Page:Cyclopaedia, Chambers - Volume 1.djvu/697

Rh Equal Figures, are those whose Area's are Equal ; whether the Figures be similar or not. See Figure.

The Segments of a Sphere, or Circle, are of an Equal Concavity, or Convexity, when they have the fame Ratio, or Proportion to the Diameters of the Spheres, or Circles whereof they are Parts. See Segment.

Equal Solids, are those which comprehend, or contain each as much as other; or whose Solidities and Capacities are Equal. See Solid.

Equal Hyperbola's, are those whose Ordinates to their indeterminate Axes, are Equal to each other; taken at Equal Distances from their Vertices. See Hyberbola.

Equal Geometrical Ratio's, are those whose least Terms are similar aliquot, or aliquant Parts of greater. See Ratio.

Equal Arithmetical Ratio's, are those wherein the Difference of the two less Terms, is Equal to the Difference of the two greater. See Ratio.

In Opticks, we fay, that Things seen under Equal Angles, are Equal. Equal Parts of the same Interval, or Magnitude, if unequally distant from the Eye, appear unequal. Equal Objects, and at Equal Distances, only the one placed directly, and the other obliquely, seem unequal;and that placed directly, the bigger. See Vision.

Equality, in Astronomy. The Circle of Equality, or the Equant, is a Circle used in the Ptolomaic Astronomy, to account for the Eccentricity of the Planets, and reduce them more easily to a Calculus. See Equant.

In Geometry, the Ratio, or Proportion of Equality, is that between two equal Numbers, or Quantities. Proportion of Equality evenly ranged, or ex quo ordinata, is that wherein two Terms in a Rank, or Series, are Proportional to as many Terms in another Series, compared to each other in the same Order, i. e. the first of one Rank to the first of another; the second to the second, &c.

Proportion of Equality evenly disturbed, call'd also ex quo Perturbata, is that wherein more than two Terms of a Rank, are proportional to as many Terms of another Rank, compared to each other, in a different and interrupted Order; viz. The first of one Rank to the second of another; the second to the third, &c.

Equality, in Algebra, is a Comparison of two Quantities, that are equal both really and representatively, i. e. both equal in Effect and Letters: And Equation, is a Comparison of two Magnitudes, equal in Effect, but unequal in Letters, to render them equal.

Equality, in Algebra, is usually denoted by two parallel Lines, as: Thus, 2+2= 4, i. e. 2 Plus 2, are equal to 4.

This Character was first introduced by Hariot. Des Cartes, and some after him, in Lieu thereof use : Thus, 2+2 4, so x-y=b+c, signifies that x minus y, is equal to b plus c.

From an Equation, we arrive at an Equality, by changing one Letter into another, whereby the two Members of the Equation, i. e. the two Quantities compared together, and connected by the Sign of Equality, are render'd Equal.

Thus, in the Equation $$\mathrm{aa}x=\mathrm{bcd}$$; supposing $$x = \frac{\mathrm{bcd}}{\mathrm{aa}}$$ we change $$x$$ into $$\frac{\mathrm{bcd}}{\mathrm{aa}}$$ and by this substitution arrive at the Equality $$\mathrm{bcd}=\mathrm{bcd}$$

In the Solution of a Numerical Problem, to be render'd rational; if there be only one Power to be equall'd to a Square, or other higher Power; it is call'd simple Equality.

When there are two Powers to be equall'd, each to a Square, it is call'd double Equality, &c.

Diophantus has given us a Method for double Equalities, and Fa. Billy, another for triple Equalities, in his Diophantus and Redivivus.

Equant, or Æqunat, in Astronomy, a Circle, imagined by Astronomers, in the Plane of the Deferent, or Eccentric for the regulating and adjusting of certain Motions of the Planets. Sec Deferent.

Equation or Equation, in Algebra, an Expression of the same Quantity, in two different, that is, diffimilar, but Equal Terms or Denominations. As when we say, 2. 3=4+2; that is, twice three is equal to four and two. Sec Qunatity.

Stiselius defines Equation to be the Ratio of Equality, between two Quantities differently denominated: As when we fay 3 Shill. = 36 Pence. Or 50 Shill. = 2 Lib. 10 Shill, = 600 Pence, = 2400 Farth. Or, b=d+c. Or, $$ 12 = \frac{2-P}{5}$$ &c.

Hence, the Reduction of two heterogeneous, or dissimilar Quantities to the fame Value, i. e. to an Equality, is called the bringing them to an Equation

The Character, or Sign of an Equation, is x or See Character.

The Resolving of Problems, by Means of Equations, is the Business of Algebra. See Algebra.

The Terms of an Equation, are the several Quantities, or Parts, of which an Equation is composed; connected together by the Signs and Thus, in the Equation b+c= d; the Terms are b, c and d. And the Tenor of the Equation is, that some Quantity, represented by b is equal to two others represented by c and d. See Terms of Equations.

The Root of an EQUATION, is the Value of the unknown Quantity in the Equation. E. gr. if $$a^2 + b^2 =x$$; the Root will be (a + b). See Roots of Equations.

Equations are divided with Regard to the Powers of the unknown Quantities, into Simple, Quadratic, Cubic, &c.

A Simple Equation, is that wherein the unknown Quantity is only of one Dimension, or the first Power. As, x= (a+b): 2. See SIMPLE Equation.

A Quadratic Equation, is that wherein the unknown Quantity is of two Dimensions, or the second Power.

As, $$x^2=ab$$. See Quadratic Equation.

A Cubic Equation, is that wherein the unknown Quantity is of three Dimensions. As $$x^3=a^3+b^3$$, &c. See Cubic Equation.

If the unknown Quantity be of four Dimensions, as $$a^4$$ b, the Equation is call'd a Biquadratic; if of 5, a Surdesolid, &c. See Power.

Equations are consider'd two Ways: Either, as the ultimate Conclusions we arrive at in the Solution of Problems or as Means, by the Help whereof, we arrive at those final Solutions. See Solution and Problem. An Equation of the first Kind, consists only of one unknown Quantity, intermix'd with other known Quantities. Those of the latter Kind, consist of several unknown Quantities, which are to be compared, and connected together, till out of them all arise a new Equation, wherein the one unknown Quantity fought, is mix'd with the known. To get the Value of which unknown Quantity, the Equa- tion is generally turn'd, and transform'd various Ways, till it be brought as low, and render'd as simple as possible.

The Doctrine and Practice of Equations; that is, the Solution of Questions by Equations, consists of several Steps, or Parts, viz. 1°. The denominating of the several Quantities, or expressing them in proper Signs, or Symbols. 2°. The bringing the Quantities thus denoted to an Equation. 3°. The reducing that Equation to its loweſt and simplest Terms. To which 4°. may be added the constructing of the Equation, or representing it in Geometrical Lines: We proceed to each in its Order.

With Regard to the first; a Question, or Problem, being proposed; we conceive the Thing fought, or required, as already done: And accordingly note, or express it by one of of the Vowels, as a, or more usually, by one of the last Letters the Alphabet, x y or z, noting the other known Quantities, by the Consonants, or the beginning Letters of the Alphabet, b, c, d, &c. See Quantity, Character, Species.

The Question being thus atated in Species; it is consider'd whether, or no, it be subject to any Restrictions, i. e. whether it be determinate, or no; which is found by these Rules.

1°. If the Quantities fought, or required, be more than the Number of Equations given, or contain'd in the Question; it is indeterminate, and capable of innumerable Solutions. The Equations are found, if they be not expresly contain'd in the Problem it self, by the Theorems of the Equality of Quantities.

2°. If the Equations given, or contain'd in the Problem, be just equal in Number with the unknown Quantities; the Queſtion is determinate, or has one only Solution.

3°. If the unknown Quantities be fewer than the given Equations, the Question is yet more limited, and fome- times discovers it self impossible, by fome Contradiction between the Equations. See Determinate, &c.

To bring Questions to EQUATIONS.

Now, to bring a Question to an Equation, that is, to bring the several mediate Equations, to one final one; the principal Thing to be attended to, is to express all the Conditions thereof, by so many Equations. In order As when to which, it is to be consider'd, whether the Propositions, or Sentences, wherein it is expressed, be all of them fit to be noted in Algebraic Terms; as our Conceptions use to be in Latin, or Greek Characters. And if so, as is generally the Case in Queſtions about Numbers, or abstract Quantities; then let Names be given both to the known and unknown Qualities, as far as Occasion requires: And thus the Drift of the Question may be couch'd, as we may