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Let ABC be a right
B angled triangle. With any radius at pleasure
T describe an
M which shall be the measure of the angle c. At the point i draw a per
А. pendicularly to the line 0
L CL, terminating at the line ct. Lt is evidently the tangentof the arc ml, since it is a perpendicular to the radius at one extremity of the arc, and is terminated by the radius which passes through the other extremity. It is also the tangent of the angle c.
The equiangular and similar triangles clt and cab give the proportion
CL : LT :: CA : AB or,
R : tan c::b: Let the angle of the roof in the above problem be 200 and the semi-breadth 25 feet, then
1:tan 200 : : 25 : 0 whence,
tan 200 x 25
1 Had the angle B been used instead of c, the resulting proportion would have been
R : tan B::C:6 Both proportions may be expressed together in common language thus: Radius : the tangent of one of the acute angles of a right angled triangle :: the side adjacent that angle : the side opposite. This last rule applied to the problem at art. 11, gives
1: tan 300 : : 200 : 0 whence,*
C= tan 300 x 200 .57735 x 200 115.47000 c is the height of the tower.
* The tangent is found from Table III. by dividing the sine by the cosine. (Art 32.) Should the cotangent be required, divide the cosine by the sine, (Art 31.) To find the secant divide 1 by the cosine. (Art 33.) For the cosecant, divide 1 by the sine. (Art 35.)
If the side c were given and the angle B, the side b might be found in the same manner, using the proportion which contains the angle B.
42. We have now exhibited all the cases which can possibly occur in the solution of right angled triangles, with some specimens of their application. The right angle of the triangle is fixed; and any two of the five remaining parts being given, the other three may be found. Let the student select at pleasure any two of the five parts, the two selected to be considered as given, and he will find the case for solution with which he will then be presented, included in the exam. ples above.
The operations in the cases already exhibited, though of the most simple kind, nevertheless involve multiplications, which, from the number of places of figures, are somewhat tedious. In more complicated cases this evil would be much increased.
On this account it is customary to employ in trigonometrical calculations, that ingenious invention of Lord Napier's for facilitating numerical calculations, the table of logarithms ;* before explaining the use of which we shall give a tolerably full exposition of the
THEORY OF LOGARITHMS.
43. The logarithm of any given number is the exponent of the power to which it is necessary to raise some particular number in order to produce the given number. Thus, let 10 be the number raised to the power; then 2 is the logarithm of 100, because 102 100 and 3 is the logarithm of 1000, because 103 = 1000. Every given number will have a corresponding logarithm or exponent of the power to which it is necessary to raise 10 in order to produce the given number.
The number 10, which is the only number that does not change in the above equalities, is called a constant.
* Table I. at the end.
Should the constant number which has been employed be changed for another, the logarithms of numbers would be different from those derived by the use of the first constant. Logarithms derived from different constants are said to belong to different systems of logarithms, and the constant number belonging to each system is called the base of that system. The system most in use has the number 10 for a base, and is called the common system. The relation which this number sustains to the decimal system of notation will readily suggest some reasons for its selection; it will be found, as we proceed, to have many advantages.
44. If 6 be the base of a system, na number, and I its logarithm, then by the definition
b =n If we put b in the place of n, this equation becomes
b? b Here l is evidently equal to 1. Hence the logarithm of the base of every system is 1. If in the equation
b =n we make n = | we have
b1 = 1 Here l is evidently equal to zero. (Davies' Bourdon, art. 52.) Hence in every system log. of 1= : 0.
45. Suppose now the system be the common system ; b will be equal to 10. If we substitute for n all possible numbers successively, we shall have a series of equations like the following,
1 102 : 2
&c. In the first I is the common logarithm of 1, in the second of 2, in the third of 3, &c. If l be made the unknown quantity, and these equations be successively resolved, we shall have the common logarithms of all numbers.* If now a table be
* The method of resolving them is given at Art. 238, Alg. Davies' Bourdon.
formed having the series of natural numbers 1, 2, 3, 4, &c.in one column, and their logarithms calculated as above placed in a second column against them, this would be a table of logarithms. The tables in actual use do not differ from such an one in principle, though some arrangements are adopted in them to avoid unnecessary repetitions.
46. A better method of calculating tables of logarithms, is by means of series. A series for the purpose will be found at the top of the last page of Art. 253 in the Algebra. It is as follows, except that l', which is the value that I takes when A of art. 219, is made equal to 1, has been changed back to 1 and a has been replaced,
5' Making in this z=1 we have
1 1 1 -71 A2
3.33 Butll = 0 (art. 44.) hence the second member above is the logarithm of 2.
This method does not give the logarithm with perfect accuracy, since it is impossible to employ all the terms of the series which extends to infinity. But it will be observed that the terms, as we advance in the series, become very small fractions, and will soon be too insignificant to make any material difference in the sum of the whole.
Calculations which furnish very nearly, but not exactly, the true value of quantities, are called approximations. The resolution of the exponential equation is by approximation, and the logarithms of numbers in general, can be found in no other way.*
Again making z= 2, the first member becomes 13 - 12. But 12 has been just found, and may be substituted; and being transposed to the second member, the whole second member
* For the method which was actually employed in calculating the first tables, see the article entitled "Invention des Logarithmes,” in the introduction to the tables of Callet. Also article Logarithms, in the Endinburgh Encyclopedia.
will then be the logarithm of 3. Making again z=3 we should have log. of 4 in terms of log. 3 (which has just been found) and a series. Thus we might proceed to find the logarithms of all numbers.* It will be perceived that the letter A remains in all the results, as indeed it ought; for we are supposing no particular system of logarithms, and the value which is given to A, will determine the system to which the above results shall refer. The calculation would be most simple, and the labor of making a table of logarithms least, on the supposition of A = 1. This is the value of A, which belongs to the
-1 Naperian system, or that of Napier, the author of the first tables. If a be any number besides 1, then a different system from the Naperian will be furnished by the series, and the new value of a will be the quantity by which it is necessary to multiply the Naperian logarithm of a number to have its logarithm in the new system. This quantity is called the modulus of the new system.
The modulus is constant for the same system. If a table of logarithms be required with a given modulus, it may be formed from a table of Naperian logarithms, by multiplying each logarithm in the Naperian table, by the given modulus.
We shall show how the modulus may be found for the common system, and this will point out the method for every other. The method would in fact, be in all respects the same, changing com. log. in the following investigation, into the lograrithm of the system under consideration.
By the definition, if a be the modulus, and n any number, then
A X nap. log. n=com. log. n. for n substitute the Naperian base, then
A X nap. log. of nap. base = com. log. of nap. base. but nap. log. nap. base = 1. Substituting this value in the first member of the last expression, it becomes
com. log. of nap. base. From which it appears that the inodulus of the common system, is equal to the common logarithm of the base of the
* The constructing of tables by means of a formula in this way, is called tabulating the formula.