APPLICATE

, Applicata, Ordinate Applicate, in Geometry, is a right line drawn to a curve, and bisected by its diameter. This is otherwise called an Ordinate, which see.

Applicate Number. See Concrete.

, the act of applying one thing to another, by approaching or bringing them nearer together. So a longer space as measured by the continual application of a less, as a foot or yard by an inch, &c. And motion is determined by a successive application of any thing to different parts of space.

Application is sometimes used, both in Arithmetic and Geometry, for the rule or operation of division, or what is similar to it in geometry. Thus 20 applied to, or divided by 4, gives 5. And a rectangle ab, applied to a line c, gives the 4th proportional ab/c, or another line which, with the given line c, will contain another rectangle which shall be equal to the given rectangle ab. And this is the sense in which Euclid uses the term, lib. 6, pr. 28.

, in Geometry, is also used for the act or supposition of putting or placing one figure upon another, to find whether they be equal or unequal; which seems to be the primary way in which the mind first acquires both the idea and proof of equality. And in this way Euclid, and other geometricians, demonstrate some of the first or leading properties in geometry. Thus, if two triangles have two sides in the one triangle equal to two sides in the other, and also the angle included by the same sides equal to each other; then are the two triangles equal in all respects: for by conceiving the one triangle placed on the other, it is proved that they coincide or exactly agree in all their parts. And the same happens if, of two triangles, one side and the two adjacent angles of the one triangle, are equal, respectively, to one side and the two corresponding angles of the other. Thus also it may be proved that the diameter of a circle divides it into two equal parts, as also that the diagonal of a square or parallelogram bisects or divides it into two equal parts.

Application of one science to another, as of Algebra to Geometry, is said of the use made of the principles and properties of the one for augmenting and perfecting the other. Indeed all arts and sciences mutually receive aid from each other. But the application here meant, is of a more express and immediate nature; as will appear by what follows.

Application of Algebra or of Analysis to Geometry. The first and principal applications of algebra, were to arithmetical questions and computations, as being the first and most useful science in all the concerns of human life. Afterwards algebra was applied to geometry and all the other sciences in their turn. The application of algebra to geometry, is of two kinds; that which regards the plane or common geometry, and that which respects the higher geometry, or the nature of curve lines.

The first of these, or the application of algebra to common geometry, is concerned in the algebraical solution of geometrical problems, and finding out theorems in geometrical figures, by means of algebraical investigations or demonstrations. This kind of appli- | cation has been made from the time of the most early writers on algebra, as Diophantus, Lucas de Burgo, Cardan, Tartalea, &c, &c, down to the present times. Some of the best precepts and exercises of this kind of application, are to be met with in Newton's Universal Arithmetic, and in Thomas Simpson's Algebra and Select Exercises. Geometrical Problems are commonly resolved more directly and easily by algebra, than by the geometrical analysis, especially by young beginners; but then the synthesis, or construction and demonstration, is most elegant as deduced from the latter method. Now it commonly happens that the algebraical solution succeeds best in such problems as respect the sides and other lines in geometrical figures, and on the contrary, those problems in which angles are concerned, are best effected by the geometrical analysis. Newton gives these, among many other remarks on this branch. Having any problem proposed; compare together the quantities concerned in it; and, making no difference between the known and unknown quantities, consider how they depend upon, or are related to, one another; that we may perceive what quantities, if they are assumed, will, by proceeding synthetically, give the rest, and that in the simplest manner. And in this comparison, the geometrical figure is to be feigned and constructed at random, as if all the parts were actually known or given, and any other lines drawn that may appear to conduce to the easier and simpler solution of the problem. Having considered the method of computation, and drawn out the scheme, names are then to be given to the quantities entering into the computation, that is, to some few of them, both known and unknown, from which the rest may most naturally and simply be derived or expressed, by means of the geometrical properties of figures, till an equation be obtained, by which the value of the unknown quantity may be derived by the ordinary methods of reduction of equations, when only one unknown quantity is in the notation; or till as many equations are obtained as there are unknown letters in the notation.

For example, suppose it were required to inscribe a square in a given triangle. Let ABC be the given triangle; and feign DEFG to be the required square; also draw the perpendicular BP of the triangle, which will be given, as well as all the sides of it. Then, considering that the triangles BAC, BEF are similar, it will be proper to make the notation as follows, viz, making the base AC=b, the perpendicular BP=p, and the side of the square DE or EF=x. Hence then ; consequently, by the proportionality of the parts of those two similar triangles, viz, BP : AC :: BQ : EF, it is p : b :: p-x: x; then, multiply extremes and means &c, there arises , or , and the side of the square sought; that is, a fourth proportional to the base and perpendicular, and the sum of the two, taking this sum for the first term, or AC + BP : BP :: AC : EF.

The other branch of the application of algebra to geo- metry, was introduced by Descartes, in his Geometry, which is the new or higher geometry, and respects the nature and properties of curve lines. In this branch, the nature of the curve is expressed or denoted by an algebraic equation, which is thus derived: A line is conceived to be drawn, as the diameter or some other principal line about the curve; and upon any indesinite points of this line other lines are erected perpendicularly, which are called ordinates, whilst the parts of the first line cut off by them, are called abscisses. Then, calling any absciss x, and its corresponding ordinate y, by means of the known nature, or relations of the other lines in the curve, an equation is derived, involving x and y, with other given quantities in it. Hence, as x and y are common to every point in the primary line, that equation, so derived, will belong to every position or value of the absciss and ordinate, and so is properly considered as expressing the nature of the curve in all points of it; and is commonly called the equation of the curve.

In this way it is found that any curve line has a peculiar form of equation belonging to it, and which is different from that of every other curve, either as to the number of the terms, the powers of the unknown letters x and y, or the signs or co- efficients of the terms of the equation. Thus, if the curve line HK be a circle, of which HI is part of the diameter, and IK a perpendicular ordinate: then put HI=x, IK=y, and p=the diameter of the circle, the equation of the circle will be . But if HK be an ellipse, an hyperbola, or parabola, the equation of the curve will be different, and for all the four curves, will be respectively as follows, viz, where t is the transverse axis, and p its parameter. And, in like manner for other curves.

This way of expressing the nature of curve lines, by algebraic equations, has given occasion to the greatest improvement and extension of the geometry of curve lines; for thus, all the properties of algebraic equations, and their roots, are transferred and added to the curve lines, whose abscisses and ordinates have similar properties. Indeed the benefit of this sort of application is mutual and reciprocal, the known properties of equations being transferred to the curves they represent; and, on the contrary, the known properties of curves transferred to their representative equations. See Curves.

Application of Geometry to Algebra. Besides the use and application of the higher geometry, namely, of curve lines, to detecting the nature and roots of equations, and to the finding the values of those roots by the geometrical construction of curve lines, even common geometry may be made subservient to the purposes of algebra. Thus, to take a very plain and simple instance, if it were required to square the binomial a+b; | by forming a square, as in the annexed figure, whose side is equal to a+b, or the two lines or parts added together denoted by the letters a and b; and then drawing two lines parallel to the sides, from the points where the two parts join, it will be im- mediately evident that the whole square of the compound quantity a+b, is equal to the squares of both the parts, together with two rectangles under the two parts, or a² and b² and 2ab, that is the square of a+b is equal to a²+b²+2ab, as derived from a geometrical figure or construction. And in this very manner it was, that the Arabians, and the early European writers on algebra, derived and demonstrated the common rule for resolving compound quadratic equations. And thus also, in a similar way, it was, that Tartalea and Cardan derived and demonstrated all the rules for the resolution of cubic equations, using cubes and parallelopipedons instead of squares and rectangles. And many other instances might be given of the use and application of geometry in algebra.

Application of Algebra and Geometry to Mechanics. This is founded on the same principles as the application of algebra to geometry. It consists principally in representing by equations the curves described by bodies in motion, by determining the equation between the spaces which the bodies describe, when actuated by any forces, and the times employed in describing them, &c. A familiar instance also of the application of geometry to mechanies, may be seen at the article ACCELERATION, where the perpendiculars of triangles represent the times, the bases the velocities, and the areas the spaces described by bodies in motion; a method first given by Galileo. In short, as velocities, times, forces, spaces, &c, may be represented by lines and geometrical figures; and as these again may be treated algebraically; it is evident how the principles and properties, of both algebra and geometry, may be applied to mechanics, and indeed to all the other branches of mixt mathematics.

Application of Mechanies to Geometry. This consists chiefly in the use that is sometimes made of the centre of gravity of figures, for determining the contents of solids described by those figures.

Application of Geometry and Astronomy to Geography. This consists chiefly in three articles. 1st, In determining the figure of the globe we inhabit, by means of geometrical and astronomical operations. 2d, In determining the positions of places, by observations of latitudes and longitudes. 3d, In determining, by geometrical operations, the positions of such places as are not far distant from one another.

Geometry and Astronomy are also of great use in Navigation.

Application of Geometry and Algebra to Physics or Natural Philosophy. This application we owe to Newton, whose philosophy may therefore be called the geometrical or mathematical philosophy; and upon this application are founded all the physico-mathematical sciences. Here a single observation or experiment will often produce a whole science: so when we know, as we do by experience, that the rays of light, in reflect- ing, make the angle of incidence equal to the angle of reflexion; we thence deduce the whole science of catoptrics: for that experiment once admitted, catoptrics become a science purely geometrical, since it is reduced to the comparison of angles and lines given in position. And the same in many other sciences.

APPLICATION of one thing to another, in general, is employed to denote the use that is made of the former, to understand or to perfect the latter. Thus, the application of the cycloid to pendulums, means the use made of the cycloidal curve for improving the doctrine and use of pendulums.

APPLY. This term is used two different ways, in geometry.

1st, It signifies to transfer or place a given line, either in a circle or some other figure, so that the extremities of the line shall be in the perimeter of the figure.

2d, It is also used to express division in geometry, or to find one dimension of a rectangle, when the area and the other dimension are given. As the area ab applied to the line c, is ab/c.