, in Geometry, is either the quarter or 4th part of a circle, or the 4th part of its circumference; the arch of which therefore contains 90 degrees.

Quadrant also denotes a mathematical instrument, of great use in astronomy and navigation, for taking the altitudes of the sun and stars, as also taking angles in surveying, heights-and-distances, &c.

This instrument is variously contrived, and furnished with different apparatus, according to the various uses it is intended for; but they have all this in common, that they consist of the quarter of a circle, whose limb or arch is divided into 90° &c. Some have a plummet suspended from the centre, and are furnished either with plain sights, or a telescope, to look through.

1. The Common, or Surveying Quadrant.—This instrument ABC, fig. 1, pl. 24, is made of brass, or wood, &c; the limb or arch of which BC is divided into 90°, and each of these farther divided into as many equal parts as the space will allow, either diagonally or otherwise. On one of the radii AC, are fitted two moveable sights; and to the centre is sometimes also annexed a label, or moveable index AD, bearing two other sights; but instead of these last sights, there is sometimes fitted a telescope. Also from the centre hangs a thread with a plummet; and on the under side or face of the instrument is fitted a ball and socket, by means of which it may be put into any position. The general use of it is for taking angles in a vertical plane, comprehended under right lines going from the centre of the instrument, one of which is horizontal, and the other is directed to some visible point. But besides the parts above described, there is often added on the face, near the centre, a kind of compartment EF, called a Quadrat, or Geometrical Square, which is a kind of separate instrument, and is particularly useful in Altimetry and Longimetry, or Heights-and-Distances.

This Quadrant may be used in different situations; in each of them, the plane of the instrument must be set parallel to that of the eye and the objects whose angular distance is to be taken. Thus, for observing heights or depths, its plane must be disposed vertically, or perpendicular to the horizon; but to take horizontal angles or distances, its plane must be disposed parallel to the horizon.

Again, heights and distances may be taken two ways, viz, by means of the fixed sights and plummet, or by the label; as also, either by the degrees on the limb, or by the Quadrat. Thus, fig. 2 pl. 24 shews the manner of taking an angle of elevation with this Quadrant; the eye is applied at C, and the instrument turned vertically about the centre A, till the object R be seen through the sights on the radius AC; then the angle of elevation RAH, made with the horizontal line KAH, is equal to the angle BAD, made by the plumb line and the other radius of the Quadrant, and the quantity of it is shewn by the degrees in the arch BD cut off by the plumb line AD.

See the use of the instrument in my Mensuration, under the section of Heights-and-Distances.

2. The Astronomical Quadrant, is a large one, usu- | ally made of brass or iron bars; having its limb EF (fig. 3 pl. 24) nicely divided, either diagonally or otherwise, into degrees, minutes, and seconds, if room will permit, and furnished either with two pair of plain sights or two telescopes, one on the side of the Quadrant at AB, and the other CD moveable about the centre by means of the screw G. The dented wheels I and H serve to direct the instrument to any object or phenomenon.

The application of this useful instrument, in taking observations of the sun, planets, and fixed stars, is obvious; for being turned horizontally upon its axis, by means of the telescope AB, till the object is seen through the moveable telescope, then the degrees &c cut by the index, give the altitude &c required.

To find the Sun's Altitude by this instrument. Turn your back to the sun, holding the staff of the instru- ment with the right hand, so that it be in a vertical plane passing through the sun; apply one eye to the sight-vane, looking through that and the horizonvane till the horizon be seen; with the left hand slide the quadrantal arch upwards, till the solar spot or shade, cast by the shade-vane, fall directly upon the spot or slit in the horizon-vane; then will that part of the quadrantal arch, which is raised above G or S (according as the observation respects either the solar spot or shade) shew the altitude of the sun at that time. But for the meridian altitude, the observation must be continued, and as the sun approaches the meridian, the sea will appear through the horizon-vane, which completes the observation; and the degrees and minutes, counted as before, will give the sun's meridian altitude: or the degrees counted from the lower limb upwards will give the zenith distance.

4. Adams's Quadrant, differs only from Cole's Quadrant, just described, in having an horizontal vane, with the upper part of the limb lengthened; so that the glass, which casts the solar spot on the horizonvane, is at the same distance from the horizon-vane as the sight-vane at the end of the index.

5. Collins's or Sutton's Quadrant, (fig. 8 pl. 24) is a stereographic projection of one quarter of the sphere between the tropics, upon the plane of the ecliptic, the eye being in its north pole; and fitted to the latitude of London. The lines running from right to left, are parallels of altitude; and those crossing them are azimuths. The smaller of the two circles, bounding the projection, is one quarter of the tropic of Capricorn; and the greater is a quarter of the tropic of Cancer. The two ecliptics are drawn from a point on the left edge of the Quadrant, with the characters of the signs upon them; and the two horizons are drawn from the same point. The limb is divided both into degrees and time; and by having the sun's altitude, the hour of the day may here be found to a minute. The quadrantal arches next the certre contain the calendar of months; and under them, in another arch, is the sun's declination. On the projection are placed several of the most remarkable fixed stars between the tropics; and the next below the projection is the Quadrant and line of shadows.

But note, that if the sun's altitude be less than what it is at 6 o'clock, the operation must be performed among those parallels above the upper horizon; the bead being rectified to the winter ecliptic.

6. Davis's Quadrant, the same as the BACKSTAFF; which see.

7. Gunner's Quadrant, (fig. 6 pl. 24), sometimes called the Gunner's Square, is used for elevating and pointing cannon, mortars, &c, and consists of two branches either of wood or brass, between which is a quadrantal arch divided into 90°, and furnished with a thread and plummet.

The use of this instrument is very easy; for if the longer branch, or bar, be placed in the mouth of the piece, and it be elevated till the plummet cut the degree necessary to hit a proposed object, the thing is done.

Sometimes on the sides of the longer bar, are noted the division of diameters and weights of iron balls, as also the bores of pieces.

8. Gunter's Quadrant, so called from its inventor Edmund Gunter, (fig. 4 pl. 24) beside the apparatus of other Quadrants, has a stereographic projection of the sphere on the plane of the equinoctial; and also a calendar of the months, next to the divisions of the limb; by which, beside the common purposes of other Quadrants, several useful questions in astronomy, &c, are easily resolved.

Use of Gunter's Quadrant. — 1. To find the sun's meridian altitude for any given day, or conversely the day of the year answering to any given meridian altitude. Lay the thread to the day of the month in the scale next the limb; then the degree it cuts in the limb is the sun's meridian altitude. And, contrariwise, the thread being set to the meridian altitude, it shews the day of the month.

2. To find the hour of the day. Having put the bead, which slides on the thread, to the sun's place in the ecliptic, observe the sun's altitude by the Quadrant; then if the bead be laid over the same in the limb, the bead will fall upon the hour required. On the contrary, laying the bead on a given hour, having first rectified or set it to the sun's place, the degree cut by the thread on the limb gives the altitude.

Note, the bead may be rectified otherwise, by bringing the thread to the day of the month, and the bead to the hour-line of 12.

3. To find the sun's declination from his place given; and the contrary. Bring the bead to the sun's place in the ecliptic, and move the thread to the line of declination ET, so shall the bead cut the degree of declination required. On the contrary, the bead being adjusted to a given declination, and the thread moved to the ecliptic, the bead will cut the sun's place.

4. The sun's place being given, to find the right ascension; or contrariwise. Lay the thread on the sun's place in the ecliptic, and the degree it cuts on the limb is the right ascension sought. And the converse.

5. The sun's altitude being given, to find his azimuth; and contrariwise. Rectify the bead for the time, as in the second article, and observe the sun's altitude; bring the thread to the complement of that altitude; then the bead will give the azimuth sought, among the azimuth-lines.

9. Hadley's Quadrant, (fig. 7 pl. 24) so called from its inventor John Hadley, Esq. is now universally used as the best of any for nautical and other observations.

This instrument consists of the following particulars: 1. An octant, or the 8th part of a circle, ABC. 2. An index D. 3. The speculum E. 4. Two horizontal glasses, F, G. 5. Two screens, Kand K. 6. Two sight-vanes, H and I.

The octant consists of two radii, AB, AC, strengthened by the braces L, M, and the arch BC; which, though containing only 45°, is nevertheless divided into 90 primary divisions, each of which stands for degrees, and are numbered 0, 10, 20, 30, &c, to 90; beginning at each end of the arch for the convenience of numbering both ways, either for altitudes or zenith distances: also each degree is subdivided into minutes, by means of a vernier. But the number of these divisions varies with the size of the instrument.

The index D, is a flat bar, moveable about the centre of the instrument; and that part of it which slides over the graduated arch, BC, is open in the middle, with Vernier's scale on the lower part of it; | and underneath is a screw, serving to fasten the index against any division.

The speculum E is a piece of flat glass, quick silvered on one side, set in a brass box, and placed perpendicular to the plane of the instrument, the middle part of the former coinciding with the centre of the latter: and because the speculum is fixed to the index, the position of it will be altered by the moving of the index along the arch. The rays of an observed object are received on the speculum, and from thence reflected on one of the horizon glasses, F or G; which are two small pieces of looking glass placed on one of the limbs, their faces being turned obliquely to the speculum, from which they receive the reflected rays of objects. This glass F has only its lower part silvered, and set in brass-work; the upper part being left transparent to view the horizon. The glass G has in its middle a transparent slit, through which the horizon is to be seen. And because the warping of the materials, and other accidents, may distend them from their true situation, there are three screws passing through their feet, by which they may be easily replaced.

The screens are two pieces of coloured glass, set in two square brass frames K and K, which serve as screens to take off the glare of the sun's rays, which would otherwise be too strong for the eye; the one is tinged much deeper than the other; and as they both move on the same centre, they may be both or either of them used: in the situation they appear in the figure, they serve for the horizon-glass F; but when they are wanted for the horizon-glass G, they must be taken from their present situation, and placed on the Quadrant above G.

The sight-vanes are two pins, H and I, standing perpendicularly to the plane of the instrument: that at H has one hole in it, opposite to the transparent slit in the horizon-glass G; the other, at I, has two holes in it, the one opposite to the middle of the transparent part of the horizon-glass F, and the other rather lower than the quick-silvered part: this vane has a piece of brass on the back of it, which moves round a centre, and serves to cover either of the holes.

Of the Observations.—There are two sorts of observations to be made with this instrument: the one is when the back of the observer is turned towards the object, and therefore called the back observation; the other when the face of the observer is turned towards the object, which is called the fore-observation.

To Rectify the Instrument for the Fore-observation: Slacken the screw in the middle of the handle behind the glass F; bring the index close to the button h; hold the instrument in a vertical position, with the arch downwards; look through the right-hand hole in the vane I, and through the transparent part of the glass F, for the horizon; and if it lie in the same right line with the image of the horizon seen on the silvered part, the glass F is rightly adjusted; but if the two horizontal lines disagree, turn the screw which is at the end of the handle backward or forward, till those lines coincide; then fasten the middle screw of the handle, and the glass is rightly adjusted.

To take the Sun's Altitude by the Fore-observation. Having fixed the screens above the horizon-glass F, and suited them proportionally to the strength of the sun's rays, turn your face towards the sun, holding the instrument with your right hand, by the braces L and M, in a vertical position, with the arch downward; put your eye close to the right-hand hole in the vane I, and view the horizon through the transparent part of the horizon-glass F, at the same time moving the index D with the left hand, till the reflex solar spot coincides with the line of the horizon; then the degrees counted from C, or that end next your body, will give the sun's altitude at that time, observing to add or subtract 16 minutes according as the upper or lower edge of the sun's reflex image is made use of.

But to get the sun's meridian altitude, which is the thing wanted for finding the latitude; the observations must be continued; and as the sun approaches the meridian the index D must be continually moved towards B, to maintain the coincidence between the reflex solar spot and the horizon; and consequently as long as this motion can maintain the same coincidence, the observation must be continued, till the sun has reached the meridian, and begins to descend, when the coincidence will require a retrograde motion of the index, or towards C; and then the observation is finished, and the degrees counted as before will give the sun's meridian altitude, or those from B will give the zenith distance; observing to add the semi-diameter, or 16′, when his lower edge is brought to the horizon; or to subtract 16′, when the horizon and upper edge coincide.

To take the Altitude of a Star by the Fore-observation. Through the vane H, and the transparent slit in the glass G, look directly to the star; and at the same time move the index, till the image of the horizon behind you, being reflected by the great speculum, be seen in the silvered part of G, and meet the star; then will the index shew the degrees of the star's altitude.

To Rectify the Instrument for the Back-observation. Slacken the screw in the middle of the handle, behind the glass G; turn the button h on one side, and bring the index as many degrees before 0 as is equal to double the dip of the horizon at your height above the water; hold the instrument vertical, with the arch downward; look through the hole of the vane H; and if the horizon, seen through the transparent slit in the glass G, coincide with the image of the horizon seen in the silvered part of the same glass, then the glass G is in its proper position; but if not, set it by the handle, and fasten the screw as before.

To take the Sun's Altitude by the Back-observation. Put the stem of the screens K and K into the hole r, and in proportion to the strength or faintness of the sun's rays, let either one or both or neither of the frames of those glasses be turned close to the face of the limb; hold the instrument in a vertical position, with the arch downward, by the braces L and M, with the left hand; turn your back to the sun, and put one eye close to the hole in the vane H, observing the horizon through the transparent slit in the horizon glass G; with the right hand move the index D, till the reflected image of the sun be seen in the silvered part of the glass G, and in a right line with the horizon; swing your body to and fro, and if the observation be well made, the sun's image will be observed to brush the horizon, and the degrees reckoned from C, or that part of the arch farthest from your | body, will give the sun's altitude at the time of observation; observing to add 16′ or the sun's semidiameter if the sun's upper edge be used, and subtract the same for the lower edge.

The directions just given, for taking altitudes at sea, would be sufficient, but for two corrections that are necessary to be made before the altitude can be accurately determined, viz, one on account of the observer's eye being raised above the level of the sea, and the other on account of the refraction of the atmosphere, especially in small altitudes.

The following tables therefore shew the corrections to be made on both these accounts.

 TABLE I. TABLE II. Dip of the Hori- Refractions of the Stars &c in zon of the Sea. Altitude. Height Dip of Appar. Refrac- Appar. Refrac- of the the Ho- Altitude Altitude Eye. rizon. in Deg. tion. in Deg. tion. Feet. ′ ″ ° ′ ″ ° ′ ″ 1 0 57 0 33 0 11 4 47 2 1 21 1/4 30 35 12 4 23 3 1 39 1/2 28 22 15 3 30 5 2 8 1 24 29 20 2 35 10 3 1 2 18 35 25 2 2 15 3 42 3 14 36 30 1 38 20 4 16 4 11 51 35 1 21 25 4 46 5 9 54 40 1 8 30 5 14 6 8 29 45 0 57 35 5 39 7 7 20 50 0 48 40 6 2 8 6 29 60 0 33 45 6 24 9 5 48 70 0 21 50 6 44 10 5 15 80 0 10
General Rules for these Corrections.

1. In the fore-observations, add the sum of both corrections to the observed zenith distance, for the true zenith distance: or subtract the said sum from the observed altitude, for the true one. 2. In the backobservation, add the dip and subtract the refraction, for altitudes; and for zenith distances, do the contrary, viz, subtract the dip, and add the refraction.

Example. By a back observation, the altitude of the sun's lower edge was found by Hadley's Quadrant to be 25° 12′; the eye being 30 feet above the horizon. By the tables, the dip on 30 feet is 5′ 14″, and the refraction on 25° 12′ is 2′ 1″. Hence

 Appar. alt. lower limb 25° 12′  0″ Sun's semidiameter, sub. 0  16   0 Appar. alt. of centre 24  56   0 Dip. of horizon, add 0   5  14 25   1  14 Refraction, subtract 0   2   1 True alt. of centre 24  59  13

In the case of the moon, besides the two corrections above, another is to be made for her parallaxes. But for all these particulars, see the Requisite Tables for the Nautical Almanac, also Robertson's Navigation, vol. 2, pa. 340 &c, edit. 1780.

10. Horodictical Quadrant, a pretty commodious instrument, and is so called from its use in telling the hour of the day. Its construction is as follows. From the centre of the Quadrant C, (fig. 5 pl. 24), whose limb AB is divided into 90°, describe seven concentric circles at any intervals; and to these add the signs of the zodiac, in the order represented in the figure. Then, applying a ruler to the centre C and the limb AB, mark upon the several parallels the degrees corresponding to the altitude of the sun, when in them, for the given hours; connect the points belonging to the same hour with a curve line, to which add the number of the hour. To the radius CA fit a couple of fights, and to the centre of the Quadrant C tie a thread with a plummet, and upon the thread a bead to slide.

Now if the bead be brought to the parallel in which the sun is, and the Quadrant be directed to the sun, till a visual ray pass through the sights, the bead will shew the hour. For the plummet, in this situation, cuts all the parallels in the degrees corresponding to the sun's altitude. And since the bead is in the parallel which the sun describes, and because hourlines pass through the degrees of altitude to which the sun is elevated every hour, therefore the bead must shew the present hour.

11. Sinical Quadrant, is one of some use in Navigation. It consists of several concentric quadrantal arches, divided into 8 equal parts by means of radii, with parallel right lines crossing each other at right angles. Now any one of the arches, as BC, (fig. 10 pl. 24) may represent a Quadrant of any great circle of the sphere, but is chiefly used for the horizon or meridian. If then BC be taken for a Quadrant of the horizon, either of the sides, as AB, may represent the meridian; and the other side, AC, will represent a parallel, or line of east-and-west; all the other lines, parallel to AB, will be also meridians; and all those parallel to AC, east-and-west lines, or parallels. Again, the eight spaces into which the arches are divided by the radii, represent the eight points of the compass in a quarter of the horizon; each containing 11° 15′. The arch BC is likewise divided into 90°, and each degree subdivided into 12′, diagonalwise. To the centre is fixed a thread, which, being laid over any degree of the Quadrant, serves to divide the horizon.

If the sinical Quadrant be taken for a fourth part of the meridian, one side of it, AB, may be taken for the common radius of the meridian and equator; and then the other, AC, will be half the axis of the world. The degrees of the circumference, BC, will represent degrees of latitude; and the parallels to the side AB, assumed from every point of latitude to the axis AC, will be radii of the parallels of latitude, as likewise the cosine of those latitudes.

Hence, suppose it be required to find the degrees of longitude contained in 83 of the lesser leagues in the parallel of 48°: lay the thread over 48° of latitude on the circumference, and count thence the 83 leagues on AB, beginning at A; this will terminate in H, allow- | ing every small interval four leagues. Then tracing out the parallel HE, from the point H to the thread; the part AE of the thread shews that 125 greater or equinoctial leagues make 6° 15′; and therefore that the 83 lesser leagues AH, which make the difference of longitude of the course, and are equal to the radius of the parallel HE, make 6° 15′ of the said parallel.

When the ship sails upon an oblique course, such course, beside the north and south greater leagues, gives lesser leagues easterly and westerly, to be reduced to degrees of longitude of the equator. But these leagues being made neither on the parallel of departure, nor on that of arrival, but in all the intermediate ones, there must be found a mean proportional parallel between them. To find this, there is on the instrument a scale of cross latitudes. Suppose then it were required to find a mean parallel between the parallels of 40° and 60°; take with the compasses the middle between the 40th and 60th degree on the scale: this middle point will terminate against the 51st degree, which is the mean parallel sought.

The chief use of the sinical Quadrant, is to form upon it triangles similar to those made by a ship's way with the meridians and parallels; the sides of which triangles are measured by the equal intervals between the concentric Quadrants and the lines N and S, E and W: and every 5th line and arch is made deeper than the rest. Now suppose a ship has sailed 150 leagues northeast-by-north, or making an angle of 33° 45′ with the north part of the meridian: here are given the course and distance sailed, by which a triangle may be formed on the instrument similar to that made by the ship's course; and hence the unknown parts of the triangle may be found. Thus; supposing the centre A to represent the place of departure; count, by means of the concentric circles along the point the ship sailed on, viz, AD, 150 leagues: then in the triangle AED, similar to that of the ship's course, find AE = difference of latitude, and DE = difference of longitude, which must be reduced according to the parallel of latitude come to.