three:-1. Only three polygons, or regular plane figures, can fill up the space about a point; viz. the equilateral triangle, the square, and the hexagon. 2. The sum of three angles of every triangle is equal to two right angles. 3. In any right-angled triangle, the square on the longest side is equal to both the squares on the two shorter sides. For the discovery of this last theorem, some authors say he offered to the gods a hecatomb, or a sacrifice of an hundred oxen. Plutarch, however, says it was only one ox; and even that is questioned by Cicero, as inconsistent with his doctrine, which forbade bloody sacrifices. The more accurate, therefore, say he sacrificed an ox made of flour, or of clay; and Plutarch even doubts whether such sacrifice, whatever it was, was made for the said theorem, or for that concerning the parabola, which it was said Pythagoras also found out.

In astronomy his inventions were many and great. It is reported, that he discovered, or maintained, the true system of the world, which places the sun in the centre, and makes all the planets revolve about him: from him it is to this day call

ed the old, or Pythagorean system; and is the same as that lately revived by Copernicus. He first discovered that Lucifer and Hesperus were but one and the same being, the planet Venus, though formerly thought to be two different stars. The invention of the obliquity of the zodiac is likewise ascribed to him. He first gave to the world the name Kosmos, from the order and beauty of all things comprehended in it; asserting, that it was made according to musical proportion: for as he held that the sun, by him and his followers termed the fiery globe of unity, was seated in the midst of the universe, and the earth and planets moving around him, so he held that the seven planets had an harmonious motion, and their distances from the sun corresponded to the musical intervals or divisions of the monochord.

Pythagoras and his followers held the transmigration of souls, making them suc cessively occupy one body after another; on which account they abstained from flesh, and lived chiefly on vegetables.

PYTHAGOREANS, a sect of ancient philosophers, so denominated from their being the followers of Pythagoras of Sa

mos.

See PYTHAGORAS.

Or q, the sixteenth letter, and twelfth consonant, of our alphabet, but is not to be found either in the Greek, old Latin, or Saxon alphabets; and, indeed, some would entirely exclude it, pretending that k ought to be used wherever this occurs. However, as it is formed in the voice in a different manner, it is undoubtedly a distinct letter; for in expressing this sound the cheeks are contracted, and the lips, particularly the under one, are put into a cannular form, for the passage of the breath.

The q is never sounded alone, but in

conjunction with u, as in quality, question, quite, quote, &c. and never ends any Eng

lish word.

As a numeral, Q stands for 500; and with a dash over it, thus, Q, for 500,000. Used as an abbreviature, q. signifies quantity, or quantum: thus, among phy

sicians, q. pl. is quantum placet, i. e. as much as you please of a thing; and q. s. quantum sufficit, i. e. as much as is ne cessary. Q. E. D. among mathematicians, is quod erat demonstrandum, i. e. which was to be demonstrated; and Q. E. F. quod erat faciendum, i. e. which was to be done; Q. D. among grammarians, is quasi dictum, i. e. as if it were said.

QUACK, a medical impostor, who, "for the good of the public," and "by the blessing of God," undertakes, with his powders, potions, or balsams, to cure "all phemy unite in picking the pockets and disorders." Thus, ignorance and blasruining the constitutions of thousands of credulous people in this and other countries. The pretension to infallibility in any one medicine, as a cure for any one disorder, is next to absurd; much more ridiculous is it then to suppose, that any

medicine will remove all kinds of complaints.

Every medicine possesses active properties, or it does not. If it be active, it must be dangerous to apply it, indiscriminately, to persons of every age, and without regard to their habits of living. An active medicine, which might be very useful in strengthening a debilitated constitution, would be highly injurious, if exhibited in an acute rheumatism, or other inflammatory disorder, and vice versa; consequently, an application of the same remedy in all cases can hardly fail of being fatal in some.

Should the medicine

be inactive, which happily is often the case, it can be of no other utility than to work upon the patient's imagination, and amuse him while his pocket is picked. See MEDICAL DICT.

QUADRANGLE, in geometry, the same with a quadrilateral figure, or one consisting of four sides and four angles. To the class of quadrangles belong the square, parallelogram, trapesium, rhombus, and rhomboides. A square is a regular quadrangle; a trapesium an irregular one.

QUADRANS, the quarter or fourth part of any thing, particularly the as, or pound.

QUADRANT, denotes a mathematical instrument, of great service in astronomy, and, consequently, in navigation, for tak ing the altitudes of the sun and stars; as also for taking angles in surveying. Those chiefly in use are, Adams's, Cole's, Gunter's, Hadley's, Sutton's, or Collins's, the horodictical, the sinical, the astronomical, and the common surveying quadrant. Many of these are made of wood, generally ebony, mounted with ivory; but such are subject to warp, which occasions those made of brass to be preferred, for very hot or very cold climates, though their expansion and contraction, under various temperaments, is some drawback on their merits; however, that being the lesser evil, and scarcely ever amounting to more than two or three seconds in the whole arch of the quadrant, cannot be considered as any great defect.

Although these instruments are generally termed quadrants, they are, in truth, but octants, since they occupy but oneeighth of a circumference; but as each of the 45° they contain on the arch actually measures two, while taking the observation, they do not receive their designation improperly. We have another description of this instrument, called the sextant, which has 60° marked on its arch,

and includes 120° in real measurement. This is peculiarly calculated for the observation of various celestial bodies, so as to ascertain their distances at any particular moment: this often could not be effected by an instrument which embraced only 90°; whereas, we rarely find any two planets suitable to the purposes of navigation, at so great a distance as 120°.

The manner in which the quadrant is held relieves it from the effect of the vessel's motion; although, in the first instance, some difficulty may occur in suiting the body to the rolling, or pitching, of a vessel, yet, in a very short time, the operator will become so habituated, as to overcome that trifling impediment.

Hadley's quadrant (or his sextant) is the only instrument, hitherto known,on which the mariner can depend for a correct observation. It may be called the " portable observatory." "The first idea of this machine originated with the celebrated Dr. Hooke; it was completed by Sir Isaac Newton, and first offered to the public by Mr. Hadley; however, it has undergone many changes since that time. The great perfection it exhibits, with respect to the accuracy of the angles it defines, is considerably enhanced by the facility with which it may be rectified; so that errors may be avoided: a matter of supreme importance, when we consider the rough usage to which the instrument is subject; and that an error of one degree in the index makes two in the observation.

Description of Hadley's Quadrant. Fig. 1. Plate XIII. Miscel. shows the quadrant, as usually constructed. The following parts compose the instrumént. B C, the arc of 45°: AD, the index, moving on a pivot, under the centre of the index-glass, E; which glass is in the exact direction of the index, and stands at right angles upon it. F, the fore-horizon glass, which receives the reflection from the indexglass. G, the back-horizon-glass. The former stands parallel with the leg, A C; the latter at right angles thereto. K, is a pivot, on which three dark glasses, or screens, move, so that any one, or more, may be placed between the index-glass and the horizon-glass, to diminish the lustre of the reflected planet. Hand I, the vanes, or sights. The arc, BC, is called the limb, or quadrantal arc; what is beyond 0, is the arc of excess: the residue of the arc usually is graduated up as far as 100°.

A large portion of the lower part of the index is open, so as to show the gradations of the arc: the lower edge is cham

fered, that it may come close down to them, and is there divided into smaller portions: this scale is called the nonius, and shows the smaller divisions in a more correct and obvious manner than could be done by the quadrantal arc, on which each degree is subdivided into no more than three equal parts of 20' each. Now the nonius being divided into 21 equal parts, shows at what portions of the arc the index cuts the division of 20 minutes; therefore it shows every minute.

THE USE OF HADLEY'S QUADRANT.

For the Fore-Observation. Bring the index close to the bottom, so that the midIdle of the Vernier's scale, or nonius, stand against 0 degrees. Hold the plane of the instrument vertical, with the arch downwards; look through the right-hand hole in the vane, and direct the sight through the transparent part of the horizon-glass, to observe the horizon. If the horizon-line, seen both in the quick-silvered part, and through the transparent part, should coincide, or make one straight line, then is the glass adjusted; but if one of the horizon-lines should stand above the other, slacken the screw in the middle of the lever, backwards or forwards, as there may be occasion, until the lines coincide: fasten the screw in the middle of the lever, and all is ready for use.

To take the Sun's Altitude. Fix the screens above the horizon-glass, using either or both of them, according to the strength of the sun's rays, by turning one or both the frames of those glasses close against the plane or face of the instrument; then your face being turned towards the sun, hold the quadrant by the braces, or by either radius, as is found most convenient, so as to be in a vertical position, with the arch downwards. Put the eye close to the right-hand hole in the vane, look at the horizon through the transparent part of the horizon-glass, at the same time sliding the index with the left hand, until the image of the sun, seen in the quicksilvered part, falls in with the edge of the horizon, taking either the upper or the under edge of the

solar image. Swing your body gently from side to side; and when the edge of the sun is observed not to cut, but to touch the horizontal line like a tangent, the observation is made. Then will the degrees on the arch, reckoning from the end next your body, give the altitude of that edge of the sun which was brought to the horizon. If the lower edge was observed, then sixteen minutes, added to the said degrees, give the altitude of the sun's centre; but if the upper edge was used, the sixteen minutes must be subtracted.

To take the Altitude of a Star. Look directly up at the star, through the vane, and transparent part of the glass, the index being close to the button: then will the image of the star, by refraction, be seen in the silvered part, right against the star seen through the other part. Move the index forward, and, as the image descends, let the quadrant descend also, to keep it in the silvered part, till it comes down in a line with the horizon, seen through the transparent part, and the observation is made.

In

To make an Artificial Horizon. Often, when the atmosphere is clear above, the horizon is so laden with vapours, as to prevent an observation being taken. such case, an artificial horizon is to be made thus: fill into any vessel, having a diameter of about three inches, and about half an inch deep, from one to two pounds of quicksilver, on which lay a metal speculum, or a piece of plain glass, whose diameter may be about one-third of an inch less than that of the surface of quicksilver in this the image of the sun may be seen distinctly. Sling the vessel so that it may remain level, and take an observation with a stained glass, which will subdue the great brilliancy of the reflection. The observation thus taken will be as correct as is taken by means of the natural horizon.

As refraction causes each ray of light to assume a curved direction, all objects, when observed, especially by means of instruments, appear with an excess of altitude beyond their actual height. The refractions, to be deducted, as follow:

1. If the object be south when observed, call the zenith-distance south, and vice versa. Then, if the zenith-distance, and the declination, be of contrary names, (that is, if the sun, or star, comes to the meridian in the north, and has south declination, or per contra,) the zenith-distance, added to the declination, gives the latitude of the place of observation; the designation will be north, or south, according as the declination may be.

2. When the zenith-distance, and the declination, are of the same name, that is, when the sun or star comes to the meridian in the north, and has north declination; or per contra; then subtract the lesser from the greater, and the remainder is the latitude.

This general rule decides whether it be north or south. When the declination is greater than the zenith-distance, the latitude is of the same name with the declination; but if less, the latitude is on the opposite side of the equator. For further particulars, see LATITUDE.

QUADRANT of altitude, is a thin piece of metal, in general applied to the globe, and marked with the degrees, from 0 to 90° when laid upon the meridian of any place, it shows its latitude or distance from the equator.

QUADRANT of a circle, or the fourth part of its circumference, is contained under two radii standing at right angles. The quadrant contains ninety degrees, and is the parent of various lines of the greatest utility in many branches of the mathematics, such as the lines of chords, of sines, of latitude, &c. See MATHEMATICAL instruments, and DIALLING.

QUADRANTS, gunner's, are made in various manners, some of them having levels; but the most simple construction, with which we are acquainted, is that made with a staff about a foot in length, having on one side a quadrant, which, by means of a pendulum of metal, shows the exact angle made by the chase, or bore. The staff being put into the muzzle of a mortar, or howitzer, so as to lay, in contact, evenly with its lower side, and the quadrant part being turned down, immediately beyond the muzzle, the penduJum-wire, which is fixed to a small pivot in the right angle, exactly at the centre, whence the quadrant was described, will be kept perpendicular by the weight attached thereto, and will thus indicate the exact elevation of the piece. The point of oscillation, i. e. the pivot, must, however, be always kept very smooth, that there may not be the least roughness,

else the action would be affected, and the index prove erroneous.

QUADRAT, a mathematical instrument, called also a geometrical square, and line of shadows; it is frequently an additional member on the face of the common quadrant, as also on those of Gunter's and Sutton's quadrants; but we shall describe it by itself, as being a dis. tinct instrument.

It is made of any solid matter, as brass, together at right angles, as represented in wood, &c. or of any four plain rules joined Plate XIII. Miscell. fig. 2, where A is the centre, from which hangs a thread with a small weight at the end, serving as a plummet. Each of the sides, BE and D E, is divided into an hundred equal parts; or, if the sides be long enough to admit of it, into a thousand parts; C and Fare two sights, fixed on the side A D. There is, moreover, an index, GH, which, when there is occasion, is joined to the centre, A, in such a manner as that it can move freely round, and remain in any given situation; on this instrument are two sights, KL, perpendicular to the right line going from the centre of the upright side, or the line of the direct or instrument. The side D E is called the termed the reclining side, or the line of upright shadows; and the inside BE is

the versed or back shadows.

To measure an accessible height, A B, (fig. 3.) by the quadrant, let the distance, BD, be measured, which suppose = 96 feet, and let the height of the observer's eye be 6 feet; then holding the instrument with a steady hand, or rather resting it on a support, let it be directed towards the summit A, so that it may be seen clearly through both sights; the perpendicular, or plumb-line, meanwhile hanging free, and touching the surface of the instrument; let now the perpendicular be supposed to cut off on the upper side, K N, 80 equal parts; it is evident, that I. KN, A C K, are similar triangles, and (by prop. 4. lib. 6. of Euclid) NK: KL :: KC (i. e. B D): C A; that is, 80: 100 :: 96: CA: therefore, by the rule of three, CA= 96 × 100

80

=

120 feet, and CB= 6 feet being added, the whole height BA is 126 feet.

If the observer's distance, as D E, be such, that, when the instrument is directed as formerly towards the summit A, the perpendicular fall on the angle P, and the distance, B E or C G, be 120 feet, C A will also be 120 feet: for P G: GH:: GC: