PARALLEL

, in Geometry, is applied to lines, figures, and bodies, which are every where equidistant from each other; or which, though infinitely produced, would never either approach nearer, or recede farther from, each other; their distance being every where measured by a perpendicular line between them. Hence,

Parallel right lines are those which, though insinitely produced ever so far, would never meet: which is Euclid's definition of them.

Newton, in Lemma 22, book 1 of his Principia, defines Parallels to be such lines as tend to a point infinitely distant.

Parallel Lines stand opposed to lines converging, and diverging.

Some define an inclining or converging line, to be that which will meet another at a finite distance, and a Parallel line, that which will only meet at an insinite distance.

As a perpendicular is by some said to be the shortest of all lines that can be drawn to another; so a Parallel is said to be the longest.

It is demonstrated by geometricians, that two lines, AB and CD, that are both Parallel to one and the same right line EF, are also Parallel to each other. And that if two Parallel lines AB and EF be cut by any other line GH; then 1st, the alternate angles are equal; viz the angle a = [angle] b, and [angle] c = [angle] d. 2d, The external angle is equal to the internal one on the same side of the cutting line; viz the [angle] e = [angle] d, and the [angle] f = [angle] b. 3d, That the two internal ones on the same side are, taken together, equal to two| right angles; viz, , or .

To draw a Parallel Line.—If the line to be Parallel to AB must pass through a given point P: Take the nearest distance between the point P and the given line AB, by setting one foot of the compasses in P, and with the other describe an arc just to touch the line in A; then with that distance as a radius, and a centre B taken any where in the line, describe another arc C; lastly, through P draw a line PC just to touch the arc C, and that will be the Parallel sought.

Otherwise.—With the centre P, and any radius, describe an arc BC, cutting the given line in B. Next, with the same radius, and centre B, describe another arc PA, cutting also the given line in A. Lastly, take AP between the compasses, and apply it from B to C; and through P and C draw the Parallel PC required.

Or, draw the line with the Parallel Ruler, described below, by laying one edge of the ruler along AB, and extending the other to the given point or distance.

When the one line is to be at a given distance from the other; take that distance between the compasses as a radius, and with two centres taken any where in the given line, describe two arcs; then lay a ruler just to touch the arcs, and by it draw the Parallel.

Parallel Planes, are every where equidistant, or have all the perpendiculars that are drawn between them, everywhere equal.

Parallel Rays, in Optics, are those which keep always at an equal distance in respect to each other, from the visual object to the eye, from which the object is supposed to be infinitely distant.

Parallel Ruler, is a mathematical instrument, consisting of two equal rulers, AB and CD, either of wood or metal, connected together by two slender cross bars or blades AC and BD, moveable about the points or joints A, B, C, D.

There are other forms of this instrument, a little varied from the above; some having the two blades crossing in the middle, and fixed only at one end of them, the other two ends sliding in groovca along the two rulers; &c.

The use of this instrument is obvious. For the edge of one of the rulers being applied to any line, the other opened to any extent will be always parallel to the former; and consequently any Parallels to this may be drawn by the edge of the ruler, opened to any extent.

Parallel Sailing, in Navigation, is the sailing on or under a Parallel of latitude, or Parallel to the equator. —Of this there are three cases.

1. Given the Distance and Difference of Longitude; to find the Latitude.—Rule. As the difference of longitude is to the distance, so is radius to the cosine of the latitude.

2. Given the Latitude and Difference of Longitude; to find the Distance.—Rule. As radius is to the cosine of the latitude, so is the difference of longitude to the distance.

3. Given the Latitude and Distance; to find the difference of longitude.—Rule. As the cosine of latitude is to radius, so is the distance to the difference of longitude.

Parallel Sphere, is that situation of the sphere where 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 circles Parallel to the horizon, as well as the sun himself, which when in the equinoctial wheels round the horizon the whole day. Also, After the sun rises to the elevated pole, he never sets for six months; and after his entering again on the other side of the line, he never rises for six months longer.

This position of the sphere is theirs only who live at the poles of the earth, if any such there be. The greatest height the sun can rise to them, is 23 1/2 degrees. They have but one day and one night, each being half a year long. See Sphere.

, or Places of Arms, in a Siege, are deep trenches, 15 or 18 feet wide, joining the several attacks together; and serving to place the guard of the trenches in, to be at hand to support the workmen when attacked.

There are usually three in an attack: the first is about 600 yards from the covert-way, the second between 3 and 400, and the third near or on the glacis. —It is said they were first invented or used by Vauban.

Parallels of Altitude, or Almacantars, are circles Parallel to the horizon, conceived to pass through every degree and minute of the meridian between the horizon and zenith; having their poles in the zenith.

, or Parallel Circles, called also Parallels of Latitude, and Circles of Latitude, are lesser circles of the sphere, Parallel to the equinoctial or equator.

Parallels of Declination, are lesser circles Parallel to the equinoctial.

Parallels of Latitude, in Geography, are lesser circles Parallel to the equator. But in Astronomy they are Parallel to the ecliptic.