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light and the image of the object come into the tube, and are thence conveyed to the eye. APEX, the vertex, top, or summit of any thing. APHELION, or APHELIUM, that point in the orbit of a planet, in which it is at its greatest distance from the sun; that is, at the ex1remity of the transverse diameter of the elliptic orbit, which is farthest distant from that focus in which the sun is placed. The aphelia of the planets are not fixed ; for their mutual actions upon each other keep those points of their orbits in a continual motion, which is greater or less in the different planets. APOGEE, in the ancient Astronomy, that point in the orbit of the sum, or a planet, which is farthest distant from the earth; and which corresponds with the aphelion of modern astronomers. It is still used to denote the greatest distance of a body from the earth. The moon, for example, is in its apogee when farthest from the earth; and its aphelion, when farthest from the sum. APOTOME, the remainder or difference between two lines or quantities, which are only commensurable in power. Such as the difference between 1 and V2, or the difference between the side and diagonal of a square. Euclid, in his tenth book, distinf.” these quantities, under six heads, viz. A potoME Prima, when the greater term is rational, and the difference of the squares of the two is a square number; as, 3– 5. A PotoME Secunda. When the less term is rational, and the square root of the difference of the squares of the two terms has to the greater term a ratio expressiBle in numbers; such is V 18–4, because the difference of the squares 18 and 16 is 2, and V2 is to J. as V1 to V.9, or as 1 to 3. Aporo Me Tertia. When both the terms are irrational, and, as in the second, the square root of the difference of their squares has to the *; term a rational ratio;
as y24–V18, for the difference of their squares 24 and 18 is G, and V6 is to V24 as V1 to V4 or as 1 to 2. Apoto MB Quarta. When the greater term is a rational number, and the square root of the differ. ence of the squares of the two terms has not a rational ratio to it; as, 4–V3, where the difference of the squares 16 and 3 is 13, and Y” has not a ratio in numbers to 4. A PotoME Quinta, when the less term is a rational number, and the square root of the difference of the squares of the two has not a rational ratio to the greater; as V6 –2, where the difference of the squares 6 and 4 is 2, and V2 to V6, or V1 to V3, or 1 to V3, is not a rational ratio. AporoME Serta, where both terms are irrational, and the square root of the difference of their squares has not a rational ratio to the greater; as V6—V2, where the difference of the squares 6 and 2 is 4, and V4 to V6, or 2 to V6, is not a rational ratio. APPARATUS, the appendages or utensils belonging to machines; as the apparatus of an air-pump, electrical machine, &c.; meaning the various detached parts which are necessary for putting the machinery in action, and for performing experiments, &c. APPARENT, in Mathematics and Astronomy, is used to signify things as they appear to us, in contradistinction from real or true; and in this respect the apparent state of things is often very different from their real state: as is the case of distance, magnitude, &c. APPARENT Conjunction of the Planets, is when a right line, sup. posed to be drawn through their centres, passes through the eye of the spectator, and not through the centre of the earth. APPARENT Diameter of an Object, is not the real length of that diameter, but the angle which it subtends at the eye, or under which it appears. This angle diminishes as the distance increases; so that a small object at a small distance may have the same appa
rent diameter as a much larger
object at a greater distance. If the objects are parallel to each other,
their real diameters are, in this case, proportional to their distancess he 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 DisTANCEAPPARENT Altitude of celestial Oojects, 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 mo, 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 notion. The motions of bodies nt a great distance, though really moving equally, or passing over equal spaces in equal times, may appear to be very unequal and in egular 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 juy 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 apparent motion differ from the true oneIf 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 disserence of the velocities: 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 appears, 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
davs. . *FPARItion, 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 Applicarb, 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 diviSiOil. 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. APPLICATION of Algebra to Geometry, is of two kinds; viz. to plane or common . and to curve lines. The first of these is concerned in the algebraical solution of geometrical problems, the investigation of geometrical figures, &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 demonstration, is the most elegant. The algebraical solution generally succeeds best in such problems as
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 hypothen use and perpendicular of a right-angled triangle, to find the sides severally. Let A B C (plate I. fig. 9,) represent the right-angled triangle, of which the base A B is given, and put A P = b, A C == x, C B = y, the sum of A C and B C = s : then we readily obtain the two following equations; viz. 1st. are—yo– bo (Euclid 1–47.) 2d. a +y= #: • * - by the quest.
Add and subtract double the last equation from the first, and we have x3 + 2ary + y? = n° -- 2pm a2 — 2xy + y^ = n° — 2pm, the roots of which are, a + y = V(n° + 2pm) and a — y = V(n° — 2pn). Whence again, a = , V(n°-H 2pm)-H A V(n°–2pm = AB, and . . . y = | N/(n°-H2pm)— * V(n°–2pm) = BC. Wherefore the three sides AB, BC, AC, are determined. In the second branch of the Application of Algebra to Geometry, or that which respects the higher geometry, or the mature 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 by them are termed abscisses. Calling the abscis ar, and its corresponding ordinate y, the known nature of the curve, or the mutual relation of the other lines in it, will furnish an equation, involving a and y, with some other letter or letters which are known. And as z and y are common to every point in the primary line, the equation, derived in this manner, will belong to every position or value of the absciss and ordi. mate; 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 as and y, or the signs of the coefficients of the terms of the equation. APPLICATION of Geometry to Algebra, is the converse of the first of the two preceding cases. It relates principally to the find