conic sections, which are the 2d order of lines, are only the 1st order of curves; &c. See CURVES and LINES. Also Newton's Enumeratio Linearum Tertii Ordinis.

ORDINATES, in the Geometry of Curve Lines, are right lines drawn parallel to each other, and cutting the curve in a certain number of points.

The parallel ordinates are usually all cut by some other line, which is called the absciss, and commonly the ordinates are perpendicular to the abscissal line. When this line is a diameter of the curve, the property of the ordinates is then the most remarkable; for, in the curves of the first kind, or the conic sections and circle, the ordinates are all bisected by the diameter, making the part on one side of it equal to the part on the other; and in the curves of the 2d order, which may be cut by an ordinate in three points, then the three parts of the ordinate, lying between these three intersections of the curve and the intersection with the diameter, have the part on one side the diameter equal to both the two parts on the other side of it. And so for curves of any order, whatever the number of intersections may be, the sum of the parts of any ordinate, on one side of the diameter, being in all cases equal to the sum of the parts on the other side of it.

The use of ordinates in a curve, and their abscisses, is to define or express the nature of the curve, by means of the general relation or equation between them; and the greatest number of factors, or the dimensions of the highest term, in such equation, is always the same as the order of the line; that equation being a quadratic, or its highest term of two dimensions, in the lines of the 2d order, being the circle and conic sections; and a cubic equation, or its highest term containing 3 dimensions, in the lines of the 3d order; and so on.

Thus, y denoting an ordinate BC, and r its absciss AB; also a, b, c, &c, given quantities: then y = ax2 + bx + c is the general equation for the lines of the 2d order; and ry2 - ey = ax3 + bx2 cxd is the equation for the lines

of the 3d order; and so on.

ORDNANCE, are all sorts of great guns, used in war; such as cannons, mortars, howitzers, &c.

B

ORFFYREUS's Wheel, in Mechanics, is a machine so called from its inventor, which he asserted to be a perpetual motion. This machine, according to the account given of it by Gravesande, in his Œuvres Philosophiques, published by Allemand, Amst. 1774, consisted externally of a large circular wheel, or rather drum, 12 feet in diameter, and 14 inches deep; being very light, as it was formed of an assemblage of deals, having the intervals between them covered with waxed cloth, to conceal the interior parts of it. The two extremities of an iron axis, on which it turned, rested on two supports. On giving a slight impulse to the wheel, in either direction, its motion was gradually accelerated; so that after two or three revolutions it acquired so great a velocity as to make 25 or 26 turns in a minute. This rapid motion it actually preserved during the space of 2 months, in a chamber of the landgrave of Hesse, the door of which was kept locked, and sealed with the landgrave's own seal. At the end of that time it was stopped, to prevent the wear of the materials. The professor, who had been an eye-witness to these circumstances, examined all the external parts of

the machine, and was convinced that there could not be any communication between it and any neighbouring room. Orffyreus however was so incensed, or pretended to be so, that he broke the machine in pieces, and wrote on the wall, that it was the impertinent curiosity of professor Gravesande which made him take this step. The prince of Hesse, who had seen the interior parts of this wheel, but sworn to secrecy, being asked by Gravesande, whether, after it had been in motion for some time, there was any change observable in it, and whether it contained any pieces that indicated fraud or deception, answered both questions in the negative, and declared that the machine was of a very simple construction.

ORGANICAL Description of Curves, is the description. of them on a plane, by means of instruments, and commonly by a continued motion. The most simple construction of this kind, is that of a circle by means of a pair of compasses. The next is that of an ellipse by means of a thread and two pins in the foci, or the ellipse and hyperbola, by means of the elliptical and hyperbolic compasses. A great variety of descriptions of this sort are to be found in Schooten De Organica Conic. Sect. in Plano Descriptione; in Newton's Arithmetica Universalis, De Curvarum Descriptione Organica; Maclaurin's Geometria Organica; Brackenridge's Descriptio Linearum Curvarum; &c.

ORGUES, or ORGANS, in Fortification, long and thick pieces of wood, shod with pointed iron, and hung each by a separate rope over the gate-way of a town, ready on any surprise or attempt of the enemy to be let down to stop up the gate. The ends of the several ropes are wound about a windlass, so as to be let down all together.

ORGUES is also used for a machine composed of several harquebusses or musket-barrels, bound together; so as to make several explosions at the same time. They are used to defend breaches and other places attacked.

ORIENT, the east, or eastern point of the horizon. ORIENT Equinoctial, is used for that point of the horizon where the sun rises when he is in the equinoctial, or when he enters the signs Aries and Libra.

ORIENT Æstival, is the point where the sun rises in the middle of summer, when the days are longest.

ORIENT Hybernal, is the point where the sun rises in the middle of winter, when the days are shortest.

ORIENTAL, situated towards the east with regard to us: in opposition to occidental or the west. ORIENTAL Astronomy, Philosophy, &c, used for those of the east, or of the Arabians, Chaldeans, Persians, Indians, &c.

ORILLON, in Fortification, a small rounding of earth, lined with a wall, raised on the shoulder of those bastions that have casemates, to cover the cannon in the retired flank, and prevent their being dismounted by the enemy. There are other sorts of orillons, properly called epaulements, or shoulderings, which are almost of a square figure.

ORION, a constellation of the southern hemisphere, with respect to the ecliptic, but half in the northern, and half on the southern side of the equinoctial, which runs across the middle of his body. The stars in this constellation are, 38 in Ptolemy's catalogue, 42 in Tycho's, 62 in Hevelius's, and 78 in Flamsteed's. But some telescopes have discovered several thousands of stars in this constellation, of which there are 2 of the first magnitude, and 4 of the second, besides a great many of the third and

Cambden at his request.-He died at Antwerp, 1598, at 71 years of age.

fourth. One of those two stars of the first magnitude is on the middle of the left foot, and is called Regel; the other is on the right shoulder, and called Betelguese; of the 4 of the second magnitude, one is on the left shoulder, and called Bellatrix, and the other three are in the belt, lying nearly in a right line and at equal distances from each other, forming what is popularly called the Yard

wand.

This constellation is one of the 48 old asterisms, and one of the most remarkable in the heavens. It is in the figure of a man, having a sword by his side, and seems attacking the bull with a club in his right hand, his left bearing a shield.

No constellation was so terrible to the mariners of the early periods, as this of Orion. He is mentioned in this way by all the Greek and Latin poets, and even by their historians; his rising and setting being attended by storms and tempests: and as the northern constellations are made the followers of the Pleiades; so are the southern ones made the attendants of Orion.

The name of this constellation is also met with in Scripture several times, viz, in the books of Job, Amos, and Isaiah. In Job it is asked, "Canst thou bind the sweet influence of the Pleiades, or loose the bands of Orion?" And Amos says, "Seek him that maketh the Seven Stars and Orion, and turneth the shadow of death into morning." ORION's River, the same as the constellation Eridanus. ORLE, ORLET, or ORLO, in Architecture, a fillet under the ovolo, or quarter-round of a capital. When it is at the top or bottom of the shaft, it is called the cincture. -Palladio also uses Orlo for the plinth of the bases of columns and pedestals.

ORRERY, an astronomical machine, for exhibiting the various motions and appearances of the sun and planets; hence often called a Planetarium. The term Orrery applied to this instrument, we are informed by Desaguliers, arose from the following circumstance:-Mr. Rowley, a mathematical instrument-maker, having got one from Mr. George Graham, the original inventor, to be sent abroad with some of his own instruments, he copied it, and after wards constructed one for the earl of Orrery. Sir Richard Steele, who knew nothing of Mr. Graham's machine, thinking to do justice to the first encourager, as well as to the inventor of such a curious instrument, called it an orrery, and gave Rowley the praise due to Mr. Graham. Desaguliers's Experim. Philos. vol. 1, pa. 430. The figure of this grand orrery is exhibited at fig. 1, pl. 24. It is since made in various other figures.

ORTEIL, in Fortification. See BERME. ORTELIUS (ABRAHAM), a celebrated geographer, was born at Antwerp, in 1527. He was well skilled in the languages and mathematics, and acquired such reputation by his skill in geography, that he was surnamed the Ptolemy of his time. Justus Lipsius, and most of the learned men of the 16th century, were our author's intimate friends. He passed some time at Oxford in the reign of Edward the 6th; and he visited England a second time in 1577.

His Theatrum Orbis Terræ was the completest work of the kind that had ever been published, and gained our author a reputation adequate to his immense labour in compiling it. He wrote also several other excellent geographical works; the principal of which are, his Thesaurus, and his Synonyma Geographica.-The learned world is also indebted to him for the Britannia, which was undertaken by

ORTHODROMICS, in Navigation, is Great-circle sailing, or the art of sailing in the arch of a great circle, which is the shortest course: for the arch of a great circle is orthodromia, or the shortest distance between two points or places. ORTHOGONIAL, in Geometry, is the same as rectangular, or right-angled. When the term refers to a plane figure, it supposes one leg or side to stand perpendicular to the other when spoken of solids, it supposes their axes to be perpendicular to the plane of the horizon.

ORTHOGRAPHIC, or ORTHOGRAPHICAL Projection of the Sphere, is the projection of its surface or of the sphere on a plane, passing through the middle of it, by an eye vertically at an infinite distance. See PROJECTION. ORTHOGRAPHY, in Geometry, is the drawing or delineating the front plan or side of any object, and of expressing the heights or elevations of every part: being so called from its delineating objects by perpendicular right lines falling on the geometrical plan; or rather, because all the horizontal lines are here straight and parallel, and not oblique as in representations of perspective.

ORTHOGRAPHY, in Architecture, is the profile or elevation of a building, showing all the parts in their true proportion. This is either external or internal.

External QRTHOGRAPHY, is a delineation of the outer face or front of a building; showing the principal wall with its apertures, roof, ornaments, and every thing visible to an eye placed before the building. And

Internal ORTHOGRAPHY, called also a Section, is a delineation or draught of a building, such as it would appear if the external wall were removed.

ORTHOGRAPHY, in Fortification, is the profile, or representation of a work; or a draught so conducted, as that the length, breadth, height, and thickness of the several parts are expressed, such as they would appear if it were perpendicularly cut from top to bottom.

ORTHOGRAPHY, in Perspective, is the front side of any place; that is, the side or plane that lies parallel to a straight line that may be imagined to pass through the outward convex points of the eyes, continued to a convenient length.

ORTIVE, or Eastern Amplitude, in Astronomy, is an arch of the horizon intercepted between the point where a star rises, and the east point of the horizon.

OSCILLATION, in Mechanics, denotes the vibration, or the reciprocal ascent and descent of a pendulum.

C

If a simple pendulum be suspended between two semicycloids, BC, CD, that have the diameter CF of the generating circle equal to half the length of the string, so that the string, as the body E oscillates, folds about them, then will the body oscillate in another cycloid BEAD,similar and equal to the former. And the time of the oscillation in any arc AE, measured from the lowest point A, is always the same constant quantity, whether that are be larger

or smaller. But the oscillations in a circle are unequal, those in the smaller arcs being less than those in the larger; and so always less and less as the ares are smaller,

but still greater than the time of oscillation in a cycloidal arc; till the circular arc becomes very small, and then the time of oscillation in it is very nearly equal to the time in the cycloid, because the circle and cycloid have the same curvature at the vertex, the length of the string being the common radius of curvature to them both at that point. The time of one whole oscillation in the cycloid, or of an ascent and descent in any arch of it, is to the time in which a heavy body would fall freely through CF or FA, the diameter of the generating circle, or through half the length of the pendulum string, as the circumference of a circle is to its diameter, that is as 3·1416 to 1. So that if 7 denote the length of the pendulum CA, and g = 16 feet 193 inches, the space through which a heavy body falls in the 1st second of time, and p31416 the circumference of a circle whose diameter is 1: then by the laws of falling bodies, it is √g : √1⁄2l : : 1":✔ the time of falling

2g

through CF or 1; therefore :P::: ✔P which

2g

16

2g' is the time of one vibration in any arch of the cycloid which has the diameter of its generating circle equal to 4. Or, by substituting the known numbers for p and q, the time of an oscillation becomes barely or very nearly, or more nearly, being the length of the pendulum in inches. And therefore this is also very nearly the time of an oscillation in a small circular arc, whose radius is / inches.

Hence the times of the oscillation of pendulums of different lengths, are directly in the subduplicate ratio of their lengths, or as the square roots of their lengths.-The more exact time of oscillating in a circular arc, when this is of some finite small length, is √! × (1+ ); where h is the height of the vibration, or the versed sine of the single arc of ascent or descent, to the radius l.

The celebrated Huygens first resolved the problem concerning the oscillations of pendulums, in his book De Horologio Oscillatorio, reducing compound pendulums to simple ones. And his doctrine is founded on this hypothesis, that the common centre of gravity of several bodies, connected together, must ascend exactly to the same height from which it fell, whether those bodies be united, or separated from one another in ascending again, provided that each begin to ascend with the velocity acquired by its descent.

This supposition was opposed by several persons, and very much suspected by others. And those even who believed the truth of it, yet thought it too daring to be admitted without proof into a science which demonstrates every thing.

At length James Bernoulli demonstrated it, from the nature of the lever; and published his solution in the Mem. Acad. des Scienc. of Paris, for the year 1703. After his death, which happened in 1705, his brother John Bernoulli gave a more easy and simple solution of the same problem, in the same Memoirs for 1714; and about the same time, Dr. Brook Taylor published a similar solution in his Methodus Incrementorum: which gave occasion to a dispute between these two mathematicians, who accused each other of having stolen their solutions. The particulars of which dispute may be seen in the Leipsic Acts for 1716, and in Bernoulli's works, printed in 1743.

Axis of OSCILLATION, is a line parallel to the horizon, supposed to pass through the centre or fixed point about

which the pendulum oscillates, and perpendicular to the plane in which the oscillation is made.

Centre of OSCILLATION, in a suspended body, is a certain point in it, such that the oscillation of the body will be made in the same time as if that point alone were suspended at that distance from the point of suspension. Or it is the point into which, if the whole weight of the body be collected, the several oscillations will be performed in the same time as before: the oscillations being made only by the force of gravity of the oscillating body. See CENTRE of Oscillation.

OSCULATION, in Geometry, denotes the contact between any curve and its osculatory circle, that is, the circle of the same curvature with the given curve, at the point of contact or of osculation. If AC be the evolute of the involute curve AEF, and the tangent CE the radius of curvature at the point E, with which, and the centre c, if the circle BEG be described; this circle is said to osculate the curve AEF in the point E, which point E M. Huygens calls

the point of osculation, or kissing point.

The line CE is called the osculatory radius, or the radius. of curvature; and the circle BEG the osculatory or kissing circle.

The evolute AC is the locus of the centres of all the cir

cles that osculate the involute curve AEF.

OSCULATION also means the point of concourse of two branches of a curve which touch each other. For example, if the equation of a curve be y = √x + √x3, it is easy to see that the curve has two branches touching one another at the point where x = 0, because the roots have each the signs + and --.

The point of osculation differs from the cusp or point of retrocession (which is also a kind of point of contact of two branches) in this, that in this latter case the two branches terminate, and pass no farther, but in the former the two branches exist on both sides of the point of osculation. Thus, in the second figure above, the point B is the osculation of the two branches ABD, EBF; but A, though it is also a tangent point, is a cusp or the point of retrocession, of AC and AB, the branches not passing beyond the point a.

OSCULATORY Circle, is the same as the circle of curvature; that is, the circle having the same curvature with any curve at a given point. See the foregoing article, Osculation, where BEG, in the last figure but one, is the osculatory circle of the curve AEF at the point E; and CE the osculatory radius, or the radius of curvature.

This circle is called osculatory, because that, of all the circles that can touch the curve in the same point, that one touches it the closest, or in such manner that no other tangent circle can be drawn between it and the curve; so that, in touching the curve, it embraces it as it were, both touching and cutting it at the same time, being on one

side at the convex part of the curve, and on the other at the concave part of it.

In a circle, all the osculatory radii are equal, being the common radius of the circle; the evolute of a circle being only a point, which is its centre. See some properties of the osculatory circle in Maclaurin's Algebra, Appendix De Linearum Geometricarum Proprietatibus generalibus Tractatus, Theor. 2, § 15, &c, treated in a pure geometrical manner.

OSCULATORY Parabola. See PARABOLA.

OSCULATORY Point, the osculation, or point of contact between a curve and its osculatory circle.

OSTENSIVE Demonstrations, such as plainly and directly demonstrate the truth of any proposition. In which they stand distinguished from apagogical ones, or reductions ad absurdum, or ad impossibile, which prove the truth proposed by demonstrating the absurdity or impossibility

of the contrary.

OTACOUSTIC, an instrument that aids or improves the sense of hearing. See AcouSTICS.

OVAL, an oblong curvilinear figure, having two unequal diameters, and bounded by a curve line returning into itself. Or a figure contained by a single curve line, imperfectly round, its length being greater than its breadth, like an egg: whence its name. The proper oval, or egg shape, is an irregular figure, being narrower at one end than at the other; in which it differs from the ellipse, which is the mathematical oval, and is equally broad at both ends. The common people confound the two together: but geometricians call the oval a false ellipse.

The method of describing an oval chiefly used among artificers, is by a cord or string, as FHf, whose length is equal to the greater diameter of the intended oval, and which is fastened by its extremes to two points or pins, y and f, planted in its longer diameter; then, holding it always stretched out as at H, with a pin or pencil carried round the inside, the oval is described: which will be so much the longer and narrower as the two fixed points are farther apart. This oval so described is the true mathematical ellipse, the points F and f being the two foci. But, in architectural designs, where great accuracy is required, the elliptic compasses are better employed. See COMPASSES Elliptical.

H

H A

Another popular way to describe an oval of a given length and breadth, is thus:-Set the given length and breadth, AB and CD, to bisect each other perpendicularly at E; with the centre c, and radius AE, describe an arc to cross AB in F and &; then with these centres, F and G, and radii AF and BG, describe two little arcs HI and KL for the smaller ends of the oval; and lastly, with the centres C and D, and radius CD, describe the arcs HK and IL, for the flatter or longer sides of the oval. But this, it is evident, does not form a true ellipse. Sometimes other points, instead of C and D, are to be taken by trial, as centres in the line CD, produced if necessary, so as to make the two last arcs join best with the two former ones.

OVAL denotes also certain roundish figures, of various