360° AP-PARENT LEVEL. In Leveling, the 2n line of level indicated by the axis of the telescope when made horizontal. The true level is a line every point of which is equally distant from the centre of the earth; hence, a line of apparent level at any point, is tangent to the line of true level through the same point; or more strictly speaking, the plane of apparent level at any point is tangent to the surface of true level passing through the same point.

hence, it is visible to the eye, at the level of the sea, 12 miles distant. If the eye of the observer is elevated above the level of the sea, the same rule must be applied to this elevation, and also to the elevation of the lighthouse, and the sum of the results will be the required result thus, if a light-house is 96 feet high, and the eye of the observer 24 feet high, the light-house is visible at a distance equal to

√ × 96 + √ × 24, or 12 + 6, or 18 miles. This rule only gives approximate results, since it does not take into account the effect of refraction which operates to increase the range of vision. See Leveling.

The application of one branch of mathematics to another, or of one science to another, consists in using the principles developed in one, for the purpose of developing or illustrating the principles of the other. For this purpose, algebra has been applied to geometry, geometry to algebra, and both to mechanics, astronomy, navigation, &c.

APPLICATION OF ALGEBRA TO GEOMETRY, consists in applying the rules and principles of algebra to the solution of geometrical problems, or the demonstration of geometrical propositions. Instances of this kind of investigation occur in the works of the earliest mathematicians, as Diophantus, Tartalea, &c., as well as in those of more recent

AP'PLI-CATE. A chord which is bisected date. by a diameter. If a curve is referred to a The algebraic solution of a geometrical diameter, and a line parallel to the chords problem, consists of three parts: 1st, exwhich it bisects, then an applicate is the pressing the conditions of the problem in same as the double ordinate through any algebraic language by means of equations. point of the diameter. APPLICATE NUMBERS are the same as con- known rules, so as to develop the relations crete numbers.

APPLICATE ORDINATE. An applicate with reference to an axis of the curve. It is the same as the double ordinate, perpendicular to an axis of the curve.

AP-PLI-CA'TION [L. applicatio]. The operation of applying one thing to another, or of comparing one thing with another by bring ing them together: thus, the length of a line is determined by applying to it some unit of measure, and determining the number of times which it contains the unit.

2d. Combining these equations by means of

between the required and known parts; and 3d. Interpreting the results, and making the necessary constructions thus indicated.

The general method of proceeding has already been indicated in treating of determinate analytical geometry.

A few examples will serve to illustrate the rule there given, as well as the different methods of translating the conditions, and interpreting the results, which cannot be reduced to any fixed and invariable rules.

1. Having given a triangle, let it be required to find the sides of an inscribed rectIn this sense, application is nearly synony-angle, such that its adjacent sides shall bear to each other a given ratio. Let ABC be the given triangle, and suppose the re

be inscribed.

Denote the base

mous with division. In Arithmetic, the term
is employed to denote the use of the princi-
ples of science in the solution of practical
problems. In Geometry, one figure is ap-
plied, or conceived to be applied to another, quired rectangle to
for the purpose of determining whether they
are equal or unequal. In this manner, many
of the fundamental principles of Geometry
have been proved: thus, it is proved, that if
two triangles have two angles and the in-
cluded side of the one, equal to two angles
and the included side of the other, each to
each, they will be equal. It is also proved,
by the method of application, that the diam-
eter of a circle divides the circle into two
equal parts.

C

F

of the given tri-
A II D B
angle by b, its alti-
tude by h. Denote the side of the rectangle
perpendicular to the base by z, and the adja-
cent side by nx, n being the given ratio.
Then, from the figure,

bh

bh nx: h -x .. x = b+nh

To construct this value of x, produce the

3. Having given an isoceles triangle, to

base AC, and on the prolongation lay off] CH' equal to nh; through H' draw H'B' find a second isoceles triangle, which shall

FB

parallel to CB, and through the vertex B draw BB' parallel to the base AC. Join B'A, and through F draw FG perpendicular Let ABC represent the given triangle, and have an equal area and an equal perimeter. to AC, and FE parallel to AC, and complete DEF the required triangle. Denote the area the rectangle FD, which will be the required of the given triangle by A, and its perimeter by p; denote the base of the first triangle by 2a, and one of its equal sides by b; then will its altitude be equal to

rectangle for, from the figure,

AH' AC : : BP B+nh : b :: h

FG, or

:

FG ;

Hence, FG is equal to the side designated by z, and FE to the side designated by nx.

The construction is therefore verified.

2. In a right-angled triangle, having given

the lengths of two lines drawn from the ver- Since the area of the second triangle is

tices of the acute angles to the middle points of the opposite sides, to find the sides of the triangle.

Let ABC represent the triangle, and AD

and CE the given

lines. Denote AD

by a, CE by b, AB by 2x, and

we have

and

A = x√ y2 — x2,
p=x+y, or, y = p-x;

whence

y2 = p2 - px + x2;

a

2'

possible, so as to see the relative advantages of each method.

2. APPLICATION OF GEOMETRY TO ALGEBRA, consists in applying the principles of geometry to the elucidation of algebraic formulas and principles. Higher geometry, and sometimes elementary geometry, may be usefully applied to the purpose of investigating the nature of the roots of equations, and also to determine the value of those roots by geometrical construction. It is also of use in the investigation of trigonometrical formulas. It is said that the Arabians discovered the rule for solving complete equations of the second degree, by the aid of geometry; and also that by the same means Tartalea and Cardan deduced and demonstrated the rules for solving cubic equations, employing for that purpose the principles of solid geometry.

The method of proceeding in this kind of

and ED is equal to (ba). Erect EG perpendicular to FD till it intersects the semi-investigation is to construct a figure such that circle described on FD as a diameter; then

Hence, K is one of the vertices at the base, and, by laying off EL EK, we find a second vertex. Now, let a circle be inscribed

each part shall represent one of the given the relation between these parts shall be the quantities in the expression, and such that same as that expressed by the algebraic expression; then from the known geometrical properties of the figure, to deduce the required relations. We annex the geometrical method of constructing the roots of equations of the first, second, third, and fourth, and higher degrees.

1. Equations of the First Degree. Let us take ar b= 0, which gives

in the first triangle, and through the points intersecting at A.
K and L draw tangents to it, forming the tri-Lay off from A on
angle KLM; this will be the triangle re- AB, the distance
quired. If the given triangle is equilateral, AB = a, and from
the construction will give only the triangle it- A on AE, the dis-
self.
tance AE= b; draw

EB; lay off from A on AB, the distance AC = 1, and draw CD parallel to BE; then is AD the representation of the value of x.

AB AE ::

AC: AD;

b

The second value of x corresponds to a second construction, which would give a triangle lying below the given triangle, which corresponds to the algebraic enunciation of For, from the figure, the problem. It may easily be constructed. Nothing but long experience can enable the student to seize upon the relations of the parts of the problems presented so as to give or the simplest solutions. It is, therefore, well 2. Equations of the Second Degree. Every to solve every given problem in every manner equation of the second degree, containing

or a

: b: : 1 AD .. AD:

AD = x.

is

but one unknown quantity, can be reduced If p<q the circle does not cut FA, the to one of the forms below, in which construction fails, and the roots of both forms are imaginary.

essentially positive.

M

Construct a parabola whose axis is AP, and whose parameter is equal to 2p. Lay off

on the axis a distance AD = a, and at D

erect a perpendicular equal to b; from its extremity C as a centre, and with a radius CM equal to r, describe the circumference of a circle cutting the parabola in the points M, M', M", and M""; from each of these points let fall a perpendicular upon the axis, and these perpendiculars will be roots of an equation of the fourth degree.

If one of these points of intersection fall at A, the perpendiculars will be roots of a cubic equation.

The following considerations will serve to determine proper values for a, b, 2p, and r, in any given case.

E A F B Draw an indefinite right line FA, and at any point as D, erect a perpendicular DC equal to q; from C as a centre, with a radius CB equal to p, describe an arc of a circle reduction, cutting FA in B and E; then is

BD = √ p2 - q2.

y* — (4pa — 4p2)y2 — 4bp2y

+ 2(a2 + b2 — r2)p = 0 . . . (1). In any given case, we reduce the equation to the form of equation (1), by depriving it of its second term, and making the coefficient of the first term 1. Then equate the remain

From D lay off on FA, in both directions, the distances DF and DA respectively, equal to p. The lines - AE and - AB will represent the first and second roots of the third form, and the lines FB and FE the first and ing coefficients and absolute term with the second roots of the fourth form.

corresponding coefficients in equation (1). Three equations will thus be found containing a, b, 2p, and r, from which we may, after assuming a value for either one, deduce corresponding values for the other. The con