NAPIERIAN LOGARITHMS.

1

1

+ + + &c.)......... = 6931472,

3 3.33

1

5.35

1

+ + ;+ &c.)........ =1·0986123,

3.53 5.55

4-2 log 2 (Art. 120)...

1

1

=1.3862944,

5=2 + + + &c.)......... =1.6094379,

9 3.93 5.95

and so on, to any extent.

As log 10 log 2+ log 5;

.. log 10 2.3025851.

COMMON LOGARITHMS.

123. In the system of common, or Briggs's logarithms, the value of A, is to be determined on the supposition that the base a is 10; and as the log of a number in one system is to its log in another as the value of A, in the former to the value of A, in the latter, it follows that

system, because the A, in words,

1 2.3025851

Napier's system is 1. In other

is the A, in the common system; and there

fore the value of 2 A, in the general formula for log (p+1), when the logarithms are those of the common system, is 86858896. Hence, making p=1, 2, 3, &c. successively, as before, we have the following common logarithms:—

The symbol A,, employed in these investigations, is called the modulus of the system of logarithms. The Napierian modulus is 1; that of the common system is 43429448.

What is now done is sufficient to show the student the practicability of constructing a table of logarithms. Algebraists have, however, discovered various expedients for reducing the labour involved in the operations of which a brief specimen is exhibited above. Those who desire to enter into a full consideration of these will find an ample account of them in the author's "Essay on the Computation of Logarithms;" and for other methods of establishing the theory of logarithms, with fuller details respecting their use and the comparative advantages of the common and Napierian systems, the learner is referred to the larger treatise on Algebra.

NOTE: page 211.-If an even number of roots of an equation exist between two values of x, as for instance between the values ama, and ab, then the substitution of these values in the equation will always lead to numerical results of the same signs: that is, the signs will be either both or both. Suppose the results are both, then, having decomposed the polynomial into factors, as in the text, if a be put for a in the rational part of each, we shall get a certain numerical result: let this result, with changed sign, be appended to the rational part: the corresponding irrational part will invariably be imaginary, for x=a, and will continue so till, in proceeding from a towards b, we arrive at a real root of the expression under the radical; so that if no such root exists, then no real roots of the equation can exist between the suspected limits. If b be employed instead of a, similar conclusions follow, in proceeding from b towards a; and we thus see how successfully the original limits, a, b, may always be contracted whenever the signs due to those limits are plus. When the signs are minus, we may adopt the same plan; the expression under the radical will in this case be always + for a=a, or a=b; and in proceeding from one of these limits to the other, will continue plus till a real root of that expression be reached; if, therefore, such real root exist in the interval, it will separate the real roots, then known also to exist, between a and b.

It is impossible, in the case supposed, that an odd number of roots of the expres sion under the radical can lie between a and b: there must be an even number or none at all, since the proposed polynomial is minus for both a and b.

Those who are acquainted with the theorem of Budan will know how to avail themselves of these principles. And we would here remark, as a circumstance entitled to especial notice in connection with the researches to which these observations refer, that whenever for any value xa, the expression under the radical becomes negative, it will become a negative number still greater if the value of the rational part of the factors, for r=a, be appended, with changed sign, to that part, and the radical be suitably modified in consequence. As an illustration of the use of this, we may take example 1, p. 208. The factors finally deduced here are

3x-2±√(-3x3+34),

which give 2 and a fraction for the smallest superior limit of the roots. Now, if this 2 be put for x in the rational part, 3x-2, the result is 4; hence, appending -4 to this part, we have

3x−6 ± √(−3.x3—24x+66),

from which we infer that even 2 is too great; and in this way we may continue to approach nearer and nearer to the root.

It will always be best, as here, to substitute in the rational part the integer next below the superior limit already determined; and in this substitution we need not confine ourselves to the pair of factors last deduced, but may take any pair we please.

Thus, in example 4, page 210, 7 is found to be a superior limit; hence, substituting 6 for a in the rational part of the factors,

we have

x2±√(−x3+29x2+27x+6),

x2-36 ±√(x3—43x2+27x+1302),

Or, to lower the degree of the ex

from which we infer that 6 is a superior limit. pression under the radical, we may take the factors

x2+1x ± √(292x2+27x+6);

and putting 6 for x in the rational part, and subtracting the numerical result, the factors become

x2+x-39 ±√(−483x2—12x+1527),

which furnish, of course, the same conclusion.

As a more striking example of the efficacy of this principle, let example (3), p. 215, be taken: the factors are

6x2—30x±√(—78x2+381x) .'. x < 5.

Putting, therefore, 4 for x in the rational part, we have

6x2-30x+24 ± √(210x2—1821x+242).'. x<1.

Hence all the positive roots must lie between 0 and 1, seeing that the factors continue imaginary from x=1 up to x=5.

In the first paragraph of this note, we have mentioned the importance of ascertaining whether or not a real root of the expression under the radical exist between the limits a, b, suspected to inclose real roots of the proposed equation. When this expression does not exceed the fourth degree, the theorem of Sturm may be readily applied to the inquiry, as is sufficiently shown in "The Analysis and Solution of Cubic and Biquadratic Equations."

We shall only add further, that, when the roots of the expression under the radical are all imaginary, and the leading sign of that expression plus, then if the rational part, equated to zero, have a real root, the proposed equation, if of an even degree, must have two real roots at least. But for the proof of this, and of other particulars of interest, reference must be made to Part I. of the "General Principles of Analysis."

ADDENDUM.-The general theory of elimination, as taught in the more advanced parts of algebra, shows that the solution of a pair of simultaneous equations, with two unknowns, is dependent on the solution of a single equation with one unknown. The principle to which the above note refers enables us to solve the reverse problem in various ways; that is, to replace an equation with one unknown by two equations with two unknowns. Thus the equation

x4+px3+qx2+rx+s=0,

is equivalent to the two equations

x2+1px+y=0,

(p2—q+2y)x2+(py—r)x+y2—s=0 (p. 186);

so that the real roots of the original equation will be indicated by the intersections of the curves of which the two latter are the equations. And the real roots of every equation of the degree 2n may, in like manner, be indicated by the intersections of two curves, one of which will be of degree n+1, and the other a parabolic curve of degree n.

THE END,