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stant take the altitudes of both horns: the difference of these two altitudes being halved and added to the least, or subtracted from the greatest, gives nearly the visible or apparent altitude of the moon's centre; and the true altitude is nearly equal to the altitude of the centre of the shadow at that time. Now we know the altitude of the shadow, because we know the place of the sun in the ecliptic, and its depression under the horizon, which is equal to the altitude of the opposite point of the eclipic in which is the centre of the shadow. And therefore, having both the true altitude of the moon and the apparent altitude, the difference of these is the parallax required. But as the parallax oft: moon increases as she approaches towards the earth, or the perigaeum of her orbit, therefore astrono. mers have made tables, which show the horizontal parallax for every degree of its anomaly. The parallax always diminishes the altitude of a phenomenon, or makes it appear lower than it would do, if viewed from the centre of the earth; and this change of the altitude may, according to the different situation of the ecliptic and equator in respect of the horizon of the spectator, cause a change of the latitude, longitude, declination, and right ascension of any phenomenon, which is called their parallax. The parallax, therefore, increases the right and oblique ascension; diminishes the descension; diminishes the northern declination and latitude in the eastern part, and increases them in the western; but increases the southern both in the eastern and western part; diminishes the longitude in the western part, and increases it in the eastern. Hence it appears, that the parallax has just opposite effects to refraction. See Refaaction. PARALLAx, annual, the change of the apparent place of a heavenly body, which is caused by being viewed from the earth in different parts of its orbit round the sun. The annual parallax of all the planets is found very considersble, but that of the fixed stars is imperceptible. PARALLAx, in levelling, denotes the angle contained between the line of the true level and that of the apparent level. PARALLEL. The subject of parallel lines, says Playfair, is one of the most dif. ficult in the Elements of Geometry. It has accordingly been treated in a great yariety of different ways, of which, perhaps, there is none which can be said to
have given entire satisfaction. The difficulty consists in converting the twentyseventh and twenty-eighth of Euclid, or in demonstrating, that parallel straight lines (or such as do not meet one another) when they meet a third line, make the alternate angles with it equal, or, which comes to the same, are equally inclined to it, and make the exterior angle equal to the interior and opposite. In order to demonstrate this proposition, Euclid assumed it as an axiom, that if a straight line meet two straight lines, so as to make the interior angles on the same side of it less than two right angles, these straight lines being continually produced, will at length meet on the side on which the angles are that are less than two right angles. This proposition, however, is not self-evident ; and ought the less to be received, without proof that the converse of it is a proposition that confessedy requires to be demonstrated. In order to remedy this defect, three sorts of methods have been adopted—a new definilion of parallel lines; a new manner of reasoning on the properties of straight lines, without any new axiom ; and the introduction of a new axiom less exceptionable than Euclid's. Playfair adopts the latter plan; but we do not perceive that his axiom is by any means self-evident upon Euclid's definition, which he retains, viz. Parallel straight lines are such as are in the same plane, and which being produced ever so far both ways do not meet. A more intelligible, and we think an equally rigid demonstration of the property of parallels may be obtained without any axiom, by means of a new definition. It may at first sight be thought, that the objection urged by Playfair against the definition in f. Simpson's first edition, must equally hold against ours; but we think that if his objection really hold good against that definition, (though we confess we cannot feel the force of it,) it is obviated by distinguishing, as ought to be done, between the distance and the measure of that distance. We must of course suppose our readers acquainted with the propositions in Euclid preceding the twenty-seventh; but to save the necessity of reference, we shall give an enunciation of those which we shall have to employ in our demonstration, in the form in which we employ them. 1. (Prop. 16.) If one side of a triangle be produced, the outward angle is greater than either of the inward oppo
site angles. 2. (Prop. 19.) The greater angle of every triangle has the greater side opposite to it. 3. (Prop. 4.) If two triangles have two sides of the one respectively equal to two sides of the other, and have the included angles equal, the other angles will be respectively equal, viz. those to which the equal sides are opposite. 4. (Prop. 15.) If two straight lines cut each other, the vertical or opposite angles will be equal. 5. (Prop. 13.) If a straight line meet another, the sum of the adjacent angles is equal to the sum of two right angles. 6. Definition. Parallel straight lines are those whose least distances from each other are every where equal. 7. Theorem I. The perpendicular drawn to a straight line from any point, is the least line that can be drawn from that point to the given line. Let C D (Plate XII. Miscell. fig. 2.) be a straight line drawn from C perpendicular to A B; and let C E be any other straight line from C to A B; them is CD less than C. E. For the angle C D E equals angle C D A by construction; and Q DA is greater than C E D (1); therefore C D Eis greater than C. E. D. Hence (?) C D is less than C. E. 8. Cor. 1. Hence the perpendicular from any point to a straight line is the true measure of the least distance of that point from that line. 9. Cor. 2. Hence (6) the perpendiculars to one of two parallel straight lines, 9m any points in the other, are every where equal to each other. , 10. Cor. 3. Hence two parallel straight lines, however for they may be produced, Can never meet. 11. Theorem II. If a line meeting two Parallelstraight lines be perpendicular to one of them, it is also perpendicular to the other. If A B (fig. 3.) be parallel to CD, and ** meet them so as to be perpendicular ;AB, it will also be perpendicular to CD, If not, draw fo perpendicular to CD, and som G draw Głi perpendicular to A B. hen since EF and G H are both perpencular to A B, and are drawn from F and G, points in C D, G H equals E F (9). *šin, since angle G H B or G H E is $oter than angle G E H (1) E G is great... G H (3). Hence é G is greater . B. F. Therefore E G is not perpenicular to c D (7); and in the same man* it may be shown, that no other line .."he drawn from the point E perpen
dicular to op without coinciding with
E. F. Therefore E F is perpendicular to C. D. 12. Theorem III. If two straight lines be perpendicular to the same straight line, they are parallel to each other. If A B (fig. 4.) and C D be both perpendicular to E F, then A B is parallel to C D. If A B be not parallel to CD, let G H, passing through the point E, be parallel to C D. Then since E F is perpendicular to CD, it is also perpendicular to G H (11). Hence angle H E F is a right angle, and therefore equal to angle B E F, the less to the greater, which is absurd. Therefore G H is not parallel to C D : and in the same manner it may be shown that no other line passing through E, and not coinciding with A B, is parallel to C D. Therefore A B is parallel to C. D. 13. Cor. Hence it appears, that through the same point no more than one line can be drawn parallel to the same straight line. It may be thought necessary to remark, that the preceding theorem pre-supposes the admission of a postulate, that through any point, not in a given straight line, a straight line may be drawn parallel to that straight line, or that straight line produced. 14. Theorem IV. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite angle on the same side; and likewise the two interior angles upon the same side, together, equal to two right angles If A B (fig. 5) be parallel to C, D, and E F cut them in the points H G, then the angle A H G equals, the alternate angle H G D ; the exterior angle E H B equals the interior and opposite angle on the same side, H G D ; and the two interior angles on the same side, B N G, and H G D are together equal to two right angles. From H draw. H K perpendicular to C D, and from G draw GI perpendicular to A B. Then since H K is perpendicular to C D, it is also perpendicular to A B (11); consequently G I is parallel to H K (12). But III and G K are perpendiculars to G I, from H and K, points in H K; therefore (9) H I equals G. K. Hence in triangles G I H, H G K, the side H I equals the side G. K. G. I equals H K (9), and the included angle G L H equals the included angle H K G ; therefore angle I H G equals angle H G K (3). Again, angle E H B equals A H G (4); therefore it equals H G D. Lastly, B N G and H G D are together equal to A H G and B HG together; and therefore (5) are equal together to the sum of two right angles. 15. Theorem V. If a straight line falling upon two other straight lines makes the alternate angles equal to one another, those two straight lines will be parallel. Let the straight line EF, (fig. 6.) which falls upon the two straight lines A B, CD, make the alternate angles A E F, EF D, equal to one another, then A B is parallel to C I). If not, through Edraw GH parallel to C D. Then the alternate angle G E F equals the alternate angle E. F. D. But A E F equals E F D ; therefore A E F is equal to G E F, the less to the greater. Hence G H is not parallel to C D ; and in like manner it may be shown that no other line passing through the point E, and not coinciding with A B, is parallel to C. D. Therefore A B is parallel to C I). 16. Cor. If a straight line, falling upon two other straight lines, makes the exterior angle equal to the interior and oppo. site one on the same side of the line; or makes the interior angles on the same side equal to two right angles; the two straight lines shall be parallel to one another. PARALLEL planes, are such planes as have all the perpendiculars drawn betwixt them equal to each other. PARALLEL rays, in optics, are those which keep at an equal distance from the visible object to the eye, which is supposed to be infinitely remote from the object. PARALLEI, ruler, an instrument consisting of two wooden, brass, &c. rulers, equally broad every where ; and so joined together by the cross blades as to open to different intervals, accede and recede, and yet still retain their parallelism. See PENTAGRAPH. PARALLELs, or parallel circles, in geography, called also parallels or circles of latitude, are lesser circles of the sphere conceived to be drawn from west to east, through all the points of the meridian, commencing from the equator to which they are parallel, and terminating with the poles. They are called parallels of latitude, because all places lying under the same parallel have the same latitude. PARALLEts of latitude, in astronomy, are lesser circles of the sphere parallel to the gliptic, imagined to pass through every dégree and minute of the colures. They are represented on the globe by the
divisions on the quadrant of altitude, in its motion round the globe, when screwed over the pole of the ecliptic. See GLoBE. PanAllels of allitude, or ALMUCANTARs, are circles parallel to the horizon, imagined to pass through every degree and minute of the meridian between the horizon and zenith, having their poles in the zenith. They are represented on the globe by the divisions on the quadrant of altitude, in its motion about the body of the globe, when screwed to the zenith. PARALLEls of declination, in astronomy, are the same with parallels of latitude, in geography. Pan Allel sphere, that situation of the sphere, wherein the equator coincides with the horizon, and the poles with the zenith and nadir. In this sphere all the parallels of the equator become parallels of the horizon; consequently, no stars ever rise or set, but all turn round in cir. cles parallel to the horizon; and the sh when in the equinoctial, wheels round the horizon the whole day. After his risis; to the elevated pole, he never sets for so months; and after his entering again on the other side of the line, never rises!" six months longer. This is the position of the sphere to such as live under to poles, and to whom the sun is never above 23° 30' high. . PALALLE, sailing, in navigation, is ho sailing under a parallel of latitude. See NAVIGATION. PARALLELEPIPED, or PARALLElo. PIPED, in geometry, a regular solid, “” prehended under six parallelograms.” opposite ones whereof are similar, Poo" lé, and equal. As paralleleppo, prisms, cylinders, &c. whose bases * heights are equal, are themselves equal A diagonal plane divides a paralleleppo into two equal prisms; so that a trialso lar prism is half a parallelepped, "P" the same base, and of the same altitu% All parallelépipeds, prisms, cylino. &c. are in a ratio compounded of ther bases and altitudes ; wherefore, if." bases be equal, they are in proportion." their altitudes, and conversely. All Po rallelepipeds, cylinders, cones, &c.”" a triplicate ratio of their homolog” sides, and also of their altitudes, Equal parallepipeds, prismo, cylinders, &c. reciprocate their bases altitudes. PARALLELISM, the situation or 4°, lity whereby any thing is denomina" parallel. See PARALLEL. • : PARALLElism of the earth's arê." astronomy, that situation of the earth's
axis, in its progress through its orbit, whereby it is still directed towards the pole-star; so that if a line be drawn parallel to its axis, while in any one position, the axis, in all other positions, will be always parallel to the same line.
This parallelism is the result of the earth’s double motion, viz. round the sun, and round its own axis; or its annual and diurnal motion; and to it we owe the vicissitudes of seasons, and the inequality of day and might.
PARALLELISM of the rows of trees. These are never seen parallel, but always inclining to each other towards the further extreme. Hence mathematicians have taken occasion to inquire in what lines the trees must be disposed to correct this effect of the perspective, and make the rows still appear parallel. The two rows must be such, as that the unequal intervals of any two opposite or correspondent trees may be seen under equal visual rays.
PARALLELOGRAM, in Geometry, a quadrilateral right-lined figure, whose opposite sides are parallel and equal to each other. It is generated by the equable motion of a right line always parallel to itself. When it has all its four angles right, and only its opposite sides equal, it is called a rectangle or oblong. When the angles are all right, and the sides equal, it is called a square. If all the sides are equal, and the angles unequal, it is called a rhombus or lozenge; and if the sides and angles be unequal, it is called a rhomboides. In every parallelogram, of what kind soever, a diagonal divides it into two equal parts; the angles diagonally opposite are equal ; the opposite angles of the same side are together equal to two right angles; and each two sides, together, greater than the diagonal. Two parallelograms on the same or equal base, and of the same height, or between the same parallels, are equal; and hence two triangles on the same base, and of the same height, are also equal. Hence, also, every triangle is half a parallelogram, upon the same or an equal base, and of the same altitude, or between the same parallels. Hence, also, a triangie is equal to a parallelogram, having the same base, and half the altitude, or half the base, and the same altitude. Parallelograms, therefore, are in a given ratio compounded of their bases and altitudes. If then the altitudes be equal, they are as the bases, and conversely. In similar parallelograms and triangles,
the altitudes are proportional to the homologous sides, and the bases are cut proportionably thereby. Hence similar parallelograms and triangles are in a duplicate ratio of their homologous sides; as also of their altitudes, and the segments of their bases; they are, therefore, as the squares of the sides, altitudes, and homologous segments of the bases. In every parallelogram, the sum of the squares of the two diagonals, is equal to the sum of the squares of the four sides. For if the parallelogram be rectangular, it follows that the two diagonals are equal ; and, consequently, the square of a diagonal, or, which comes to the same thing, the square of the hypothenuse of a right angle, is equal to the squares of the sides. See GeoMETRY. PARALLELoG RAM, or PARALLELISM, a machine for the ready reduction of designs; it is the same with the PENTAGRAPH, which see. PARAMETER, in conic sections, a constant line, otherwise called latus rectum. The parameter is said to be constant, because, in the parabola, the rectangle under it, and any absciss, is always equal to the square of the corresponding semi-ordinate; and in the ellipsis and hyperbola, it is a third proportional to the conjugate and transverse axis. } t and c be the two axes in the ellipse and hyperbola, and w and y an absciss and its ordinate in the parabola: then
t; c :: c : p === the parameter in the former; 2. a : y :: y : p === the parameter in the
The parameter is equal to the double ordinate drawn through the focus of one of the three conic sections.
PARAMECIUM, in natural history, a genus of the Vermes Infusoria class and order. Worm invisible to the naked eye, simple, pellucid, flattened, oblong. There are seven species, of which P. aurelia is rather a large animalculum, membranaceous, pellucid, and about four times longer than it is broad; the fore-part obtuse, transparent, without intestines; the hind-part replete with molecules of various sizes; the fold which goes from the middle to the apex is a striking characteristic of the species, forming a kind of triangular aperture, and giving it somewhat the appearance of a gimblet. Its motion is rectilinear, reeling or staggering, and generally vehement. They are frequently found cohering lengthwise; the lateral edges of both bodies appear bright. The may also be seen sometimes lying on one another alternately, at others adhering by the middle. They will live many months in the same water without its being renewed. They are found in the beginning of summer, in those ditches in which duck-weed abounds. P. chrysalis is found plentifully in salt water. PARAPET, in fortification, an elevation of earth, designed for covering the soldiers from the enemies’ cannon or small shot. The thickness of the paraH. is from eighteen to twenty feet ; its eight is six feet on the inside, and four or five on the outside. It is raised on the rampart, and has a slope above called the supérior talus, and sometimes the glacis, of the parapet. The exterior talus of the parapet is the slope facing the country : there is a banquette or two for the soldiers who defend the parapet to mount upon, that they may the better discover the country, fosse, and counterscarp, and fire as they find occasion. Parapet of the covert-way, or corridor, is what covers that way from the sight of the enemy, which renders it the most dangerous place for the besiegers, because of the neighbourhood of the faces, flanks, and curtains of the place. PARAPET is also a little wall raised breast high, on the banks of bridges, keys, or high buildings, to serve as a stay, and prevent people’s falling over. PARAPHRASE, an explanation of some text, in clearer and more simple terms, whereby is supplied what the author might have said or thought on the subject; such are esteemed Erasmus's Paraphrase on the New Testament, the Chaldee Paraphrase on the Pentateuch,
C. PARASANG, an ancient Persian measure, different at different times, and in different places; being sometimes thirty, sometimes forty, and sometimes fifty stadia, or furlongs. PARASITES, or Parasitical plants, in botany, such plants as are supported by the trunk or branches of other plants, from whence they receive their nourishment, and will not grow upon the ground, as the misletoe, &c. PARCENERS, in law, persons holding lands in copartnership, and who may be compelled to make division. It occurs where lands descend to the females, who all take equal shares of their deceased father’s lands.
PARCHMENT, in commerce, the
skins of sheep or goats, prepared after such a manner as to render it proper for writing upon, covering books, &c. The manufacture of parchment is begun by the skinner, and finished by the parchment-maker. The skin, having been stripped of its wool, and placed in #. limepit, in the manner described under the article SHAMMY, the skinner stretches it on a kind of frame, and pares off the flesh with an iron instrument ; this done, it is moistened with a rag, and powdered chalk being spread over it, the skinner takes a large pumice-stone, flat at bottom, and rubs over the skin, and thus scowers off the flesh; he then goes over it again with the iron instrument, moistens it as before, and rubs it again with the pumicestone without any chalk underneath: this smooths and softens the flesh-side very considerably. He then drains it again, by passing over it the iron instrument asbe: fore. The flesh-side being thus drained, by scraping off the moisture, he in the same manner passes the iron over the wool or hair-side; then stretches it tight on a frame, and scrapes the flesh-side again: this finishes its draining; and the more it is drained, the whiter it becomes. The skinner now throws on more chalk, sweeping it over with a piece of ano: skin that has the wool on, and this smooths it still further. It is now left to dry, and when dried, taken off the from? by cutting it all round. The skin, tho far prepared by the skinner, is taken out of his hands by the parchment makes, who first, while it is dry, pares it on summer (which is cast skin stretched in a frame), with a sharper instrum." than that used by the skinner, and work. ing with the arm from the top to the bottom of the skin, takes away about 9” half of its thickness. The skin, to equally pared on the flesh side, is 8. rendered smooth, by being rubbed with the pumice-stone, on a bench covered with a sack stuffed with flocks, who leaves the parchment in a condition fit for writing upon. The parings thus taken off the leather are used in making glue, size, &c. See GLUE, &c. What is called jium, is only parchment made of . skins of abortives, or at least suckin; calves. This has a much finer groin,” is whiter and smoother than parchmo but is prepared in the same manno i. cept its not being passed through to lime-pit. iy PARDON, is the remitting or of ing a felony or other offence com. against the King. Blackstone me"