle is to its circumference, as 1 to 3, 14159.

The superfices of every circle is to the square of its diameter, as 11 to 14, or as 0,7854 to 1 nearly.

The cube 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 1728 solid inches.

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

552. The length of a pendulum vibrating seconds at London, is 391 inches.

The English yard is 36 inches; the mile 1760 yards; and a degree of the earth's surface, 691 miles nearly.

The French metre is the 10 millioneth of the

distance from the equator to the north pole; and is 39,388 inches English.

The English acre is 4840 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.

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

553. The tables in which all the proportions of triangles are calculated, which have 1,000,000,000 for one of the sides, are called 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 working the proportions; addition, in working logarithms, being a substitute for multiplication, and subtraction for division.

554. Trigonometry also calculates the sides of triangles, whose sides are parts of the circles of the earth and heavens: hence, it is highly useful to the astronomer and navigator. It enables us to calculate the heights of buildings and mountains, and the distance of celestial bodies.

The projection of spherical triangles, as part of the earth or heavens, and of maps on a globular principle, is a beautiful branch of practical geometry and astronomy.

555. Logarithms are numbers in arithmetical progression; which, set with others in a geometrical progression, express their ratios or proportions to one another, as in the two following series, viz., Logarithms, O. 1. 2. 3. 4. 5. 6. Arith. Prog. Numbers, 1. 2. 4. 8. 16. 32. 64. Geom. Prog.

556. It is the peculiar and useful property of Logarithms, 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 adding 2 and 4 in the logarithmic series you have 6, which is the logarithm of the number in the lower series 64, the product of 4 times 16; and the contrary for division.

By dividing a logarithm, you find the logarithm of the root of its number; so 6, the logarithm of 64, divided by 2, gives 3, the logarithm of 8, which is the square root of 64; or divide 6 by 3, it gives 2, the logarithm of four, the cube root of 64; and so of others.

Obs. 1.-After having completed a table of logarithms for all large numbers, the tedious labour of multiplication, division, and extraction of roots, is saved by the addition, subtraction, and division of logarithms.

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

In the representation of solid bodies, buildings, &c., there are three divisions

1. Ichnography, which shews the plan or ground-work of the building.

2. Orthography, which exhibits the front or parts in a direct view.

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

Obs.-Sciagraphy, or dialling, is the art of making dials on all kinds of planes; as horizontal, erect, or de

clining, or erect and reclining. The hour-lines, the height of the stile or gnomen above the plane, the distance of the substile from the meridian, and the difference of longitude, are all calculated by spherical trigonometry.

2.--In 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, grammar, arithmetic, and book-keeping, are supposed to be acquir ed in the routine of school-business.

XXIV. Algebra z

OR

Abstract Arithmetic.

558. If, in calculations, we were to substitute letters for known numbers, and operate with them by the signs, +, -, x, and, till, by reasoning, we have acquired such a disposition of the said letters as expresses the result, we should simplify and shorten the calculation.

Such, then, is the science called Algebra. We adopt any letters of the alphabet at pleasure in place of any given numbers, and operate with them by the intervention of signs till we have the result.

Obs. For the sake of precision, it is usual to take the first letters for any known numbers, as a, b, c, d, &c.; and the unknown numbers from the last letters, as x, y, z.

559. The algebraic signs are as follow:+ More or add, as a + b, is a more b, or a added to b.