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and of the same altitude, the insisting straight lines of which are not terminated in the same straight lines in the plane opposite to the base, are equal to one another. VII. Solid parallelopipeds which are upon equal bases, and of the same altitude, are equal to one another. VIII. Solid parallelopipeds which have the same altitude, are to one another as their bases. Cor. 1. From this it is manifest, that prisms upon triangular bases, and of the same altitude, are to one another as their bases. 2. Also a prism and a parallelopiped, which have the same altitude, are to one another as their bases. IX. Solid parallelopipeds are to one another in the ratio that is compounded of the ratios of the areas of their bases, and of their altitudes. Cor. Hence all prisms are to one another in the ratio compounded of the ratios of their bases, and of their altitudes. For every prism is equal to a parallelopiped of the same altititue with it, and of an equal base. X. Solid parallelopipeds, which have their bases and altitudes reciprocally proportional, are equal; and parallelopipeds which are equal, have their bases and altitudes reciprocally proportional. Cor. Equal prisms have their bases and altitudes reciprocally proportional, and conversely. XI. Similar solid parallelopipeds are to one anoth. er in the triplicate ratio of their homologous sides. Cor. Similar prisms are to one another in the triplicate ratio, or in the ratio of the cubes of their homologous sides. XII. If two triangular pyramids, which have equal bases and altitudes, be cut by planes that are parallel to the bases, and at equal distances from them, the sections are equal to one another. Cor. 1. The sections parallel to the base of a poly. gonal pyramid are similar to the base. 2. Hence also, in polygonal pyramids of equal bases and altitudes, the sec. tions parallel to the bases, and at equal distances from them, are equal to one another. XIII. A series of prisms of the same altitude may be circumscribed about any pyramid, such that the sum of the prisms shall exceed the pyramid by a solid less than any given solid. XIV. Pyramids that have equal bases and altitudes are equal to one another. XV. Every prism having a triangular base may be divided into three pyramids that have triangular bases, and that are equal to one another. Cor. 1. From this it is manifest, that eve. ry pyramid is the third part of a prism which has the same base, and the same altitude with it; for if the base of the

prism be any other figure than a triangle, it may be divided into prisms having triangular bases. 2. Pyramids of equal altitudes are to one another as their bases; because the prisms upon the same bases, and of the same altitude, are to one another as their bases. XVI. If from any point in the circumference of the base of a cylinder, a straight line be drawn perpendicular to the plane of the base, it will be wholly in the cylindric superficies. XVII. A cylinder and a parallelopiped having equal bases and altitudes, are equal to one another. XVIII. If a cone and a cylinder have the same base and the saine altitude, the cone is the third part of the cylinder. XIX. If a hemisphere and a cone have equal bases and altitudes, a series of cylinders may be inscribed in the hemisphere, and another series may be described about the cone, having all the same altitudes with one another, and such that their sum shall differ from the sum of the hemis. phere, and the cone, by a solid less than any given solid. XX. The same things being supposed as in the last proposition, the sum of all the cylinders inscribed in the hemisphere, and described about the cone, is equal to a cylinder, having the same base and altitude with the hemisphere. XXI. Every sphere is two-thirds of the circumscribing cylinder.

LOGARITHMS.-Logarithms are the exponents of a series of powers and roots. In the system of logarithms in common use, the number which is taken for the base or radix is 10. The fractional exponents of roots, and of powers of roots, are converted into decimals, before they are inserted in the logarithmic tables. To obtain the logarithm of any number, according to Brigg's system, we have to find a power or root of 10 which shall be equal to the proposed number. The exponent of that power or root is the logarithm required. The logarithm generally consists of two parts, an integer and a decimal. The integral part is called the characteristic or index of the logarithm. The index of

the logarithm is always one less, than the number of integral figures, in the natural number whose logarithm is sought. The negative index of a logarithm shows how far the first significant figure of the natural number, is removed from the place of units, on the right. The sum of the logarithms of two numbers, is the logarithm of the product of those numbers; and the difference of the logarithms of two numbers, is the logarithm of the quotient of one of the numbers divided by the other. The decimal part of the logarithm of any number is the same as that of the number multiplied or divided by 10, 100, 1000, &c. In a series of fractions continually decreasing, the negative indices of the logarithms continually increase. All negative logarithms belong to fractions which are between 1 and 0, and all positive logarithms belong to natural numbers which are greater than one. If a series of numbers be in geometrial progression, their logarithms will be in arithmetical progression. If the logarithm of two numbers be added, the sum will be the logarithm of the product of the numbers; and if the logarithm of one number be subtracted from that of another, the difference will be the logarithm of the quotient of one of the numbers divided by the other. To multiply by logarithms: Add the logarithms of the factors; the sum will be the logarithm of the product. To divide by logarithms : From the logarithm of the dividend, subtract the logarithm of the divisor; the difference will be the logarithm of the quotient. To involve a quantity by logarithms, multiply the logarithm of the quantity, by the index of the power required. To extract the root of a quantity by logarithms: divide the logarithm of the quantity by the number expressing the root required. A power of a root may be found by first multiplying the logarithm of the given quantity into the index of the power, and then dividing the product by the number expressing the root. To find by logarithms the fourth term in a proportion, add the logarithms of the second and third terms, and from the sum subtract the logarithm of the first term.

Arithmetical complement. The difference between a giv. en number and 10, or 100 or 1000, &c. is called the arith metical complement. To obtain the arithmetical complement of any number, subtract the right hand significant figure from 10, and each of the other figures from 9. To calcu

late compound interest, find the amount of one dollar for one year; multiply its logarithm by the number of years; and to the product add the logarithm of the principal. The natural increase of population in a country may be calcula.. ted in the same manner as compound interest. An exponential equation is one in which the letter expressing the unknown quantity is an exponent. The exponential equation may be solved, after the logarithm of each side is taken, by trial and error. For this purpose, make two suppositions of the value of the unknown quantity, and find their errors; then say, As the difference of the errors, to the difference of the assumed numbers; so is the least error to the correction required in the corresponding assumed number.

TRIGONOMETRY.-Trigonometry treats of the relations of the sides and angles of triangles. It is either plane or spherical. The former treats of triangles bounded by right lines; the latter of triangles bounded by the arcs of circles. The complement of an arch or an angle, is the difference be tween the arc or angle and 90 degrees. The supplement of an arc or an angle is the difference between the arc or angle and 180 degrees. The sine of an arc is a straight line drawn from one end of the arc, perpendicular to the diame ter which passes through the other end. The sine is half the chord of double the arc. The versed sine of an arc is that part of the diameter which is between the sine and the arc. The tangent of an arc, is a straight line drawn per. pendicular from the extremity of the diameter, which passes through one end of the arc, and extending till it meets a line drawn through the other end from the centre. The secant of an arc is a straight line drawn from the centre through one end of the chord, and extended to the tangent drawn from the other end. The sine complement or cosine of an angle, is the sine of the complement of that angle. The cotangent is the tangent of the complement of the angle. Also the cosecant is the secant of the complement of the angle. The sine of 90°, the chord of 60° and the tangent of 45°, are, in any circle, each equal to the radius, and therefore equal to each other. The chord of any arch is a mean proportional between the diameter of the circle, and the versed sine of the arc. The product of radius into the versed sine of the supplement of twice a given arc, is equal to twice the square of the cosine of the arc. The product

of the sine of an arc, into the versed sine of the supplement of twice the arc, is equal to the product of the cosine of the arc, into the sine of twice the arc. In a triangle there are six parts, three sides and theee angles. The number of parts which must be given to enable us to find the other, is three, one of which must be a side. In a right angle triangle, subtracting one of the acute angles from 90° gives the other. For determining the parts of triangles which have not any of their sides equal to the tabular radius, the following proposition is used: As the radius of one circle, to the radius of any other; so is the sine, tangent or secant, in one, to the sine, tangent or secant, of the same number of degrees, in the other. In any right angle triangle, if the hypothenuse be made radius, one of the legs will be the sine of its opposite angle, and the other leg a cosine of the same angle. If either of the legs be made radius, the other leg will be a tangent of its opposite angle, and the hypothenuse will be a secant of the same angle. When a sine is required: As the tabular sine, tangent, &c. of the same name with the given side, to the given side; so is the tabular sine, tangent, &c. of the same name with the required side, to the required side. When an angle is required: As the giv en side made radius, to the tabular radius; so is another given side, to the tabular sine, tangent, &c. of the same name. To find a side, begin with a tabular number. To find an angle, begin with a side. To find a side, any side may be made radius. To find an angle, a given side must be made radius. To obtain the difference of the squares of two quantities, add the logarithm of the sum of the quantities, to the logarithm of their difference. OBLIQUE ANGLED TRIANGLES. Theorem I. In every plane triangle, the sines of the angles are as their opposite sides. Theorem II. In any plane triangte, As the sum of any two of the sides, to their difference; so is the tangent of half the sum of the opposite angles, to the tangent of half their difference. Theorem III. As the largest side, to the sum of the two others; so is the difference of the latter, to the difference of the segments made by the perpendiculars. To enable us to find the sides and angles of an oblique angled triangle, three of them must be given. These may

be either, two angles and a side, or two sides and an angle opposite one of them, or two sides and the included angle, or

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