THE FIVE REGULAR SOLIDS. 56. A SOLID IS SAID TO BE REGULAR, WHEN ALL ITS SOLID ANGLES ARE EQUAL, AND ALL ITS SIDES ARE EQUAL AND REGULAR POLYGONS. The following figures are of this description; 1. The Tetraedron, 2. The Hexaedron or cube, 3. The Octaedron, 4. The Dodecaedron, four triangles; 5. The Icosaedron, twenty triangles.* Besides these five, there can be no other regular solids. The only plane figures which can form such solids, are triangles, squares, and pentagons. For the plane angles which contain any solid angle, are together less than four right angles or 360°. (Sup. Euc. 21. 2.) And the least number which can form a solid angle is three. (Sup. Euc. Def. 8. 2.) If they are angles of equilateral triangles, each is 60°. The sum of three of them is 180°, of four 240°, of five 300°, and of six 360°. The latter number is too great for a solid angle. The angles of squares are 90° each. The sum of three of these is 270°, of four 360°, and of any other greater number, still more. The angles of regular pentagons are 108° each. The sum of three of them is 324°; of four, or any other greater number, more than 360°. The angles of all other regular polygons are still greater. In a regular solid, then, each solid angle must be contained by three, four, or five equilateral triangles, by three squares, or by three regular pentagons. 57. As the sides of a regular solid are similar and equal, and the angles are also alike; it is evident that the sides are all equally distant from a central point in the solid. If then, planes be supposed to proceed from the several edges to the center, they will divide the solid into as many equal pyramids, as it has sides. The base of each pyramid will be one of the sides; their common vertex will be the central point; and their height will be a perpendicular from the center to one of the sides. *For the geometrical construction of these solids, see Legendre's Geometry; Appendix to Books vi and VII. 64. The exponent of a power may be itself a power, as in the equation where x is the exponent of the power m, which is the exponent of the power am®. Ex. 4. Find the value of x, in the equation 93 = 1000. 3 (log. 9)=log. 1000. Therefore 3*= Then as 33.14. x(log. 3)=log. 3.14. Therefore x= log. 3.14 log. 3 log. 1000 = 4 6 29 =3.14. In cases like this, where the factors, divisors, &c. are logarithms, the calculation may be facilitated, by taking the logarithms of the logarithms. Thus the value of the fraction 4922 is most easily found, by subtracting the logarithm of the logarithm which constitutes the denominator, from the logarithm of that which forms the numerator. 4771213 42 SECTION IV. DIFFERENT SYSTEMS OF LOGARITHMS, AND COMPUTATION OF THE TABLES. 65. For the common purposes of numerical computation, Briggs' system of logarithms has a decided advantage over every other. But the theory of logarithms is an important instrument of investigation, in the higher departments of mathematical science. In its numerous applications, there is frequent occasion to compare the common system with others; especially with that which was adopted, by the celebrated inventor of logarithms, Lord Napier. In conducting these investigations, it is often expedient to express the logarithm of a number, in the form of a series. If a=N, then x is the logarithm of N. (Art. 2.) To find the value of x, in a series, let the quantities a and N be put into the form of a binomial, by making a=1+b, and N=1+n. Then (1+6)=1+n, and extracting the root y of both sides, we have (1+b)=(1+n)} By the binomial theorem (1+6);=1+ (6)+*(-1) (~) + C -1) C−2) (2.3) + &c. y y (3)+ &c. 2.3 As these expressions will be the same, whatever be the value of y, let y be taken indefinitely great; then xand y y 1, being indefinitely small, in comparison with the numbers -2, &c. with which they are connected, may be cancelled from the factors (-1).(-2), &c. (1)·(-2), &c. (Alg. 456.) leaving 1+26—~(~~)+()−G) • x1b2 X y Rejecting 1 from each side of the equation, multiplying by y, (Alg. 159.) and dividing by the compound factor into which is multiplied, we have x=Log. N= _n — — n2 + \ n 3 - n1+ &c. 3 4 b − 1b2 + 1b 3 —1⁄4ba+ &c. Or, as n=N-1, and b-a-1, Log. N= 2 3 A (N − 1 ) − (N − 1 )2 + ¦ (N − 1 ) 3 — ¦ (N − 1 ) 1 +&c. (a-1)-(a-1)2 + (a−1)3 - (a− 1) + + &c. Which is a general expression, for the logarithm of any number N, in any system in which the base is a. The numerator is expressed in terms of N only; and the denominator in terms of a only: So that, whatever be the number, the denominator will remain the same, unless the base is changed. The reciprocal of this constant denominator, viz. 1 3 (a-1)-(a-1)2+(a−1)3 (a-1)+ &c. is called the Modulus of the system of which a is the base. If this be denoted by M, then Log. N=M× ((N−1) —;(N−1)3+} (N− 1 ) 3 — } (N − 1 ) ♦ +&c. ) 66. The foundation of Napier's system of Logarithms is laid, by making the modulus equal to unity. From this condition the base is determined. Taking the equation above marked A. and making the denominator equal to 1, we have x = n − 1 n2 + \n3 — }n1 +}n5 — &c. By reverting this equation* n=x+ X2 2 2.3 2.3.4 2.3.4.5 Or, as by the notation, n+1=N=a*, 2 2.3 2.3.4 2.3.4.5 If then x be taken equal to 1, we have 1 1 1 a=1+1÷÷÷+. + + 1 + &c. 2 2.3 2.3.4 2.3.4.5 Adding the first filteen terms, we have 2.7182818284 Which is the base of Napier's system, correct to ten places of decimals. * See note D. Napier's logarithms are also called hyperbolic logarithms, from certain relations which they have to the spaces between the asymptotes and the curve of an hyperbola; although these relations are not, in fact, peculiar to Napier's system. 67. The logarithms of different systems are compared with each other, by means of the modulus. As in the series 3 (N − 1) — 1(N − 1 )2 + (N − 1 ) 3 — ¦ (N − 1 ) 1 +&c. (a− 1) — (a− 1 )3 + ¦ (a− 1 ) 3 — 1 (a− 1 ) 1 +&c. which expresses the logarithm of N, the denominator only is affected by a change of the base a; and as the value of fractions, whose numerators are given, are reciprocally as their denominators: (Alg. 360. cor. 2.) The logarithm of a given number, in one system, Is to the logarithm of the same number in another system; To the modulus of the other. So that, if the modulus of each of the systems be given, and the logarithm of any number be calculated in one of the systems; the logarithm of the same number in the other system may be calculated by a simple proportion. Thus if M be the modulus in Briggs' system, and M' the modulus in Napier's; the logarithm of a number in the former, and l' the logarithm of the same number in the latter; then, M: M'::7: 7', Therefore, 7=7'xM; that is, the common logarithm of a number, is equal to Napier's logarithm of the same, multiplied into the modulus of the common system. To find this modulus, let a be the base of Briggs' system, and e the base of Napier's; and let l.a denote the common logarithm of a, and l'.a denote Napier's logarithm of a. Then M 1::l.a: l'.a. Therefore M= 1.a l'.a But in the common system, a=10, and l.a=1. So that, M= that is, the modulus of Briggs' system, 1 7.10 is equal to 1 divided by Napier's logarithm of 10. |
