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object at a greater distance. If the objects are parallel to each other, their real diameters are, in this case, proportional to their distances. The apparent diameter also varies with the position of the object and of equal objects at equal distances, those which stand in a position most nearly perpendicular to the line of their direction from the observer, will appear to have the greatest diameter: our idea of the apparent magnitude generally, varying nearly as the optic angle. But although the optic angle be the usual or sensible measure of the apparent magnitude of an ob ject, yet habit, and the frequent experience of looking at distant objects, by which we know that they are larger than they appear, has so far prevailed upon the imagination and judgment, as to cause this likewise to have some share in our estimation of apparent magnitudes; so that these will be judged to be more than in the ratio of the optic angles.

APPARENT Distance. See Dis

TANCE.

APPARENT Altitude of celestial Objects, is effected chiefly by refraction and parallax; and that of terrestrial objects, by refraction. See those words.

APPARENT Figure, is the figure or shape which an object appears under when viewed at a distance; and is often very different from the true figure. For a straight line, viewed at a distance, may appear but as a point; a surface, as a line; and a solid, as a surface. Also these may appear of different magnitudes, and the surface and solid of different figures, according to their situation with respect to the eye; thus the arch of a circle may appear a straight line; a square, a trapezium, or even a triangle; a circle, an ellipsis; angular magnitudes, round; and a sphere, a circle. Also all objects have a tendency to roundness and smoothness, or appear less angular, as their distance is greater: for, as the distance is increased, the smaller angles and asperities first disappear; after these, the next larger; and so on, as the distance is more and more increased, the

object seeming still more and more round and smooth.

APPARENT Motion, is either that motion which we perceive in a distant body that moves, the eye at the same time being either in motion or at rest; or that motion which an object at rest seems to have, while the eye itself only is in motion.

The motions of bodies at a great distance, though really moving equally, or passing over equal spaces in equal times, may appear to be very unequal and irregular to the eye, which can only judge of them by the mutation of the angle at the eye. And motions, to be equally visible, or appear equal, must be directly proportional to the distances of the objects moving. Again, very swift motions, as, those of the luminaries, may not appear to be motions at all, like that of the hour-hand of a clock, on account of the great distance of the objects: and this will always happen, when the space actually passed over in one second of time, is less than about the 14000th part of its distance from the eye. On the other hand, it is possible for the motion of a body to be so swift, as not to appear any motion at all; as when through the whole space it describes, there constantly appears a continued surface or solid as it were generated by the motion of the object, as is the case when any thing is whirled very swiftly round, describing a ring, &c.

Also the more oblique the eye is to the line which a distant body moves in, the more will the appa. rent motion differ from the true one.

If an eye move directly forwards in one direction, any remote object at rest will appear to move in a parallel line the contrary way. But if the object move the same way, and with equal velocity, it will seem to be at rest. If it move the same way, with less velocity, it will appear to move backwards, with the difference of the veloci ties: if it move with greater velocity, it will appear to move forwards with the difference of the velocities. And when the object

has a real motion contrary to that of the eye, it appears to move backwards with the sum of the velocities. The truth of all this is experienced by persons in a boat moving on water, or in a moving carriage, making observations on distant objects in motion, or at rest. APPARENT Place of an Object, in Optics, is that in which it ap pears, when seen in or through glass, water, or other reflecting or refracting media. In most cases, it differs much from the true place. APPARENT Station, in Astronomy, the position or appearance of a planet, or comet, in the same point of the zodiac for several days.

APPARITION, in Astronomy, denotes a star or other luminary's becoming visible, which before was hid in which sense it stands opposed to Occultation.

APPLICATE, Ordinate APPLICATE, in Geometry, is a right line drawn across a curve, so as to be bisected by the diameter of it; being what we commonly call a double ordinate.

APPLICATION, in Arithmetic, is sometimes used to signify division.

APPLICATION, in Geometry, generally means the placing of one line, angle, or surface upon another, with a view to prove their equality. This method of proof is also called proof by supraposition. APPLICATION of one Science to another, signifies the use that is made of the principles of the one, for extending and perfecting the

other.

respect the sides and other lines of geometrical figures; and those geometrical problems in which angles are concerned, are best resolved by the geometrical analysis. In the solution of problems of this kind, principles can be laid down which are applicable in all cases, and much must necessarily be left to the skill and ingenuity of the analyst.

When any geometrical problem is proposed for algebraical solution, one must, in the first place, draw a figure that shall represent the parts or conditions of the problem, and regard that figure as the true one; then, having considered the nature of the problem, the figure must be prepared for solution, by producing or drawing such lines as may be thought necessary. When this is done, let the unknown line or lines, which appear to be the most easily found, and any of the known ones that may be requisite, be denoted by proper symbols; then proceed to the operation, by observing the relation that the several parts have to each other. No rules can be given for drawing or producing the lines, as mentioned above, and it would therefore be useless to attempt any directions on that head; it is practice only which will render a student ready in the solution of problems of this kind; but some idea of the method may be collected from the following examples:

Prob. 1. Given the base, and the sum of the hypothenuse and perpendicular of a right-angled triangle, to find the sides severally.

APPLICATION of Algebra to Geometry, is of two kinds; viz. to plane or common geometry, and to curve lines. The first of these is concerned in the algebraical solution of geometrical problems, the investigation of geometrical fi-equations; viz. gures, &c. This method of resolving geometrical problems is, in many cases, inore direct and easy than that of geometrical analysis; but the latter method by synthesis, or construction, and demonstra tion, is the most elegant. The algebraical solution generally succeeds best in such problems as

Let A B C (plate I. fig. 9,) represent the right-angled triangle, of which the base A B is given, and put A Pb, A C=x, CB = y, the sum of A C and BC=s; then we readily obtain the two following

1st. x2-y2= b2 (Euclid 1—47.)
2d. x + y = s

x-y=

b2

2x=s+

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2y

by the quest.

by divis.

by addit.

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by subtr.

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Prob. 2. It is required to determine the side of a square inscribed in a given triangle.

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Having thus found the breadth, the length may be obtained by dividing the area by the breadth; or Let A B C (plate I. fig. 10,) re-otherwise we have, by finding the present the given triangle," and values of y instead of x, EFGH its inscribed square. Put the base A B=b, the perpendicu lar C Da, and the side of the square G F or GH; then will CI CD-DI=a-x,

And, because of the similar triangles ABC, GFC, it will be, as AB: CD GF: CI; or,

bax :α-x

Hence multiplying extremes and

means we have

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y

y

=

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bp

px b

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ba:

ba

P

Prob. 4. In a right-angled triangle, having given the lengths of

two lines drawn from the acute

of the square required. Cor. Hence it is obvious, that in all triangles whose bases and perpendiculars are constant, the side angles to the middle of the oppoof the inscribed square will be con-site sides, to find the sides of the triangle.

stant also.

Prob. 3. Giving the area or space of a rectangle, inscribed in a given triangle, to determine the sides of the rectangle.

Let ABC (fig. 11,) represent the given triangle, and EFGH the inscribed rectangle, the area of which is given, and which let be represented by a; make CD= P, AB=b, EF=x, and ID=9; then will CI=p-y; and by similar triangles we shall have

AB: CD EF : CI, or
b : p = x : p-y;

whence pabp-by,

aya, the arc a.

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and ...

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=AB.

The first gives г=

;

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15
3a+3b2
15

AC.

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Prob. 5. In a right-angled triangle, having given the hypothennse and side of the inscribed square, to find the base and perpendicular.

Let ABC (Fig. 9) be the proposed triangle, AC the given hypothenuse, and BD the given square.

Make A Ch, FD=DE=s, AB | Add and subtract double the last =x, RC=y. equation from the first, and we have

Then we have

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Whence xy-sx—sy + s2=s2; or... xy=s (x+y).

By adding double this to the first equation, we obtain

x2+2xy + y2= h2+2s (x+y), or (x+y)-2s (x + y) == h2; whence x+y=s±√(h2+s2). Now ay being known, make it =n, then we have

x2 + y2 = h2

x + y = n Square the second, and subtract it from double the first, and we obtain

x2-2xy + y2 = 2h2 — n2. By extracting, a — y = √(2h2—n2). Again •••x+y=n; therefore,

x=

and y=

as required.

n+√(2h2_n2)

2

n—√(2h2—n2)

2

AB;

= BC,

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x + y = √(n2 + 2pn) x − y = √(n2. 2pn). Whence again, x={√(n2+ 2pn) + √(n2 = 2pn=AB, y= √(n2 + 2pn) — √(n2 = 2pn) BC. Wherefore the three sides AB, BC, AC, are determined.

and

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In the second branch of the Application of Algebra to Geometry, or that which respects the higher geometry, or the nature and properties of curve lines, the nature of the curve is expressed, or denoted by an algebraical equation, which is formed as follows: a line is conceived to be drawn to represent the diameter, or some other principal line of the curve; and upon this line, at any indefinite points, are erected perpendiculars, which are called ordinates; and the parts of the first line cut off Prob. 6.-Having given the sum by them are termed abscisses. of the three sides of a right-angled Calling the abscis x, and its cortriangle, and the perpendicular, responding ordinate y, the known let fall from the right-angle upon nature of the curve, or the mutual the hypothenuse, to find the three relation of the other lines in it, sides of the triangle. will furnish an equation, involvLet ABC (fig. 12,) be the proposing x and y, with some other leted triangle, of which the sum of ter or letters which are known. the sides and perpendicular BD are And as x and y are common to given. Make the sum of the sides every point in the primary line, s, the perpendicular =p, AB, the equation, derived in this manBC y, AC=z. ner, will belong to every position or value of the absciss and ordi. nate; and may be properly considered as expressing the nature of the curve in all points of it, and is usually called the equation of the curve. Hence every particular curve will appear to have an appropriate equation, differing from that of every other; either as to the number of the terms, the powers of the unknown quantities r and y, or the signs of the coefficients of the terms of the equation.

Then we have ≈ + y + z =s,
by the question x2 + y2 = x2
(Eucl. 47, 1.) : z=p:y, by sim.

trians.; or

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xy=pz.

Now, add double this last equation to the second, then

x2+2xy + y2= x2 +2pz x2+2xy + y2= s2 — 2z+z2, by transposing z in the first equa tion and squaring. Whence, +2pz=s2—2sz + 2,

2=

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or, = AC. 2p+ 28 Now 2 being known, make it n; then the second and third equations become

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n?

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APPLICATION of Geometry to Algebra, is the converse of the first of the two preceding cases. It relates principally to the find

ing the roots of an equation by a lits point of regression, or vertex, geometrical construction.

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

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 Projec-through equal spaces in equal tiles, &c.

times, according to any law of gravity. Maupertuis also resolved the same problem, in the case of a body descending in a medium, the

APPLICATION of Mechanics to Geometry, consists chiefly in the use that is sometimes made of the centre of gravity of figures, for de-resistance of which is proportiontermining the contents of solids described by those figures.

ate to the square of the velocity.

Method of APPROACHES, a term APPLICATION of Geometry and used by Dr. Wallis, in his Algebra, Astronomy to Geography, princi- to denote a method of resolving pally consists in the three follow- certain problems, relating to square ing articles; viz. in determining numbers, &c.; which is done by by geometrical and astronomical first assigning certain limits to the operations the figure of the terres- quantities required, and then aptrial globe; in finding the positions proaching nearer and nearer till a of places by their observed lati- coincidence is obtained. This metude and longitude; and in deter-thod was invented by Dr. Pell, for mining, by geometrical operations, the solution of equations of the the positions of places that are not form x2 — ay2 = 1; which provery remote from one another. blem was proposed by Fermat, as Astronomy and geography are a challenge to all the English maagain applicable to the theory of thematicians of his time; viz. to navigation. 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.

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 mathe matical philosophy; and upon this application are founded all the phyico-mathematical sciences.Hence 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.

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.

APPROXIMATION 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 de gree being very laborious by the direct rules, mathematicians have endeavoured to find methods of approximating the roots: of these, APPROACH. The Curve of equa- Newton's rule is the most popuble Approach, is of such a nature,lar, and is founded on the followthat a body descending by the sole ing principles: power of its own gravity approach- If any two numbers, being subes the horizon equally in equal stituted for the unknown quantity times. This curve has been found, in an equation, give results with by Bernoulli, Varignon, Mauper- opposite signs, an odd number of tuis, and others, to be the second roots must be between these numcubical parabola, so placed that]bers.

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