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The above cut represents a Pentagon or Polygon of five sides, inscribed within a circle.
562. Solids are cubes regular and irregular; spheres; cylinders ; cones; pyramids; and spheroids; or elliptical spheres.
A cone cut obliquely to the base, forms an ellipsis ; perpendicularly through the side, an hyperbola; and parallel to the side, a parabola.
Angles are the corners, formed by the meeting of two lines.
A right angle is, when the lines are perpendicular to each other; an acute angle is less than a right angle ; and an obtuse angle is greater than a right angle.
SQUARE, AND DIAGONAL.
563. Parallel lines are those, which are equi-distant; diagonals are those which cross figures from one angle to another.
Tangents are lines that only touch a circle in one point.
Every circle is equal to 360 degrees; the three angles of every triangle, are equal to 180 degrees; the angles of every quadrangle, are equal to 360 degrees.
564. By means of a scale and a pair of compasses, all kinds of figures may be readily drawn.
Triangles contain six parts; viz. 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 means of it are performed most problems in astronomy, geography, navigation, and surveying.
It is founded on the great principle that all triangles which have equal angles, have all their sides in equal proportion. This is the foundation of tables for calculating triangles.
565. In every triangle, the three angles together contain 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 from the proportions of
triangles ; whose base or 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, 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 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, in a moment, be understood by attending to the following diagram :
D A is the diameter; C G is the radius; B F is the sine ; C F is the co-sine. Or, C A is the radius ; A E is the tangent; C Eis 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 exact 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 rule of proportion.
• 566. Superficial contents are ascertained by multiplying the length by the breadth ; and solid contents, by multiplying the length, breadth, and depth, together.
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 ; superficies in the proportion of their squares; and solids of their cubes.
567. Every diameter of a circle is to its circumference, as 1 to 3,14159.
The superficies of every circle is to the square of its diameter, as 11 to 14, or as 0.7854 to 1. The contents of every sphere is to the cube of its diameter, as 0.5236 to 1.
Every square foot contains 144 square inches.
282 cubic inches are a gallon of ale ; and 231, of wine.
568. The length of a pendulum vibrating seconds at London, is 39 inches.
The English yard is 36 inches ; the mile 1760 yards; and a degree of the earth's surface, 694 miles nearly.
The French metre is the 10 millionth of the distance from the equator to the north 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.
Obs.-As the preceding numbers 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
569. The tables in which all the proportions of triangles are calculated, which have 1,000,000,000 for the radius, 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 making the proportions ; addition, in working logarithms, being a substitute for multiplication, and subtraction for division, so that the process is finished in a moment.
570. 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 one of the most beautiful branches of practical geometry and astronomy.
571. 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, 0. 1. 2. 3. 4. 5. 6. Arith. Prog.
572. 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 extract 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 4, the cube root of 64; and so of others.
Obs. Having, therefore, 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.
573. Perspective is that part of the mathematics, which gives rules for delineating objects on a plain su