Page:Cyclopaedia, Chambers - Supplement, Volume 2.djvu/837

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ride oh one of thehorfcs, as there was no other feat for him: the tifual place for him being all armed with knives, as was likc- ytik the hinder part of the chariot. There were no fcythes pointing down to the earth, either From the beam or axle- tree -, but thefe were fixed at the head of the axle-tree in fuch a manner as to be moveable by means of a rope, and thereby .could be raifed or let down, and drawn forward or let fall backward bv relaxing the rope. Pitifc. Lex. Ant. m voc. CURVATURE (Cyd.) — The theory of the curvature of lines is of great ufe in geometry, and in the phyfico-mathematical fciences. Hence mathematician --have treated this fubject fully. We fhall here infert as much of this doctrine as feems necefTary to enable beginners to form a juft notion of the fubject, and refer thofe who defire a farther knowledge to }Ar. Mac Laurin's Treatifc of Fluxions, whom we have in this, as in many other articles, followed ; becaufe he every where endeavours to avoid that air of paradox and myftery ■which has been more than once made a reproach to modern mathematicians.

Any two right lines applied to each other, perfectly coincide ; and the rectitude of lines admits of no variety. Arches of equal circles applied upon each other, perfectly coincide like- wife ; and the curvature is uniform in all the parts of the fame, or of .equal circles. Arches of unequal circles cannot fee applied upon each other fo as to coincide ; but when they touch each other, the arch of the greater circle is lefs inflect- ed from the common tangent, and paffes betwixt it and the arch of the lefTer circle, through the angle of contacts form- ed by them, and is therefore lefs curve. Any two arches of curve lines touch each other when the fame right line is the tangent of both at the fame point ; but when they are applied upon each other in this manner, they never perfectly coincide, unlefs they be iimilar arches of equal and fimilar figures : and the curvature of lines admits of indefinite variety. As the curvature is uniform in a given circle, and may be varied at pleafure in circles by increafing or diminifhing their diameters, their flexure or curvature will therefore ferve for meafuring that of other lines. There is but one right line that can be the tangent of a given arch of a curve at the fame point ; but an indefinite variety of circles may touch it there ; and thefe have various degrees of more and lefs intimate contact with it. And as of all the right lines that can be drawn through a given point in the arch of a curve, that is the tangent which touches the arch fo clofely, that no right line can be drawn between them ; fo of all the circles that touch a curve in any given point, that is faid to have the fame curvature with it, which touches it fo clofely that no circle can be drawn through the point of contact between them, all other circles pafling either within or without them both.

This circle is called the circle of curvature ; its centre, the centre of curvature ; and its femidiameter, the ray or radius of curvature, belonging to the point of contact. It is alfo call- ed, cfpecially by foreign mathematicians, the ofculatory circle. The arch of this circle cannot coincide with the arch of the curve, but it is fufficient to denote it the circle of curvature, that no other can pafs between them ; as the tangent of the arch of a curve cannot coincide with it, but is applied to it fo that no right line can be drawn between them. As in aii figures, rectilinear ones excepted, the pofition of the tangent Is continually varying; fo the curvature is continual- ly varying in all curvilinear figures, the circle only excepted. As the curve is feparated from its tangent by its flexure or purvature, fo it is feparated from the circle of curvature in confequence of the increafe or decreafe of its curvature ; and as its curvature is greater or lefs, according as it is more or lefs inflected from the tangent, fo the variation of curvature is the greater or lefs according as it is more or lefs feparated from the circle of curvature.

It is manifeft, that there is but one circle of curvature be- longing to an arch of a curve at the fame point. For if there were two fuch circles, any circle defcribed between thefe through that point, would pafs between the curve and circle of curvature ; againff. the fuppofition.

When any two curves touch each other in fuch a manner that no circle can pafs between them, they muff, have the fame curvature ; for the circle that touches the one fo clofely that no circle can pafs between them, muff, touch the other in the fame ^manner.

It appears from the demonftrations of geometricians, that circles may touch curve lines in this manner ; that there may be indefinite degrees of more or lefs intimate contact between the curve and the circle of curvature ; and that a conic fection may be defcribed that fhall have the fame curvature with a given line at a given point, and the fame variation of curva- ture^ or a contact of the fame kind with the circle of curva- ture.

If we conceive the tangent of any propofed curve to be a bafe, and that a new line be defcribed, whofe ordinate is a third pro- portional to the ordinate and bale of the firft; this new line will determine the chord of the circle of curvature, by its in- terfection with the ordinate at the point of contact ; and by the tangent of the angle in which it cuts that circle, it will meafure the variation of the curvature* The lefs this angle

APtfEWD.

is, the clofer is the contact of the curve and circle of curva- ture ; and of this contact there may be indefinite degrees. To give an example : let any curve EMH(fig. I:) and a circle ERB touch the right line ET on the fame fide at E; let any right line TK, parallel to the chord EB, meet the tan- gent in T, EMH in M, and a curve BKF that paffes through B in K : then if the rectangle MTK be always equal to the fquare of E T, the curvature of E M H at E will be the fame as that of the circle ERB ; and the contact EM and ER will always be the clofer, the lefs the angle is that is contained at B by the curve BKF, and the circle of curva- ture BQE.

Fig. I.

For it is demonftrabie from the elements of geometry (fee Mac Laurin's Fluxions, art. 366. j that all the circles that can be defcribed through E fall without both ER and EM, or with- in them both, and no circle whatever can pafs between them when the rectangle MTK is always equal to the fquare of ET, and the curve m which K is always found, paffes thro' B, and confequently the circle ERB, and the curve E M, have the fame curvature at E.

Now let Em {fig. II.), any other curve touching ET in E, ZX&fk B another curve pafling through B meet TK in m and k, and let the rectangle mT k be iikewife equal to the fquare of ET; then will the curvature ofE?n at E be the fame as that of the curve EM, as has been mentioned. But the rectangles mT k> MTK, RTQ_ being equal to each other, and their fides therefore in a reciprocal proportion to each other, it is plain that if the arch B^ pafs between BK the arch of the curve BKF, and BQ_the arch of the circle BQE; the curve E?;z muff, pafs between EM the arch of the curve EMM, and ER the arch of the circle of curvature ERB : fo that Em muft have a clofer contact with this circle than EM has with it ; and the lefs the angle is that is form- ed by the curve FKB and the circle of curvature E QB at B, the clofer is the contact at E of the curve EMH and the circle of curvature ERB. Thus the curve BKF, by its in- terfection with EB, determines the curvature of EM; and by the angle in which it cuts the circle of curvature it deter- mines the degree of contact of EM and that circle, the angle BET and the right line ET being given.

Fig. II.

Hence it follows, that the contact of the curve EMH, and tft. circle of curvature, is clofeft when the curve BK touches th arch BQin B, the angle BET being given; but it is far" theft from this, or is moft open, when B K touches th right line EB in B.

Hence, alfo, there may be indefinite degrees of more and more intimate contact between a circle and a curve. The firft degree is, when the fame right line touches them both in the fame point; and a contact of this fort may take place betwixt any circle and any arch of a curve. The fe- cond is when the curve EMH, and circle ERB have the fame curvature, and the tangents of the curve BKF and circle BQ_E interfect each other at B in any aflignable angle. The. contact of the curve E M and circle of curvature, E R, at E is of the third degree, or order, and their ofculation is of the fecond, when the curve BKF touches the circle BQE at B, but fo as not to have the fame curvature with it. The contact is of the fourth degree, or order, and their ofculation of the third, when the curve BKF has the fame curvature with the circle BQE at B, but fo : as that their contact is F only