FIGURE

, in general, denotes the surface or terminating extremes of a body.——All finite bodies have some figure, form, or shape; whence, figurability is reckoned among the essential properties of body, or matter: abody without Figure, would be an infinite body.

Figures

, in Architecture and Sculpture, denote representations of things made in solid matter; such as statures, &c.

Figures

, in Arithmetic, are the numeral characters, by which numbers are expressed or written, as the ten digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. These are usually called the Arabic, and Indian figures, from which people it is supposed they have been derived. They were brought into Europe by the Moors of Spain, and into England about 1130, as Dr. Wallis apprehends: see his Algebra, pa. 9. However, from some ancient dates, supposed to consist wholly or in part of Arabian figures, some have concluded that these Figures, originally Indian, were known and used in this country at least as early as the 10th century. The oldest date discovered by Dr. Wallis, was on a chimney piece, at Helmdon, in Northamptonshire, thus M133, that is 1133. Other dates discovered fince, are 1090, at Colchester, in Essex; M16 or 1016, at Widgel-hall, near Buntingford, in Hertfordshire; 1011 on the north front of the parish church of Rumsey in Hampshire; and 975 over a gate-way at Worcester.

Dr. Ward, however, has urged several objections against the antiquity of these dates. As no example occurs of the use of these figures in any ancient manuscript, earlier than some copies of Johannes de Sacro Bosco, who died in 1256, he thinks it strange that these Figures should have been used by artificers so long before they appear in the writings of the learned; and he also disputes the fact. The Helmdon date, according to him, should be 1233; the Colchester date 1490; that at Widgel-hall has in it no Arabic Figures, the 1 and 6 being I and G, the initial letters of a name; and the date at Worcester consists, he supposes, of Roman numerals, being really MXV. Martyn's Abridg. Philos. Trans. vol. 9, pa. 420.

Mr. Gibbon observes (in his History of the Decline and Fall of the Roman Empire, vol. v. pa. 321). that “under the reign of the caliph Waled, the Greek language and characters were excluded from the accounts of the public revenue. If this change was productive of the invention or familiar use of our present numerals, the Arabic characters or cyphers, as they are commonly styled, a regulation of office has promoted the most important discoveries of arithmetic, algebra, and the mathematical sciences.”

On the other hand it may be observed that, “according to a new, though probable notion, maintained by M. de Villaison) Anecdota Græca, tom. ii. p. 152, 157), our cyphers are not of Indian or Arabic invention. They were used by the Greek and Latin arithmeticians long before the age of Boethius. After the extinction of science in the west, they were adopted in the Arabic versions from the original manuscripts, | and restored to the Latins about the eleventh century.

Figure

, in Astrology, a description, draught, or construction of the state and disposition of the heavens, at a certain point of time; containing the places of the planets and stars, marked down in a Figure of 12 triangles, called houses. This is also called a Horoscope, and Theme.

Figure

, of an Eclipsc, in Astronomy, denotes a representation upon paper &c, of the path or orbit of the sun or moon, during the time of the eclipse; with the different phases, the digits eclipsed, and the beginning, middle, and end of darkness, &c.

Figure

, or Delineation, of the full moon, such as, viewed through a telescope with two convex glasses, is of considerable use in observations of eclipses, and conjunctions of the moon with other luminaries. In this Figure are usually represented the maculæ or spots of the moon, marked by numbers; beginning with the spots that usually enter first within the shade at the time of the eclipses, and also emerge the first.

Figure

, in Conic Sections, according to Apollonius, is the rectangle contained under the latus-rectum and the transverse axis, in the ellipse and hyperbola.

Figure

, in Fortification, is the plan of any fortified place; or the interior polygon, &c. When the sides, and the angles, are all equal, it is called a regular Figure; but when unequal, an irregular one.

Figure

, in Geomancy, is applied to the extremes of points, lines, or numbers, thrown or cast at random: on the combinations or variations of which, the sages of this art found their divinations.

Figure

, in Geometry, denotes a surface or space inclosed on all sides; and is either superficial or solid; superficial when it is inclosed by lines, and solid when it is inclosed or bounded by surfaces.

Figures are either straight, curved, or mixed, according as their bounds are straight, or curved, or both.— The exterior bounds of a Figure, are called its sides; the lowest side, its base; and the angular point opposite the base, the vertex of the Figure; also its height, is the distance of the vertex from the base, or the perpendicular let fall upon it from the vertex.

For Figures, equal, equiangular, equilateral, circumscribed, inscribed, plane, regular, irregular, similar, &c; see the respective adjectives.

Apparent Figure, in Optics, that Figure, or shape, under which an object appears, when viewed at a distance. This is often very different from the true figure; for a straight line viewed at a distance may appear but as a point; a surface as a line; a solid as a surface; and a crooked figure as a straight one. Also, each of these may appear of different magnitudes, and some of them of different shapes, according to their situation with regard to the eye. Thus an arch of a circle may appear a straight line; a square or parallelogram, a trapezium, or even a triangle; a circle, an ellipsis; angular magnitudes, round; a sphere, a circle; &c.

Also any small light, as a candle, seen at a distance in the dark, will appear magnified, and farther off than it really is. Add to this, that when several objects are seen at a distance, under angles that are so small as to be insensible, as well as each of the angles subtended by any one of them, and that next to it; then all these objects appear not only as contiguous, but as constituting and seeming but one continued magnitude.

Figure of the Sines, Cosines, Versed-sines, Tangents, or Secants, &c, are Figures made by conceiving the circumference of a circle extended out in a right line, upon every point of which are erected perpendicular ordinates equal to the Sines, Cosines, &c, of the corresponding arcs; and then drawing the curve line through the extremity of all these ordinates; which is then the Figure of the Sines, Cosines, &c.

It would seem that these Figures took their rise from the circumstance of the extension of the meridian line by Edward Wright, who computed that line by collecting the successive sums of the secants, which is the same thing as the area of the Figure of the secants, this being made up of all the ordinates, or secants, by the construction of the Figure. And in imitation of this, the Figures of the other lines have been invented. By means of the Figure of the secants, James Gregory shewed how the logarithmic tangents may be constructed, in his Exercitationes Geometricæ, 4to, 1668.

Let ADB &c (fig. 1) be the circle, AD an arc, DE its sine, CE its cosine, AE the versed sine, AF the tangent, GH the cotangent, CF the secant, and CH the cosecant. Draw a right line aa equal to the whole circumference ADGBA of the circle, upon which lay off also the lengths of several arcs, as the arcs at every 10°, from 0 at a, to 360° at the other end at a; upon these points raise perpendicular ordinates, upwards or downwards, according as the sine, cosine, &c, is affirmative or negative in that part of the circle; lastly, upon these ordinates set off the length of the sines, cosines, &c, corresponding to the arcs at those points of the line or circumference aa, drawing a curve line through the extremities of all these ordinates; which will be the Figure of the sines, cosines, versedsines, tangents, cotangents, secants, and cosecants, as in the annexed Figures. Where it may be observed, that the following curves are the same, viz, those of the sines and cosines, those of the tangents and cotangents, and those of the secants and cosecants; only some of their parts a little differently placed. Fig. 1. Sines. Cosines. Versedsines. | Tangents. Cotangents. Secants. Cosecants.

It may be known when any of these lines, viz, the sines, cosines, &c, are affirmative or negative, i. e. to be set upwards or downwards, by observing the following general rules for those lines in the 1st, 2d, 3d, and 4th quadrants of the circle.

The sines	in the 1st and 2d	are	affirmative,
	in the 3d and 4th		negative:
The cosines	in the 1st and 4th	are	affirmative,
	in the 2d and 3d		negative:
The tangents	in the 1st and 3d	are	affirmative,
	in the 2d and 4th		negative:
The cotangents	in the 1st and 3d	are	affirmative,
	in the 2d and 4th		negative:
The secants	in the 1st and 4th	are	affirmative,
	in the 2d and 3d		negative:
The cosecants	in the 1st and 2d	are	affirmative,
	in the 3d and 4th		negative:

And all the versedsines are affirmative.

Draw any ordinate de; putting r = the radius AC of the given circle, x = ad or AD any absciss or arc, and y = de its ordinate, which will be either the sine DE = s, cosine CE = c, versedsine AE = v, tangent AF = t, cotangent GH = t, secant CF = s, or cosecant CH = s, according to the nature of the particular construction. Now, from the nature of the circle, are obtained these following general equations, expressing the relations between the fluxions of a circular arc and its sine, or cosine, &c. . And these also express the relation between the absciss and ordinate of the curves in question, each in the order in which it stands; where x is the common absciss to all of them, and the respective ordinates are s, c, v, t, t, s, and s. And hence the area &c, of any of these curves may be found, as follows:

1. In the Figure of Sines.—Here x = ad, and s = the ordinate de; and the equation of the curve, as above, is . Hence the fluxion of the area, or sx^. is ; the correct fluent of which is the rectangle of radius and vers. i. e. - or + as s is increasing or decreasing; which is a general expression for the area ade in the Figure of sines. When s = 0, as at a or b, this expression becomes 0 or 2r²; that is 0 at a, and 2r² = the area aeb; or r² = the area of afg when ad becomes a quadrant af.

2. In the Figure of Cosines.—Here x = ad and c = de; and the equation of the curve is . Hence the fluxion of the area is ; and the fluent of this is , the rectangle of radius and sine, for the general area adec. When s =r, or c = 0, this becomes r² = the area afc, whose absciss af is equal to a quadrant of the circumference; the same as in the Figure of the sines, upon an equal absciss.

3. In the Figure of Versedsines.—Here x = ad, and v = de; and the equation of the curve is . Hence the fluxion of the area is ; and the fluent of this is for the area ade in the Figure of versed sines. When AD or ad is a quadrant AG or af, this becomes the area afg. And when AD or ad is a semicircle ab, it becomes 3.1416r² = the area abg in the Figure of versedsines.

4. In the Figure of Tangents.—Here x = ad, and t = de; and the equation of the curve is . Hence the fluxion of the area is ; and the correct fluent of this is (1/2)r² X hyp. log. of hyp. log. of hyp. log. of s/r. And hence the Figure of the tangents may be used for constructing the logarithmic secants; a property that was remarked by Gregory at the end of his Exercit. Geomet.

When ad becomes a quadrant af, t being then insinite, this becomes infinite for the area afg. And the same for the Figure of cotangents, beginning at f instead of a. |

5. For the Figure of the Secants.—Here x = ad, and s = de; and the equation of the curve is . Hence the fluxion of the area is ; the fluent of which is r² X hyp. log. of for the general area ade. And when ad becomes the quadrant af, this expression becomes infinite for the area afg.

The same process will serve for the Figure of cosecants, beginning at f instead of a.

From hence the meridional parts in Mercator's chart may be calculated for any latitude AD or ad: For the merid. parts: are to the arc of latitude AD :: as the sum of the secants: to the sum of as many radii or :: as the area ade: to ad X radius ac or AD X AC in the first sigure.