« PreviousContinue »
ing the roots of an equation by a geometrical construction. APPLICATION of Algebra and Geometry to Mechanics. This is found on the same principles as the Application of Algebra to Geometry; and consists principally in representing, by equations, the curves described by bodies in motion; as in the theory of Projectiles, &c. APPLICATION of Mechanics to Geometry, 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. o of Geometry and Astronomy to Geography, princiily consists in the three following articles; viz. in determining by geometrical and astronomical operations the figure of the terrestrial globe; in finding the positions of places by their observed latitude and longitude; and in determining, by geometrical operations, the positions of places that are not very remote from one another. Astronomy and geography are again applicable to the theory of navigation. APPLICATION of Geometry and Algebra to Physics or Natural Philosophy. For this application we are indebted to Sir Isaac Newton, whose philosophy may, therefore, be called the geometrical or mathematical philosophy; and upon this application are founded all the floo. sciences.— ence a single observation or experiment will often produce a whole science. Having ascertained, by experience, that the rays of light, by reflection, make the angle of incidence equal to that of reflection, we hence deduce the whole science of Captoptics. The case is also the same in many other sciences. APPROACH. The Curve of equable Approach, is of such a nature, that a body descending by the sole power of its own gravity approaches the horizon equally in equal times. This curve has been found, by Bernoulli, Varignon, Maupertuis, and others, to be the second cubical parabola, so placed that
its point of regression, or vertex, is uppermost, and the descending body must commence its motion in it with a certain determinate velocity. Varignon rendered the question more general by investigating the curve which a body might describe in vacuo, so as to approach through a given point through equal spaces in equal times, according to any law of gravity. Maupertuis also resolved the same problem, in the case of a body descending in a medium, the resistance of which is proportionate to the square of the velocity. Method of APP Roach Es, a term used by Dr. Wallis, in his Algebra, to denote a method of resolving certain problems, relating to square numbers, &c.; which is done by first assigning certain limits to the quantities required, and then approaching nearer and nearer till a coincidence is obtained. This method was invented by Dr. Pell, for the solution of equations of the form a2 — aye = 1; which problenn was proposed by Fermat, as a challenge to all the English mathematicians of his time; viz. to find rational and integral values of a and y, in the above equation, for every value of a, except when it is a complete square. APPROXIMATION, in Algebra and Arithmetic, is the method of approaching nearer and nearer to the quantity sought, when there is no method of obtaining the exact value. A PPROXIMATION to the Roots of Equations. As there is no direct method of determining the roots of equations beyond those of the fourth degree, and even in those of the third and fourth degree being very laborious by the direct rules, mathematicians have endeavoured to find methods of approximating the roots: of these, Newton’s rule is the most popular, and is founded on the following principles: If any two numbers, being substituted for the unknown o in an equation, give results with opposite signs, an odd number of roots must be between these numbers.
This appears from the property of the absolute term, and from this obvious maxim, that if a number of quantities be multiplied together, and the sign of an odd number of them be changed, the sign of the product is changed. For when a positive number is substituted for a, the result, is the absolute term of an equation, whose roots are less than the roots of the given equation by that quantity. If the result has the same sign as the given absolute term, then from the property of this term, either none, or an even number only, have had their signs changed by the transformation; but if the result has an opposite sign to that of the given absolute term, the sign of an odd number of the positive roots must have been changed.
In the first case, then, the quantity substituted must have been either greater than each of an even number of the positive roots of the given equation, or less than any of them; in the second case, it must have been greater than each of an odd number of the positive roots. An odd number of the positive roots must, therefore, lie between them, when they give results with opposite signs. The same observation is to be extended to the substitution of negative quantities and the negative roots.
Let it, for example, be proposed to find an approximate value of a in the equation
ing towards the roots of equations aphelion, and the lower apsis the perihelion; or, according to the ancient astronomy, the apogee and perigee. The diameter which joins these two points is called the line of the apsides, and is supposed to pass through the centre of the orbit of the planet, and the centre of the sun or earth; in the modern astronomy, this line makes the longest or transverse axis of the elliptical orbit of the planet; and in its line is counted the eccentricity of the orbit.
And in the same way we might find the particular formula answering to an equation of any other degree.
Application of the preceding Formulae.—In order to this it may be observed, that a' is an approximate value of a, found first by trial, as near the true root as possible, which in all cases ought to be true to the nearest integer. Then substituting this value of a ', a nearer approximate value of a will be obtained, which will in all cases double the number of figures. Then, considering this as a new value of ar', another still nearer value of a will be determined; and so on.
In stating the degree of approximation to be that of doubling the numb; of figures in the assumed
root, we are rather under than over the real state of convergency; for generally, if the first supposition be a simple integer, the first approximation may be obtained to three places, which, though it be not always correct in the last place, yet by employing it as a new approximation, the next value of a will, in most cases, be found true to six or seven fignres, which is a degree of convergency that cannot be obtained by any other rule that we are acquainted with, at the same time that the operation is much more simple than by any other rule whatever. An example will sufficiently illustrate the preceding remarks. Eacample.—Find the value of a in the cubic equation a:3+3a* + 3a: = 130; Assume a = 4, then by formula (1)
MATHEM ATICAL AND PHYSICAL SCIENCE.
Yolo to the above definition, the lines of the greatest and least distance are supposed to lie in the same straight line; which is not always, precisely the case, as the two frequently make an angle with each other; and what this angle differs from 180°, is called the motion of the line of the apsides; and when this is less than 180°, the motion is said to be contrary to the order of the signs; and when it is greater than 180°, the motion is said to be according to the order of the signs.
AQUARIUS, in Astronomy, the 11th sign of the zodiac, beginning
from Aries; its character is :.
AQUEDUCT, a conduit of water, in Architecture and Hydraulics, is a construction of stone and timber built on uneven ground, to preserve the level of water, and to conduct it through canals from one place to another. ARCH, or ARC, in Geometry, a part of a curve line: as of a circle, ellipse, &c. Circular Anc, is any part of the arc of a circle, and by which the magnitudes of angles are compared; an angle being said to contain so many degrees, minutes, &c. as are contained in the arc which subtends it. Co-centric ARcs, are such as have the same centre. Equal ARcs, are such arcs of the same circle, or of equal circles, which have the same measure, or the same number of degrees, minutes, &c. Similar ARcs, are those which have the same measure, but belonging to different circles. The lengths of similar arcs are to each other ot)
As since similar arcs are to each other as their radii, it is obvious, that having the length of any arc given to radius 1, the length of a similar arc may be found for any other radius, by multiplying the length of the first arc by the given radius. Or, since -01745329 is the length of an arc of 1°, to radius 1; the length of any arc, of which the measure is given, will be found by multiplying the number of degrees by 0.1745329; and that product again by the given radius. ARCH in Astronomy, has various denominations, according to the circle of which it is a part. Diurnal Arch of the Sun, is part of a circle described by the sun in his course between rising and setting. His nocturnal arch is that described between setting and ris. 1.11g. The latitude and elevation of the pole are measured by are: of the ineridian, and the longitude by an arch of a parallel circle. ARCH of Progression, or Direc: tion, is an arch of the ecliptic, which a planet seems to pass over, when its motion is direct, or according to the order of the signs. . ARCH of Retrogradation, is an arch of the ecliptic described when a planet is retrograde, or moves contrary to the order of the signs: Anch of Position, or Angle of Position, is the same with the horary
angle. Kon of Vision, is the sun's depth below the horizon at which a pla: net or star, before hid in his rays, begins to appear. This arch is different for different planets; being for Mercury 10°, Venus 5”, Mars 11!", jupiter 10°, Saturn 11”: a star of the 1st magnitude 12°,2d magnitude iás, oc. This angle is not, however, constant in all cases for the same planet, but varies a little with the latitude and declination, &c. With respect to Venus, it is sometimes reduced to 0, as she is at times visible when the sun is some degrees above the horizon: Arch of Equilibrium, in the Theory of Bridges, is that which is in equilibrio in all its parts, and therefore equally strong throughout, having no tendency to break in one part more than another. It is not of any determinate curve, but varies according to the figure of the extrados; every different extrados requiring a particular intrados, so that the thickness, in every part may be proportional to the pressure. If the arch were equally thick throughout, the cate. mary curve would be the arch of equilibration; but as this can seldon, or never happen, it is a mistaken idea to suppose this curve the best in all cases. It therefore appears, that when the upper side of the wall is a straight horizontal line, the equation of the curve is thus expressed:
y (a + m). — no falls, will be an assymptote to the extrados. Hence the extrados, in the case of a circular arch, is a curve of the fourth order, very much., resembling the conchoid of Nicomedes, and that it coincides very nearly with the curve, in which a road is usually carried over a bridge. This holds good, of whatever portion of the circle the arch is supposed to consist. ARCTIC Circle, in Astronomy, a small circle of the sphere parallel to the equator, and distant 23° 28′ from the arctic or northern pole. Arctic Pole, the northern pole of the world. AREA, in Geometry, is the superficial measure or surface of any figure. The areas of similar plane figures are to each other as the square of their like sides, or other lineal dimensions. AREOMETER, an instrument for measuring the density or gravity of fluids. It is now commonly made of glass; consisting of around hollow ball, which terminates in a long slender neck, hermetically sealed at top; there being first as much mercury put into it, as will serve to balance or keep it swim