B and DCA are equal to two right angles=DCE + DCA (by 13. 1.): whence the angle B-DCE. But, likewise, the angle E-DAC (by 5. 1.)=DAB, and DC=DB: Wherefore the triangles BAD and CED are congruous, and so CE is equal to AB. Q. E. D. PROP. X. THEOR. Fig. 33. LET the arcs AB, BC, CD, DE, EF, &c. be equal; and let the subtenses of the arcs, AB, AC, AD, AE, &c. be drawn; then will AB:AC::AC: AB +AD :: AD:AC+AE::AE: AD+AF::AF: AE+AG. 4 Let AD be produced to H, AE to I, AF to K, and AG to L, so that the triangles ACH, ADI, AEK, AFL, be isosceles ones: Then because the angle BAD is bisected, we shall have DHAB (by the last Prop.): so likewise EI=AC, FK=AD, also GL=AE. But the isosceles triangles ABC, ACH, ADI, ΑΕΚ, AFL, because of the equal angles at the bases, are equiangular: Wherefore it will be, as AB: AC::AC: (AH=) AB+AD::AD: (AI=) AC+AE::AE: (AK=) AD+ AF::AF: (AL=)AE+AG. Q. E. D. Cor. 1. Because AB is to AC, as radius is to double the cosine of the arc AB, (by Coroll. Prop. 4.) it will also be, as radius is to double the cosine of the arc AB, so is AB:AC::AC: AB+ AD :: AD: AC+ AE::AE: AD++ AF, &c. Now let each of the arcs AB, BC, CD, &c. be 2'; then will AB be the sine of one minute, AC the sine of 2 minutes, AD the sine of 3 minutes, & AE the sine of 4 minutes, &c. Whence, if the sines of one and two minutes be given, we may easily find all the other sines in the following manner. Let the cosine of the arc of one minute, that is, the sine of the arc of 89 deg. 59', be called Q; and make the following analogies; R:2Q:: Sin. 2': S. 1' + S. 3'. Wherefore the sine of 3 minutes will be given. Also, R. : 2 Q : : S. 3': S. 2' + S. 4'. Wherefore the S. 4' is given. And R.: 2Q:: S. 4: S. 3' + S. 5'; and so the sine of 5' will be had. Likewise, R.: 2Q:: S. 5: S. 4' + S. 6'; and so we shall have the sine of 6'. And in like manner, the sines of every minute of the quadrant will be given. And bécause the radius, or the first term of the analogy, is unity, the operations will be with great ease and expedition calculated by multiplication, and contracted by addition. When the sines are found to 60 degrees, all the other sines may be had by addition only, by Cor. 1. Prop. 6. The sines being given, the tangents and secants may be found from the following analogies (see Figure 3, for the definitions); because the triangles BDC, BAE, BHK, are equiangular, we have BD: DC:: BA: AE; that is, Cos.: S.:: R.: T. CD: BC:: BH: BK; that is, 1. THE iudices or exponents of a series of numbers is geoinetrical progression, proceeding from 1, are also called the logarithms of the numbers in that series.* Thus if a denote any number, and the geometrical series, 1, al, a2, a3, a*, &c. be produced by actual multiplication, then 1, 2, 3, 4, &c. are called the logarithms of the first, second, third, and fourth powers of a respectively. Consequently, if, in the above, a be equal to the number 2, then I is the logarithm of 2, 2 is the logarithm of 4, 3 is the logarithm of 8, 4 is the logarithm of 16, &c. But if a be equal to 10, then 1 is the logarithm of 10, 2 is the logarithm of 100, 3 is the logarithm of 1000, 4 is the logarithm of 10000, &c. The series may be continued both ways from 1. Thus 11 α, α3, 11 1, a, a2, a3, a', &c. constitute a series in geometri a2, a, 1 cal progression, and, agreeable to the established notation in algebra, the indices, or logarithms, are -4, -3, -2, - 1, 0, 1, 2, 3, 4, &c. If a be equal to the number 2, then 1 - 4 is the logarithm of -3 is the logarithm of 16 2 is the logarithm of, 4 1 is the logarithm of 1 8 1 0 is the 2 logarithm of 1, 1 is the logarithm of 2, &c. If a be equal to 10, then - 4 is the logarithm of rithm of 1 1000' 1 10 1 10000 - 3 is the loga garithm of O is the logarithm of 1, and 1 is the logarithm of 10, &c. 2. From the above it is evident, that the logarithms of a series of numbers in geometrical progression, constitute a series of numbers in arithmetical progression. Beginning with 1, and proceeding towards the right hand, the terms in the geometrical series are produced by multiplication, but their corresponding logarithms are produced by addition. On the contrary, beginning with 1, and proceeding towards the left hand, the terms in the geometrical progression are produced by division, but their corresponding logarithms are produced by subtraction. * The reader ought to be acquainted with arithmetical and geometrical progression and the binomial theorem, before he enters on a perusal of any account of logarithms. 3. The same observations apply to logarithms when they 11 are fractions. Thus if a denote any number, 1 1 1 2 3 4 1, 1, 1, an an an : 그 1, an, ar, an, a", &c. constitute a series of numbers 19 0, 1, 2, 3, 4, &c. are the logarithms; and it is evident that nnnn the assertions in the last article hold true, both with respect to the numbers in geometrical progression and their corresponding logarithms. As a and n may be taken at pleasure, it follows that numbers in very different geometrical progressions may have the same logarithms; and that the same series of numbers in geometrical progression may have different series of logarithms corresponding to them. 3 4. If a be an indefinitely small decimal fraction, and successive powers of 1 + a be raised, then the excess of any power of 1 + a above that immediately preceding it will be indefinitely small. Thus let a = 00000000001, and then 1 + a2= 1.0000000000200000000001; and 1+a= 1.000000000030000000000300000000001; and proceeding by actual multiplication to obtain higher. powers of 1.00000000001, it will be found that the difference between two suuccssive powers is very small. If instead of supposing, as above, that a = 00000000001, we suppose it only one millionth part of this value, then the successive powers of 1 + a will differ from one another by much smaller decimal fractions. 5. If therefore a be indefinitely small and successive powers of 1 + a be raised, a series of numbers in geometrical progression will be produced, of which the common numbers 2, 3, 4, 5, &c. will become terms. For on every multiplication by 1 + a, an indefinitely small addition is made to the power multiplied, and by this indefinitely small addition, the next higher power is produced. Some power of 1 + a will therefore be equal to the number 2, or so nearly equal to it that they may be considered as equal. Continuing the advancement of the powers of 1 + a, the numbers 3, 4, 5, &c. for the same reasons, will fall into the series. 6. The sum of the logarithms of any two numbers is equal to the logarithm of the product of the same two numbers. Thus if 1 + a raised to the nth power be equal to the number N, and if 1 + a raised to the mth power be equal to the number M, then, by the preceding articles, n is the logarithm of 1 + an or of its equal N, and for the same reason m is the logarithm of M. Hence it follows that n + m = the logarithm of N x M, for N×M=1+ax i + a/m = 1 + a) + m by the nature of indices. If the logaritlim of N be subtracted from the logarithm of M, the difference is equal to the logarithm of the quotient which arises from the division of M by N. For M 1+am = 1 + alm-n, by the nature of indices. The addition of logarithms, therefore, answers to the multiplication of the natural numbers to which they belong; and the subtraction of logarithms answers to the division by the natural numbers to which they belong. 7. If the logarithms of a series of natural numbers be all multiplied by the same number, the several products will have the last-mentioned properties of logarithms. Thus if the indices of all the powers of 1 + a be multiplied by l, then using the notation stated in the last article, the logarithm of N is nl, and the logarithm of M is ml, and the logarithm of N×Misnl+ml; for N×M=1+anix1+Qml = 1 + ani+ml, by the nature of indices. Also ml - nl = M M the logarithm of N' 1+ami ni=1+ami-ni. Hence for N the products arising from the multiplication of I into the indices of the powers of 1 + a, are termed logarithms, as are also all numbers which have the properties stated at the end of articles 6. It is on account of these properties that logarithms are so very useful in calculations of the highest importance. 8. If the indices of the powers of 1+ a, be multiplied by a, the products are called the hyperbolic logarithms of the numbers equal to the powers of 1+a. Thus if the number N be equal to 1 + a", then na is the hyperbolic logarithm of N; and if the number M be equal to 1 + am, then ma is the hyperbolic logarithm of M. Hyperbolic logarithms are not those in common use, but they can be calculated with less labour than any other kind, and common logarithms are obtained from them. |