Chapter III: Part I: Military System and Schools in France (2)

Notions on surveying for content (_arpentage_).--Method of decomposition into triangles.--Simpler method of decomposition into trapezoids.--Surveyor’s cross.--Practical solution, when the ground is bounded, in one or more parts, by a curved line.

The area of a regular polygon is measured by half of the product of its perimeter by the radius of the inscribed circle.--The area of a circle is measured by half of the product of the circumference by the radius.--The areas of circles are to each other as the squares of the radii.

The area of a sector of a circle is measured by half of the product of
the arc by the radius.--Measure of the area of a segment of a circle.

2. OF PLANES AND BODIES TERMINATED BY PLANE SURFACES.

Conditions required to render a right line and a plane respectively
perpendicular.

Of all the lines which can be drawn from a given point to a given plane, the perpendicular is the shortest, and the oblique lines are longer in proportion to their divergence from the foot of the perpendicular.

Parallel right lines and planes.--Angles which have their sides parallel, and their openings turned in the same direction, are equal, although situated in different planes.

Dihedral angle.--How to measure the ratio of any dihedral angle to the
right dihedral angle.

Planes perpendicular to each other.--The intersection of two planes
perpendicular to a third plane, is perpendicular to this third plane.

Parallel planes.--when two parallel planes are cut by a third plane the intersections are parallel.--Two parallel planes have their perpendiculars common to both.

The shortest distance between two right lines, not intersecting and
not parallel.

Two right lines comprised between two parallel planes are always divided into proportional parts by a third plane parallel to the first two.

Trihedral angle.--The sum of any two of the plane angles which compose
a trihedral angle is always greater than the third.

The sum of the plane angles which form a convex polyhedral angle is
always less than four right angles.

If two trihedral angles are formed by the same plane angles, the dihedral angles comprised between the equal plane angles are equal.--There may be absolute equality or simple symmetry between the two trihedral angles.

_Of polyhedrons._

If two tetrahedrons have each a trihedral angle composed of equal and similarly arranged triangles, these tetrahedrons are equal. They are also equal if two faces of the one are equal to two faces of the other, are arranged in the same manner, and form with each other the same dihedral angle.

When the triangles which form two homologous trihedral angles of two tetrahedrons are similar, each to each, and similarly disposed, these tetrahedrons are similar. They are also similar if two faces of the one, making with each other the same angle as two faces of the other, are also similar to these latter, and are united by homologous sides and summits.

Similar pyramids.--A plane parallel to the base of a pyramid cuts off from it a pyramid similar to it.--To find the height of a pyramid when we know the dimension of its trunk with parallel bases.

Sections made in any two pyramids at the same distance from these
summits are in a constant ratio.

Parallelopipedon.--Its diagonals.

Any polyhedron can always be divided into triangular pyramids.--Two bodies composed of the same number of equal and similarly disposed triangular pyramids, are equal.

_Similar polyhedrons._

The homologous edges of similar polyhedrons are proportional; as are also the diagonals of the homologous faces and the interior diagonals of the polyhedrons.--The areas of similar polyhedrons are as the squares of the homologous edges.

_Measure of volumes._

Two parallelopipedons of the same base and of the same height are
equivalent in volume.

If a parallelogram be constructed on the base of a triangular prism, and on that parallelogram, taken as a base, there be constructed a parallelopipedon of the same height as the triangular prism, the volume of this prism will be half of the volume of the parallelopipedon.--Two triangular prisms of the same base and the same height are equivalent.

Two tetrahedrons of the same base and the same height are equivalent.

A tetrahedron is equivalent to the third of the triangular prism of
the same base and the same height.

The volume of any parallelopipedon is equal to the product of its base by its height.--What must be understood by that enunciation.--The volume of any prism is equal to the product of its base by its height.

The volume of a tetrahedron and that of any pyramid are measured by
the third of the product of the base by the height.

Volume of the truncated oblique triangular prism.

The volumes of two similar polyhedrons are to each other as the cubes
of the homologous edges.

3. OF ROUND BODIES.

_Of the right cone with circular base._

Sections parallel to the base.--Having the dimensions of the trunk of
a cone with parallel bases, to find the height of the entire cone.

The area of a right cone is measured by half of the product of the circumference of its circular base by its side.--Area of a trunk of a right cone with parallel bases.

Volume of a pyramid inscribed in the cone.--The volume of a cone is measured by the third of the product of the area of its base by its height.[2]

[Footnote 2: The volume of the cone is derived from that of the
pyramid; and it is to be noted that the demonstration applies to
the cone with closed base, whatever the figure of that base.]

Which of the preceding properties belong to the cone of any base
whatever?

_Of the right cylinder with circular base._

Sections parallel to the base.

The area of the convex surface of the right cylinder is measured by the product of the circumference of its base by its height.--This is also true of the right cylinder of any base.

Measure of the volume of a prism inscribed in the cylinder.--The volume of a right cylinder is measured by the product of the area of its base by its height.--This is also true of any cylinder, right or oblique, of any base whatever.

_Of the sphere._

Every section of the sphere, made by a plane, is a circle.--Great
circles and small circles.

In every spherical triangle any one side is less than the sum of the other two. The shortest path from one point to another, on the surface of the sphere, is the arc of a great circle which joins the two given points.

The sum of the sides of a spherical triangle, or of any spherical
polygon, is less than the circumference of a great circle.

Poles of an arc of a great or small circle.--They serve to trace arcs
of circles on the sphere.

Every plane perpendicular to the extremity of a radius is tangent to
the sphere.

Measure of the angle of two arcs of great circles.

Properties of the polar or supplementary triangle.

Two spherical triangles situated on the same sphere, or on equal spheres, are equal in all their parts, 1º when they have an equal angle included between sides respectively equal; 2º when they have an equal side adjacent to two angles respectively equal; 3º when they are mutually equilateral; 4º when they are mutually equiangular. In these different cases the triangles may be equal, or merely symmetrical.

The sum of the angles of any spherical triangle is less than six, and
greater than two, right angles.

The lune is to the surface of the sphere as the angle of that lune is
to four right angles.

Two symmetrical spherical triangles are equivalent in surface.

The area of a spherical triangle is to that of the whole sphere as the excess of the sum of its angles above two right angles is to eight right angles.

When a portion of a regular polygon, inscribed in the generating circle of the sphere, turns around the diameter of that circle, the convex area engendered is measured by the product of its height by the circumference of the circle inscribed in the generating polygon.--The volume of the corresponding polygonal sector is measured by the area thus described, multiplied by the third of the radius of the inscribed circle.

The surface of a spherical zone is equal to the height of that zone multiplied by the circumference of a great circle.--The surface of the sphere is quadruple that of a great circle.

Every spherical sector is measured by the zone which forms its base, multiplied by the third of the radius. The whole sphere is measured by its surface multiplied by the third of its radius.[3]

[Footnote 3: Numerical examples on the areas and volumes of the
round bodies, including the area of a spherical triangle, will be
required by the examiners. The calculations will be made by
logarithms.]

III. ALGEBRA.

Algebra[4] is not, as are Arithmetic and Geometry, indispensable to every one. It should be very sparingly introduced into the general education of youth, and we would there willingly dispense with it entirely, excepting logarithms, if this would benefit the study of arithmetic and geometry. The programme of it which we are now to give, considers it purely in view of its utility to engineers, and we will carefully eliminate every thing not necessary for them.

[Footnote 4: The true distinction between ALGEBRA and ARITHMETIC
is so commonly overlooked that it maybe well to present it here,
in the words of Comte. “The complete solution of every question of
calculation is necessarily composed of two successive parts, which
have essentially distinct natures. In the first, the object is to
_transform_ the proposed equations, so as to make apparent the
manner in which the unknown quantities are formed by the known
ones; it is this which constitutes the _Algebraic_ question. In
the second, our object is _to find the value_ of the formulas thus
obtained; that is, to determine directly the values of the numbers
sought, which are already represented by certain explicit
functions of given numbers; this is the _Arithmetical_ question.
Thus the stopping-point of the algebraic part of the solution
becomes the starting-point of the arithmetical part.

“ALGEBRA may therefore be defined as having for its object the
_resolution of equations_; taking this expression in its full
logical meaning, which signifies the transformation of _implicit_
functions into equivalent _explicit_ ones. In the same way
ARITHMETIC may be defined as intended for _the determination of
the values of functions_. Henceforth, therefore, we may call
ALGEBRA _the Calculus of Functions_, and ARITHMETIC _the Calculus
of Values_.”]

Algebraical calculation presents no serious difficulty, when its students become well impressed with this idea, that every letter represents a number; and particularly when the consideration of negative quantities is not brought in at the outset and in an absolute manner. These quantities and their properties should not be introduced except as the solution of questions by means of equations causes their necessity to be felt, either for generalizing the rules of calculation, or for extending the meaning of the formulas to which it leads. CLAIRAUT pursues this course. He says, “I treat of the multiplication of negative quantities, that dangerous shoal for both scholars and teachers, only after having shown its necessity to the learner, by giving him a problem in which he has to consider negative quantities independently of any positive quantities from which they are subtracted. When I have arrived at that point in the problem where I have to multiply or divide negative quantities by one another, I take the course which was undoubtedly taken by the first analysts who have had those operations to perform and who have wished to follow a perfectly sure route: I seek for a solution of the problem which does not involve these operations; I thus arrive at the result by reasonings which admit of no doubt, and I thus see what those products or quotients of negative quantities, which had given me the first solution, must be.” BEZOUT proceeds in the same way.

We recommend to teachers to follow these examples; not to speak to their pupils about negative quantities till the necessity of it is felt, and when they have become familiar with algebraic calculation; and above all not to lose precious time in obscure discussions and demonstrations, which the best theory will never teach students so well as numerous applications.

It has been customary to take up again, in algebra, the calculus of fractions, so as to generalize the explanations given in arithmetic, since the terms of literal fractions may be any quantities whatsoever. Rigorously, this may be well, but to save time we omit this, thinking it better to employ this time in advancing and exercising the mind on new truths, rather than in returning continually to rules already given, in order to imprint a new degree of rigor on their demonstration, or to give them an extension of which no one doubts.

The study of numerical equations of the first degree, with one or several unknown quantities, must be made with great care. We have required the solution of these equations to be made by the method of _substitution_. We have done this, not only because this method really comprehends the others, particularly that of _comparison_, but for this farther reason. In treatises on algebra, those equations alone are considered whose numerical coefficients and solutions are very simple numbers. It then makes very little difference what method is used, or in what order the unknown quantities are eliminated. But it is a very different thing in practice, where the coefficients are complicated numbers, given with decimal parts, and where the numerical values of these coefficients may be very different in the same equation, some being very great and some very small. In such cases the method of _substitution_ can alone be employed to advantage, and that with the precaution of taking the value of the unknown quantity to be eliminated from that equation in which it has relatively the greatest, coefficient. Now the method of _comparison_ is only the method of substitution put in a form in which these precautions cannot be observed, so that in practice it will give bad results with much labor.

The candidates must present to the examiners the complete calculations of the resolution of four equations with four unknown quantities, made with all the precision permitted by the logarithmic tables of Callet, and the proof that that precision has been obtained. The coefficients must contain decimals and be very different from one another, and the elimination must be effected with the above precautions.

The teaching of the present day disregards too much the applicability of the methods given, provided only that they be elegant in their form; so that they have to be abandoned and changed when the pupils enter on practice. This disdain of practical utility was not felt by our great mathematicians, who incessantly turned their attention towards applications. Thus the illustrious Lagrange made suggestions like those just given; and Laplace recommended the drawing of curves for solving directly all kinds of numerical equations.

As to literal equations of the first degree, we call for formulas sufficient for the resolution of equations of two or three unknown quantities. Bezout’s method of elimination must be given as a first application of that fruitful method of indeterminates. The general discussion of formulas will be confined to the case of two unknown quantities. The discussion of three equations with three unknown quantities, _x_, _y_, and _z_, in which the terms independent of the unknown quantities are null, will be made directly, by this simple consideration that the system then really includes only two unknown quantities, to wit, the ratios of _x_ and _y_, for example, to _z_.

The resolution of inequalities of the first degree with one or more unknown quantities, was added to equations of the first degree some years ago. We do not retain that addition.

The equations of the second degree, like the first, must be very carefully given. In dwelling on the case where the coefficient of _x_^{2} converges towards zero, it will be remarked that, when the coefficient is very small, the ordinary formula would give one of the roots by the difference of two numbers almost equal; so that sufficient exactness could not be obtained without much labor. It must be shown how that inconvenience may be avoided.

It is common to meet with expressions of which the maximum or the minimum can be determined by the consideration of an equation of the second degree. We retain the study of them, especially for the benefit of those who will not have the opportunity of advancing to the general theory of maxima and minima.

The theory of the algebraic calculation of imaginary quantities, given _à priori_, may, on the contrary, be set aside without inconvenience. It is enough that the pupils know that the different powers of √-1 continually reproduce in turn one of these four values, ±1, ±√-1. We will say as much of the calculation of the algebraic values of radicals, which is of no use. The calculation of their _arithmetical_ values will alone be demanded. In this connection will be taught the notation of fractional exponents and that of negative exponents.

The theory of numbers has taken by degrees a disproportionate development in the examinations for admission; it is of no use in practice, and, besides, constitutes in the pure mathematics a science apart.

The theory of continued fractions at first seems more useful. It is employed in the resolution of algebraic equations, and in that of the exponential equation _a_^{_x_}=_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 _f_(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.

PROGRAMME OF ALGEBRA.

_Algebraic calculation._

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 two unknown quantities.--Symbols m/o and o/o.

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 _ax^{2} + bx + 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 _x^{2} + px + q_ into factors of the first degree.--Relations between the coefficients and the roots of the equation _x^{2} + 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)^{m}_.

Limit towards which _(1 + 1/m)^{m}_ 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 from 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 _x_ is the limit towards which tends the ratio of the increment of the function to the increment _h_ of the variable, in proportion as _h_ 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 _x_ varies in a continuous manner.--When two numbers _a_ and _b_ substituted in an entire function _f(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 _x_ 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 + b √-1_, 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 _m_^{th} 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 - b √-1_.

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 _m_^{th} degree, and differences of different orders between the results are formed, the differences of the _m_^{th} 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_ being alone exempt from these difficulties, we should seek, as far as possible, to resolve all questions by means of it. Thus, let us suppose that we know the hypothenuse and one of the sides of a right-angled triangle, the direct determination of the included angle will be given by a cosine, which will be wanting in exactness if the hypothenuse of the triangle does not differ much from the given side. In that case we should begin by calculating the third side, and then use it with the first side to determine the desired angle by means of its tangent. When two sides of a triangle and the included angle are given, the tangent of the half difference of the desired angles may be calculated with advantage; but we may also separately determine the tangent of each of them. When the three sides of a triangle are given, the best formula for calculating an angle, and the only one never at fault, is that which gives the tangent of half of it.

The surveying for plans, taught in the course of Geometry, employing only graphical methods of calculation, did not need any more accurate instruments than the chain and the graphometer; but now that trigonometry furnishes more accurate methods of calculation, the measurements on the ground require more precision. Hence the requirement for the pupil to measure carefully a base, to use telescopes, verniers, etc., and to make the necessary calculations, the ground being still considered as plane. But as these slow and laborious methods can be employed for only the principal points of the survey, the more expeditious means of the plane-table and compass will be used for the details.

In spherical trigonometry, all that will be needed in geodesy should be learned before admission to the school, so that the subject will not need to be again taken up. We have specially inscribed in the programme the relations between the angles and sides of a right-angled triangle, which must be known by the students; they are those which occur in practice. In tracing the course to be pursued in the resolution of the three cases of any triangles, we have indicated that which is in fact employed in the applications, and which is the most convenient. As to the rest, ambiguous cases never occur in practice, and therefore we should take care not to speak of them to learners.

In surveying, spherical trigonometry will now allow us to consider cases in which the signals are not all in the same plane, and to operate on uneven ground, obtain its projection on the plane of the horizon, and at the same time determine differences of level.

It may be remarked that Descriptive Geometry might supply the place of spherical trigonometry by a graphical construction, but the degree of exactitude of the differences of level thus obtained would be insufficient.

PROGRAMME OF TRIGONOMETRY.

1. PLANE TRIGONOMETRY.

Trigonometrical lines.--Their ratios to the radius are alone considered.--Relations of the trigonometric lines of the same angle.--Expressions of the sine and of the cosine in functions of the tangent.

Knowing the sines and the cosines of two arcs _a_ and _b_, to find the sine and the cosine of their sum and of their difference.--To find the tangent of the sum or of the difference of two arcs, knowing the tangents of those arcs.

Expressions for sin.2_a_ and sin.3_a_; cos.2_a_ and cos. 3_a_;
tang.2_a_ and tang.3_a_.

Knowing sin._a_ or cos._a_, to calculate sin.½_a_ and cos.½_a_.

Knowing tang._a_, to calculate tang.½_a_.

Knowing sin._a_, to calculate sin.⅓_a_--Knowing cos._a_, to
calculate cos.⅓_a_.

Use of the formula cos._p_+cos._q_ = 2cos.½(_p + q_)cos.½(_p - q_), to render logarithms applicable to the sum of two trigonometrical lines, sines or cosines.--To render logarithms applicable to the sum of two tangents.

Construction of the trigonometric tables.

Use in detail of the tables of Callet.--Appreciation, by the proportional parts, of the degree of exactness in the calculation of the angles.--Superiority of the tangent formulas.

_Resolution of triangles._

Relations between the angles and the sides of a right-angled triangle, or of any triangle whatever.--When the three angles of a triangle are given, these relations determine only the ratios of the sides.

Resolution of right-angled triangles.--Of the case in which the
hypothenuse and a side nearly equal to it are given.

Knowing a side and two angles of any triangle, to find the other
parts, and also the surface of the triangle.

Knowing two sides _a_ and _b_ of a triangle and the included angle C, to find the other parts and also the surface of the triangle.--The tang.½(A - B) may be determined; or tang.A and tang.B directly.

Knowing the three sides _a_, _b_, _c_, to find the angles and the surface of the triangle.--Employment of the formula which gives tang.½A.

_Application to surveying for plans._

Measurement of bases with rods.

Measurement of angles.--Description and use of the circle.--Use of the telescope to render the line of sight more precise.--Division of the circle.--Verniers.

Measurement and calculation of a system of triangles.--Reduction of
angles to the centres of stations.

How to connect the secondary points to the principal system.--Use of
the plane table and of the compass.

2. SPHERICAL TRIGONOMETRY.

Fundamental relations (cos._a_ = cos._b_ cos._c_ + sin._b_ sin._c_
cos.A) between the sides and the angles of a spherical triangle.

To deduce thence the relations sin.A : sin.B = sin._a_ : sin._b_; cot._a_ sin._b_ - cot.A sin.C = cos._b_ cos.C, and by the consideration of the supplementary triangle cos.A = -cos.B cos.C + sin.B sin.C cos._a_.

Right-angled triangles.--Formulas cos._a_ = cos._b_ cos._c_; sin._b_ =
sin._a_ sin.B; tang._c_ = tang._a_ cos.B, and tang._b_ = sin._c_ tang.B.

In a right-angled triangle the three sides are less than 90°, or else two of the sides are greater than 90°, and the third is less. An angle and the side opposite to it are both less than 90°, or both greater.

Resolution of any triangles whatever:

1º Having given their three sides _a_, _b_, _c_, or their three angles
A, B, C.--Formulas tang.½_a_ and tang.1/2A, calculable by logarithms:

2º Having given two sides and the included angle, or two angles and
the included side.--Formulas of Delambre:

3º Having given two sides and an angle opposite to one of them, or two angles and a side opposite to one of them. Employment of an auxiliary angle to render the formulas calculable by logarithms.

Applications.--Survey of a mountainous country.--Reduction of the base and of the angles to the horizon.--Determination of differences of level.

Knowing the latitude and the longitude of two points on the surface of
the earth, to find the distance of those points.

V. ANALYTICAL GEOMETRY.

The important property of homogeneity must be given with clearness and simplicity.

The transformation of co-ordinates must receive some numerical applications, which are indispensable to make the student clearly see the meaning of the formulas.

The determination of tangents will be effected in the most general manner by means of the derivatives of the various functions, which we inserted in the programme of algebra. After having shown that this determination depends on the calculation of the derivative of the ordinate with respect to the abscissa, this will be used to simplify the investigation of the tangent to curves of the second degree and to curves whose equations contain transcendental functions. The discussion of these, formerly pursued by laborious indirect methods, will now become easy; and as curves with transcendental equations are frequently encountered, it will be well to exercise students in their discussion.

The properties of foci and of the directrices of curves of the second degree will be established directly, for each of the three curves, by means of the simplest equations of these curves, and without any consideration of the analytical properties of foci, with respect to the general equation of the second degree. With even greater reason will we dispense with examining whether curves of higher degree have foci, a question whose meaning even is not well defined.

We retained in algebra the elimination between two equations of the second degree with two unknown quantities, a problem which corresponds to the purely analytical investigation of the co-ordinates of the points of intersection of two curves of the second degree. The final equation is in general of the fourth degree, but we may sometimes dispense with calculating that equation. A graphical construction of the curves, carefully made, will in fact be sufficient to make known, approximately, the co-ordinates of each of the points of intersection; and when we shall have thus obtained an approximate solution, we will often be able to give it all the numerical rigor desirable, by successive approximations, deduced from the equations. These considerations will be extended to the investigation of the real roots of equations of any form whatever with one unknown quantity.

Analytical geometry of three dimensions was formerly entirely taught within the Polytechnic school, none of it being reserved for the course of admission. For some years past, however, candidates were required to know the equations of the right line in space, the equation of the plane, the solution of the problems which relate to it and the transformation of co-ordinates. But the consideration of surfaces of the second order was reserved for the interior teaching. We think it well to place this also among the studies to be mastered before admission, in accordance with the general principle now sought to be realized, of classing with them that double instruction which does not exact a previous knowledge of the differential calculus.

We have not, however, inserted here all the properties of surfaces of the second order, but have retained only those which it is indispensable to know and to retain. The transformation of rectilinear co-ordinates, for example, must be executed with simplicity, and the teacher must restrict himself to giving his pupils a succinct explanation of the course to be pursued; this will suffice to them for the very rare cases in which they may happen to have need of them. No questions will be asked relating to the general considerations, which require very complicated theoretical discussions, and especially that of the general reduction of the equation of the second degree with three variables. We have omitted from the problems relating to the right line and to the plane, the determination of the shortest distance of two right lines.

The properties of surfaces of the second order will be deduced from the equations of those surfaces, taken directly in the simplest forms. Among these properties, we place in the first rank, for their valuable applications, those of the surfaces which can be generated by the movement of a right line.

PROGRAMME OF ANALYTICAL GEOMETRY.

1. GEOMETRY OF TWO DIMENSIONS.

Rectilinear co-ordinates.--Position of a point on a plane.

Representation of geometric loci by equations.

Homogeneity of equations and of formulas.--Construction of algebraic
expressions.

Transformation of rectilinear co-ordinates.

Construction of equations of the first degree.--Problems on the right
line.

Construction of equations of the second degree.--Division of the curves which they represent into three classes.--Reduction of the equation to its simplest form by the change of co-ordinates.[5]

[Footnote 5: The students will apply these reductions to a
numerical equation of the second degree, and will determine the
situation of the new axes with respect to the original axes, by
means of trigonometrical tables. They will show to the examiner
the complete calculations of this reduction and the trace of the
two systems of axes and of the curves.]

Problem of tangents.--The coefficient of inclination of the tangent to the curve, to the axis of the abscissas, is equal to the derivative of the ordinate with respect to the abscissa.

_Of the ellipse._

Centre and axes.--The squares of the ordinates perpendicular to one of the axes are to each other as the products of the corresponding segments formed on that axis.

The ordinates perpendicular to the major axis are to the corresponding ordinates of the circle described on that axis as a diameter, in the constant ratio of the minor axis to the major.--Construction of the curve by points, by means of this property.

Foci; eccentricity of the ellipse.--The sum of the radii vectors drawn to any point of the ellipse is constant and equal to the major axis.--Description of the ellipse by means of this property.

Directrices.--The distance from each point of the ellipse to one of the foci, and to the directrix adjacent to that focus, are to each other as the eccentricity is to the major axis.

Equations of the tangent and of the normal at any point of the ellipse.[6]--The point in which the tangent meets one of the axes prolonged is independent of the length of the other axis.--Construction of the tangent at any point of the ellipse by means of this property.

[Footnote 6: They will be deduced from the property, previously
demonstrated, of the derivative of the ordinate with respect to
the abscissa.]

The radii vectores, drawn from the foci to any point of the ellipse, make equal angles with the tangent at that point or the same side of it.--The normal bisects the angle made by the radii vectores with each other.--This property may serve to draw a tangent to the ellipse through a point on the curve, or through a point exterior to it.

The diameters of the ellipse are right lines passing through the centre of the curve.--The chords which a diameter bisects are parallel to the tangent drawn through the extremity of that diameter.-- Supplementary chords. By means of them a tangent to the ellipse can be drawn through a given point on that curve or parallel to a given right line.

Conjugate diameters.--Two conjugate diameters are always parallel to supplementary chords, and reciprocally.--Limit of the angle of two conjugate diameters.--An ellipse always contains two equal conjugate diameters.--The sum of the squares of two conjugate diameters is constant.--The area of the parallelogram constructed on two conjugate diameters is constant.--To construct an ellipse, knowing two conjugate diameters and the angle which they make with each other.

Expression of the area of an ellipse in function of its axes.

_Of the hyperbola._

Centre and axes.--Ratio of the squares of the ordinates perpendicular
to the transverse axes.

Of foci and of directrices; of the tangent and of the normal; of diameters and of supplementary chords.--Properties of these points and of these lines, analogous to those which they possess in the ellipse.

Asymptotes of the hyperbola.--The asymptotes coincide with the diagonals of the parallelogram formed on any two conjugate diameters.--The portions of a secant comprised between the hyperbola and its asymptotes are equal.--Application to the tangent and to its construction.

The rectangle of the parts of a secant, comprised between a point of the curve and the asymptotes, is equal to the square of half of the diameter to which the secant is parallel.

Form of the equation of the hyperbola referred to its asymptotes.

_Of the parabola._

Axis of the parabola.--Ratio of the squares of the ordinates
perpendicular to the axis.

Focus and directrix of the parabola.--Every point of the curve is equally distant from the focus and from the directrix.--Construction of the parabola.

The parabola may be considered as an ellipse, in which the major axis is indefinitely increased while the distance from one focus to the adjacent summit remains constant.

Equations of the tangent and of the normal.--Sub-tangent and sub-normal. They furnish means of drawing a tangent at any point of the curve.

The tangent makes equal angles with the axis and with the radius vector drawn to the point of contact.--To draw, by means of this property, a tangent to the parabola, 1º through a point on the curve; 2º through an exterior point.

All the diameters of the parabola are right lines parallel to the axis, and reciprocally.--The chords which a diameter bisects are parallel to the tangent drawn at the extremity of that diameter.

Expression of the area of a parabolic segment.

* * *

Polar co-ordinates.--To pass from a system of rectilinear and rectangular co-ordinates to a system of polar co-ordinates, and reciprocally.

Polar equations of the three curves of the second order, the pole being situated at a focus, and the angles being reckoned from the axis which passes through that focus.

Summary discussion of some transcendental curves.--Determination of
the tangent at one of their points.

Construction of the real roots of equations of any form with one unknown quantity.--Investigation of the intersections of two curves of the second degree.--Numerical applications of these formulas.

2. GEOMETRY OF THREE DIMENSIONS.

The sum of the projections of several consecutive right lines upon an axis is equal to the projection of the resulting line.--The sum of the projections of a right line on three rectangular axes is equal to the square of the right line.--The sum of the squares of the cosines of the angles which a right line makes with three rectangular right lines is equal to unity.

The projection of a plane area on a plane is equal to the product of
that area by the cosine of the angle of the two planes.

Representation of a point by its co-ordinates.--Equations of lines and
of surfaces.

Transformation of rectilinear co-ordinates.

_Of the right line and of the plane._

Equations of the right line.--Equation of the plane.

To find the equations of a right line, 1º which passes through two given points, 2º which passes through a given point and which is parallel to a given line.

To determine the point of intersection of two right lines whose
equations are known.

To pass a plane, 1º through three given points; 2º through a given point and parallel to a given plane; 3º through a point and through a given right line.

Knowing the equations of two planes, to find the projections of their
intersection.

To find the intersection of a right line and of a plane, their
equations being known.

Knowing the co-ordinates of two points, to find their distance.

From a given point to let fall a perpendicular on a plane; to find the
foot and the length of that perpendicular (rectangular co-ordinates).

Through a given point to pass a plane perpendicular to a given right
line (rectangular co-ordinates).

Through a given point, to pass a perpendicular to a given right line; to determine the foot and the length of that perpendicular (rectangular co-ordinates).

Knowing the equations of a right line, to determine the angles which that line makes with the axes of the co-ordinates (rectangular co-ordinates).

To find the angle of two right lines whose equations are known
(rectangular co-ordinates).

Knowing the equation of a plane, to find the angles which it makes
with the co-ordinate planes (rectangular co-ordinates).

To determine the angle of two planes (rectangular co-ordinates).

To find the angle of a right line and of a plane (rectangular
co-ordinates).

_Surfaces of the second degree._

They are divided into two classes; one class having a centre, the
other not having any. Co-ordinates of the centre.

Of diametric planes.

Simplification of the general equation of the second degree by the
transformation of co-ordinates.

The simplest equations of the ellipsoid, of the hyperboloid of one sheet and of two sheets, of the elliptical and the hyperbolic paraboloid, of cones and of cylinders of the second order.

Nature of the plane sections of surfaces of the second order.--Plane sections of the cone, and of the right cylinder with circular base.--Anti-parallel section of the oblique cone with circular base.

Cone asymptote to an hyperboloid.

Right-lined sections of the hyperboloid of one sheet.--Through each point of a hyperboloid of one sheet two right lines can be drawn, whence result two systems of right-lined generatrices of the hyperboloid.--Two right lines taken in the same system do not meet, and two right lines of different systems always meet.--All the right lines situated on the hyperboloid being transported to the centre, remaining parallel to themselves, coincide with the surface of the asymptote cone.--Three right lines of the same system are never parallel to the same plane.--The hyperboloid of one sheet may be generated by a right line which moves along three fixed right lines, not parallel to the same plane; and, reciprocally, when a right line slides on three fixed lines, not parallel to the same plane, it generates a hyperboloid of one sheet.