ponential equation a=b. But these methods are entirely unsuited to practice, and we therefore omit this theory.

The theory of series, on the contrary, claims some farther developments. Series are continually met with in practice; they give the best solutions of many questions, and it is indispensable to know in what circumstances they can be safely employed.

We have so often insisted on the necessity of teaching students to calculate, as to justify the extent of the part of the programme relating to logarithms. We have suppressed the inapplicable method of determining logarithms by continued fractions, and have substituted the employment of the series which gives the logarithm of n+1, knowing that of n. To exercise the students in the calculation of the series, they should be made to determine the logarithms of the numbers from 1 to 10, from 101 to 110, and from 10,000 to 10,010, the object of these last being to show them with what rapidity the calculation proceeds when the numbers are large; the first term of the series is then sufficient, the variations of the logarithms being sensibly proportional to the variations of the numbers, within the limits of the necessary exactness. In the logarithmic calculations, the pupils will be exercised in judging of the exactness which they may have been able to obtain: the consideration of the numerical values of the proportional parts given in the tables is quite sufficient for this purpose, and is beside the only one which can be employed in practice.

The use of the sliding rule, which is merely an application of logarithms, gives a rapid and portable means of executing approximately a great number of calculations which do not require great exactness. We desire that the use of this little instrument should be made familiar to the candidates. This is asked for by all the professors of the "School of application," particularly those of Topography, of Artillery, of Construction, and of Applied Mechanics, who have been convinced by experience of the utility of this instrument, which has the greatest possible analogy with tables of logarithms.

Before entering on the subjects of higher algebra, it should be remembered that the reductions of the course which we have found to be so urgent, will be made chiefly on it. The general theory of equations has taken in the examinations an abnormal and improper development, not worth the time which it costs the students. We may add, that it is very rare to meet a numerical equation of a high degree requiring to be resolved, and that those who have to do this, take care not to seek its roots by the methods which they have been taught. These methods moreover are not applicable to transcendental equations, which are much more frequently found in practice.

The theory of the greatest common algebraic divisor, in its entire generality, is of no use, even in pure science, unless in the elimination between equations of any degree whatever. But this last subject being omitted, the greatest common divisor is likewise dispensed with.

It is usual in the general theory of algebraic equations to consider the derived polynomials of entire functions of x. These polynomials are in fact useful in several circumstances, and particularly in the theory of equal roots; and in analytical geometry, they serve for the discussion of curves and the determination of their tangents. But since transcendental curves are very often encountered in practice, we give in our programme the calculation of the derivatives of algebraic and fractional functions, and transcendental functions, logarithmic, exponential, and circular. This has been long called for, not only because it must be of great assistance in the teaching of analytical geometry, but also because it will facilitate the elementary study of the infinitesimal calculus.

We have not retrenched any of the general ideas on the composition of an entire polynomial by means of factors corresponding to its roots. We retain several theorems rather because they contain the germs of useful ideas than because of their practical utility, and therefore wish the examiners to restrict themselves scrupulously to the programme.

The essential point in practice is to be able to determine conveniently an incommensurable root of an algebraic or transcendental equation, when encountered. Let us consider first an algebraic equation.

All the methods which have for their object to separate the roots, or to approximate to them, begin with the substitution of the series of consecutive whole numbers, in the first member of the equation. The direct substitution becomes exceedingly complicated, when the numbers substituted become large. It may be much shortened, however, by deducing the results from one another by means of their differences, and guarding against any possibility of error, by verifying some of those results, those corresponding to the numbers easiest to substitute, such as 10, 20. The teacher should not fail to explain this to his pupils. Still farther: let us suppose that we have to resolve an equation of the third degree, and that we have recognized by the preceding calculations the necessity of substituting, between the numbers 2 and 3, numbers differing by a tenth, either for the purpose of continuing to effect the separation of the roots, or to approximate nearer to a root comprised between 2 and 3. If we knew, for the result corresponding to the substitution of 2, the first, second, and third differences of the results of the new substitutions, we could thence deduce those results themselves with as much simplicity, as in the case of the whole numbers. The new third difference, for example, will be simply the thousandth part of the old

third difference. We may also remark that there is no possibility of error, since, the numbers being deduced from one another, when we in this way arrive at the result of the substitution of 3, which has already been calculated, the whole work will thus be verified.

Let us suppose again that we have thus recognized that the equation has a root comprised between 2.3 and 2.4; we will approximate still nearer by substituting intermediate numbers, differing by 0.01, and employing the course just prescribed. As soon as the third differences can be neglected, the calculation will be finished at once, by the consideration of an equation of the second degree; or, if it is preferred to continue the approximations till the second differences in their turn may be neglected, the calculation will then be finished by a simple proportion.

When, in a transcendental equation ƒ(X)=0, we have substituted in f(X) equidistant numbers, sufficiently near to each other to allow the differences of the results to be neglected, commencing with a certain order, the 4th, for example, we may, within certain limits of x, replace the transcendental function by an algebraic and entire function of x, and thus reduce the search for the roots of f (X)=0 to the preceding theory.

Whether the proposed equation be algebraic or transcendental, we can thus, when we have obtained one root of it with a suitable degree of exactness, continue the approximation by the method of Newton.

Algebraic calculation.

PROGRAMME OF ALGEBRA.

Addition and subtraction of polynomials.-Reduction of similar terms. Multiplication of monomials.-Use of exponents.-Multiplication of polynomials. Rule of the signs.-To arrange a polynomial.-Homogeneous polynomials.

Division of monomials. Exponent zero.-Division of polynomials. How to know if the operation will not terminate.-Division of polynomials when the dividend contains a letter which is not found in the divisor.

Equations of the first degree.

Resolution of numerical equations of the first degree with one or several unknown quantities by the method of substitution.-Verification of the values of the unknown quantities and of the degree of their exactness.

Of cases of impossibility or of indetermination.

Interpretation of negative values.-Use and calculation of negative quantities. Investigation of general formulas for obtaining the values of the unknown quantities in a system of equations of the first degree with two or three unknown quantities.-Method of Bezout.-Complete discussion of these formulas for the case of cwo unknown quantities.-Symbols and

Discussion of three equations with three unknown quantities, in which the terms independent of the unknown quantities are null.

Equations of the second degree with one unknown quantity.
Calculus of radicals of the second degree.

Resolution of an equation of the second degree with one unknown quantity.— Double solution.-Imaginary values.

When, in the equation a+b+c=0, a converges towards 0, one of the roots increases indefinitely.-Numerical calculation of the two roots, when a is very small. Decomposition of the trinomial x2+p+q into factors of the first degree.-Relations between the coefficients and the roots of the equation +px+q=0..

Trinomial equations reducible to the second degree.

Of the maxima and minima which can be determined by equations of the second degree.

Calculation of the arithmetical values of radicals.

Fractional exponents.-Negative exponents.

Of series.

Geometrical progressions.-Summation of the terms.

What we call a series.-Convergence and divergence.

A geometrical progression is convergent, when the ratio is smaller than unity; diverging, when it is greater.

The terms of a series may decrease indefinitely and the series not be converging. A series, all the terms of which are positive, is converging, when the ratio of one term to the preceding one tends towards a limit smaller than unity, in proportion as the index of the rank of that term increases indefinitely.-The series is diverging when this limit is greater than unity. There is uncertainty when it is equal to unity. In general, when the terms of a series decrease indefinitely, and are alternately positive and negative, the series is converging.

Combinations, arrangements, and permutations of m letters, when each combination must not contain the same letter twice.

Development of the entire and positive powers of a binomial.-General terms. Development of (a+b√−1)”.

Limit towards which (1+1)" tends, when m increases indefinitely.

Summation of piles of balls.

Of logarithms and of their uses.

All numbers can be produced by forming all the powers of any positive number, greater or less than one.

General properties of logarithms.

When numbers are in geometrical progression, their logarithms are in arithmetical progression.

How to pass from one system of logarithms to another system.

Calculation of logarithms by means of the series which gives the logarithm of n+1, knowing that of n.-Calculation of Napierian logarithms.-To deduce froin them those of Briggs. Modulus.

Use of logarithms whose base is 10.-Characteristics.-Negative characteristics. Logarithms entirely negative are not used in calculation.

A number being given, how to find its logarithm in the tables of Callet. A logarithm being given, how to find the number to which it belongs.-Use of the proportional parts. Their application to appreciate the exactness for which we can answer. Employment of the sliding rule.

Resolution of exponential equations by means of logarithms.

Compound interest. Annuities.

Derived functions.

Development of an entire function F (x+h) of the binomial (x+h).-Derivative of an entire function.-To return from the derivative to the function.

The derivative of a function of a is the limit towards which tends the ratio of the increment of the function to the increment h of the variable, in proportion as ▲ ́ tends towards zero.

Derivatives of trigonometric functions.

Derivatives of exponentials and of logarithms.

Rules to find the derivative of a sum, of a product, of a power, of a quotient of functions of x, the derivatives of which are known.

Of the numerical resolution of equations.

Changes experienced by an entire function f(x) when a varies in a continuous manner. When two numbers a and b substituted in an entire function ƒ (x) give results with contrary signs, the equation f(x)=0 has at least one real root not comprised between a and b. This property subsists for every species of function which remains continuous for all the values of a comprised between a and b.

An algebraic equation of uneven degree has at least one real root.-An algebraic equation of even degree, whose last term is negative, has at least two real roots. Every equation f(x)=0, with coefficients either real or imaginary of the form a+b1, admits of a real or imaginary root of the same form. [Only the enunciation, and not the demonstration of this theorem, is required.]

If a is a root of an algebraic equation, the first member is divisible by x-a. An algebraic equation of the mth degree has always m roots real or imaginary, and it cannot admit more.-Decomposition of the first members into factors of the first degree. Relations between the coefficients of an algebraic equation and its roots. When an algebraic equation whose coefficients are real, admits an imaginary root of the form a+b=1, it has also for a root the conjugate expression a

In an algebraic expression, complete or incomplete, the number of the positive roots cannot surpass the number of the variations; consequence, for negative

roots.

Investigation of the product of the factors of the first degree common to two entire functions of x.-Determination of the roots common to two equations, the first members of which are entire functions of the unknown quantity.

By what character to recognize that an algebraic equation has equal roots.-How we then bring its resolution to that of several others of lower degree and of unequal roots.

Investigation of the commensurable roots of an algebraic equation with entire coefficients.

When a series of equidistant numbers is substituted in an entire function of the mth degree, and differences of different orders between the results are formed, the differences of the mth order are constant.

Application to the separation of the roots of an equation of the third degree Having the results of the substitution of -1, 0, and +1, to deduce therefrom, by means of differences, those of all other whole numbers, positive or negative.-The progress of the calculation leads of itself to the limits of the roots.-Graphical representation of this method.

Substitution of numbers equidistant by a tenth, between two consecutive whole numbers, when the inspection of the first results has shown its necessity.-This substitution is effected directly, or by means of new differences deduced from the preceding.

How to determine, in continuing the approximation towards a root, at what moment the consideration of the first difference is sufficient to give that root with all desirable exactness, by a simple proportion.

The preceding method becomes applicable to the investigation of the roots of a transcendental equation X=0, when there have been substituted in the first member, numbers equidistant and sufficiently near to allow the differences of the results to be considered as constant, starting from a certain order.-Formulas of interpolation.

Having obtained a root of an algebraic or transcendental equation, with a certain degree of approximation, to approximate still farther by the method of Newton.

Resolution of two numerical equations of the second degree with two unknown quantities.

Decomposition of rational fractions into simple fractions.

IV. TRIGONOMETRY.

In explaining the use of trigonometrical tables, the pupil must be able to tell with what degree of exactness an angle can be determined by the logarithms of any of its trigonometrical lines. The consideration of the proportional parts will be sufficient for this. It will thus be seen that if the sine determines perfectly a small angle, the degree of exactness, which may be expected from the use of that line, diminishes as the angle increases, and becomes quite insufficient in the neighborhood of 90 degrees. It is the reverse for the cosine, which may serve very well to represent an angle near 90 degrees, while it would be very inexact for small angles. We see, then, that in our applications, we should distrust those formulas which give an angle by its sine or cosine. The tangent