Now if y become o, we have fy x (bf b2 + f b3—} b1+&c.)=n~} n2 + } n3 — n2 + &c. 4 [n— { n2 + } n 3 — 2 4 n1 + &c. hence a, the log. (1+n)=; but n=N-1 and b-r-1, therefore, by substitution, the above expression becomes (N-1) — (N-1)2 + (N−1)3— (N−1)1+&c. 14. Let 1 3 --M. (r—1)—§ (r—1)o + } (r−1)3—1 (r—1)1 +&c. This quantity M, which evidently depends upon the base r, is called the modulus of the particular system of logarithms to which it belongs. As it is obvious the series n―n2 + n3 — ¦ n1 + } n3· &c. will not converge when n is any whole number greater than unity, before proceeding to the calculation of the logarithms of any particular system, it will be proper to show the manner in which the value of x in the last article may be expressed in a converging series. This may be effected by means of the following process, in which M is substituted for the quantity 1 thus, (r—1) — (r—1)2 + } (r−1)3—1 (r—1)1 + &c.; Log. (1+1)=M (n~} n2 + } n3 — } n1 + } n3—&c.) In the above for n put —n, and then Log. (1-n) =M (—n— r2-} n3—} n1—} n3—&c.) Subtract (4) from (3), then log. (1 + n) — log. (1 — n) —1+" _2 M (n + } n3 +} n3 + ‡ n2 + &c.) 1+n Log. N=2 M { (N = 1) + + (N = 1)2 + + (N = 1)2+&c. } ⋅ (6) Log. N~1−2 M (2 N−1 + 3(2 N—1)3 † 5(2 N−1)3 =2 1 1 Log. N—log. (N—1) −2 M (2 N−1+3(2 N—1)3+5(2 Ñ—1) 2 M (2 N + 1 + 3(2N + 1)3 + 5 (2N+1)+&c.) +log. N. (8) M(2N"+1+3(2N”+1)3+5(2 Nm+1)s+&c. * By means of this formula (9) la the logarithm of a quantity exceeding unity by a very small fraction may be readily found. See examples in the explanation of the tables. Since the log. of 1=0, this last series which converges very rapidly, will give the logarithms of all the natural numbers with facility in succession. To these theorems might have been added others still more convenient, but they are sufficient for ordinary cases. 1 15. Before proceeding to compute a table of logarithms, some value must be assigned to M. Since the value of r is arbitrary, let it be so assumed that or M shall ́(r—1)—↓ (n—1)2 + } (r−1 )3—&c. be equal to 1, that adopted by Napier. Taking series (8) we have (art. 6.) Log. 1 1 1 2=2 + + +&c. to 8 terms 3=2 3 34 5.35 4=2 log. 2 (art.) 12) ) =0.0000000 -0.6931472 =1.0986123 =1.3862944 =1.6094379 =1.7917595 5=2(3+3.95 +596 +&c.) +log. 4 5.95 6=log. 2+log. 3 (art. 10) 1 8=3 log. 2 (art. 12) 9=2 log. 3 (art. 12) 1 10=log. 2+log. 5 (art. 10) &c. In this manner the Napierean logarithms of all the natural numbers may be found. As their accuracy, however, depends upon those immediately preceding, being derived successively from each other, it would be necessary to check the computations in the actual construction of a table of logarithms by some independent formula, such as (6), though this in large numbers would be rather inconvenient from its slow convergency. 16. To find the value of r, the base, in this system recourse must be had to the series (3) art. (14). If log. (1+n) or log. N be put land M=1, we have In-n2 + } n3 — { n1 +, &c.; reverting this series, and 1+n, or N=1+1+12+235 + 14, &c. Now let l=1, then the number whose logarithm is 1, that is, the base 1 1 r=1+1+2+ 2.3+2.3.4+, &c. =2.7182818. To prevent confusion, however, we shall always designate the base or radix of this system by R, retaining r for that of the common logarithms. Hence R-2.718,281,828,46. These are also called hyperbolic logarithms from their application to the quadrature of the hyperbola; but this designation is improper, as any system may be similarly employed. 17. When we have the logarithm of a number N for any particular value of r, the base, we can readily obtain the logarithm of the same number in every other system. Since, art. (5), when the base is r we have r=N, we shall likewise have RX-N when the base is R, in which x is different from X, therefore, RX=p2. Now taking the logarithms relatively to the system whose base is r then but l.rx by hypothesis, and l.RX=X/.R, art. (12), whence X l.R=x, or X 7. R I.N 7. R X= But if R is the base, X will be the logarithm of N in the system having that base, and designating this by L.N. to distinguish it from the other, we shall have L. N=; (12.) consequently we obtain the logarithm of N in the second system, by dividing its logarithm taken in the first system by the logarithm of the base of the second system. Again, from formula (12) we get (13) L.Nxl. R=1. N Hence in every system the logarithm of any number is the product of its Napierean logarithm by the logarithm of R, called the modulus. 1. N L.N Also since 7. R, there exists between 1.N and L.N a constant ratio represented by l.R I.N Since we have by formula (12) L.N=R, as N=10, then art. (15.) (14.) 18. It is now easy to construct a table of common logarithms whose base r=10, for by formula (13) we have l.N=l.R × L.N, but 1.R=M=0.4342944819; consequently 1.N=0.4342944819 × L.N. It therefore is only necessary to substitute this value for M in any of the series formerly given for the computation of the Napierean logarithms to obtain the common; thus, if in series (8) for 2M we substitute its value 0.86858896 we shall have 1 1 1 5+&c. log. (N+1)=0.86858896(2N+1+3(2N+1)+5(2N+1)6 +log. N, and making N successively 1, 2, 3, &c. Log. 1= 0.0000000 1 1 1 =0.3010300 1 1 1 &c.)+ log. 2 =0.4771213 4=2 log. 2. =0.6020600 5—86858896(+3.95 7=86858896(13+3(13)5+5(13)s+, &c.)+log.6.=0.8450980 8-3 log. 2 10= -0.9030900 -0.9542425 1.0000000 19. After Lord Napier had computed his first tables of logarithms it occurred to him that it would be proper to change the radix R-2.7182818 to r=10, at the same time making the logarithms of integers positive, and those of fractions negative, (art. 8.), as more conformable to the denary scale notation, and more convenient in practice. It appears that Mr Henry Briggs had also conceived the idea of changing the radix and had computed logarithms on a plan somewhat less commodious, by making the logarithms of integers negative, and those of fractions positive, which, upon a personal communication with Lord Napier, he rejected, and finally adopted his lordship's views. He soon afterwards published the first thousand logarithms of this kind, under the title of Logarithmorum Chilias Prima. SECTION III. Of the Trigonometrical Lines, called Sines, Tangents, &c. 20. THE Egyptians and Chaldeans began to study astronomy at a very early period. As the determination of the relations and distances of the heavenly bodies involve the mensuration of lines and angles, it was necessary to invent some method of ascertaining the value of these quantities, at least in an approximate manner, before any useful results could be obtained. Some of the more elementary propositions in geometry must have been discovered in the most remote antiquity, and the inventive genius of the Greeks filled up the general outline. The properties of geometrical figures thus acquired, would, without doubt, be applied to the mensuration of several magnitudes, and the distances of various points in space. About six hundred years before the Christian era, Thales measured the heights of the pyramids in Egypt by means of their shadows, a method which depends upon the proportionality of the sides of similar triangles. This simple property forms the basis of modern trigonometry. If, for example, a pole or gnomon be set perpendicular to the horizontal plane, it will, in a clear day, when the sun is not vertical, cast a shadow to a given distance, while any other high object, such as a steeple near it, will do the same. If straight lines be conceived to be drawn from the top of these objects to the extremity of each of their shadows, it is evident that, unless they are very distant, by this means triangles nearly similar will be formed, whose sides are proportional; that is, as the shadow of the gnomon is to its height, so is the shadow of the object to its height. Now, suppose the length of the shadow of the gnomon to be made the radius with which an arc of a circle is described commencing at the bottom of the gnomon, and, as will be afterwards explained, measuring the angle between the horizontal line and the line from the extremity of the shadow to the top of the gnomon, that gnomon will, by the principles of geometry, be a tangent to the circle. Whence the former proportion becomes as the radius is to the tangent of the angle of elevation, so is the length of the shadow of the object to its height. It would thus require the length of the shadow of the pole or gnomon to be measured each time any height was determined. This, however, might be avoided by having the measure of a set of triangles whose sides, to an assumed radius, and a corresponding series of angles, are previously determined by computation. By this means, in such cases, it is only necessary to measure the angle of elevation of the object, at a given point, and its distance from it, and comparing it with one of those computed triangles equiangular to it, to determine, in a manner similar to the former, the height of the object. It is obvious that the same principles may be applied to objects situated in any plane, whether vertical, horizontal, or oblique. Several series of triangles of the kind now mentioned have been actually computed and arranged in tables under the designation of Trigonometrical Tables. These were not accomplished at once, but were the improvements of successive ages. Hipparchus, about 150 years before the Christian era, supposed similar triangles to be inscribed in circles, and employed in his computations the chords subtending the arcs measuring them in sexagesimal parts of the radius. Nearly 300 years afterwards, Ptolemy, in his Mayaλn Zurtağıs, recomputed the chords, but in his Analemma employs the half chords instead of the chords approaching very nearly to the use of sines, afterwards introduced by the Arabians. Some notions of the tangents, secants, and versed sines, were, towards the beginning of the tenth century, entertained by the more learned Arabians. About the commencement of the fifteenth century the sciences began to be cultivated in Europe, where the greatest progress has been made. At that period Müller invented the tangents, and shortly affer Maurolycus produced his table of secants. These were all in natural numbers to a given radius now generally taken at unity, and, therefore, their application was in many cases troublesome. To remove this inconvenience as far as possible, Napier, about 1614, invented his logarithms, which have brought them perhaps to the last degree of perfection. Hipparchus, who has been followed by most of the moderns, employed the circle to measure angles. He supposed the whole circumference to be divided into 360 equal parts, each called a degree. The degree was divided into 60 equal parts called minutes, and the minute into 60 equal parts called seconds, and the sexagesimal division was continued, though now the fractions of seconds are commonly expressed in decimals, which are more convenient for calculation.* Whence the semicircle contains 180 degrees and the quadrant 90. As four right angles can be constituted about a point, 90 degrees must be the measure of a right angle. For the purposes of abbreviation, a degree is marked with a small circle, a minute with one accent, a second with two accents, &c. Thus, 57° 17′ 44′′.806, denotes 57 degrees, 17 minutes, 44 seconds, and .806 the decimal, whose value is 806 thousandths of a second. This, being an arc whose length is equal to the radius, as will be afterwards explained, is also expressed in degrees and in decimal parts of a degree, thus 57°2957795, a mode of using it which, in some cases, has its advantages. The number of these parts, in either case, contained in the arc between the lines constituting the angle, of which arc the angular point is the centre, indicates the measure of that angle accordingly. DEFINITIONS. 21. If two straight lines intersect one another, in the centre of a circle, the arc of the circumference intercepted between them is called *The French have lately adopted the centesimal division in some of their works, which, in many cases, is preferable to the sexagesimal. The whole circle is divided into 400 degrees, each degree into 100 minutes, and the centesimal division is continued. Hence the semicircle contains 200 degrees, the quadrant 100, and the ratio of the centesimal to the sexagesimal degree is as 9 to 10. To convert sexagesimal degrees into centesimal, add of the arc to itself. B |