353 



CURTATE. 



CURVE. 



354 



Will. IV. c. 99, much of the business of his office lias entirely ceased ; 

 and the commissioners appointed under the 1 Will. IV. c 58, having 

 recommended the abolition of the office, this was recently effected on 

 the death of the last Cursitor baron, 19 & 20 Viet. c. 86. 



CURTATE (shortened), a term sometimes applied in geometry or 

 astronomy to a line projected orthographically upon a plane fPRO- 



JECTIOX.J 



CURTEIN, or CURTANA, the name given to the first or pointless 

 sword, carried before the kings of England at their coronation ; also 

 called the sword of King Edward the Confessor. It is mentioned by 

 both these names in Matthew Paris, under the year 1236, when detail- 

 ing the marriage ceremonial of Henry III. In ancient times it was 

 the privilege of the earls of Chester to bear this sword before the 

 king, which, in an emblematical form, is considered as the sword of 

 mercy. 



!UR\ ATURE, a mathematical term expressive of the comparative 

 degree of bending which takes place near the different points of a 

 curve. If we imagine a point to describe a curve, that is, to be 

 continually changing the direction of it motion, it may change this 

 direction either more or less rapidly, that is, describe a line which is 

 more or less curved. As in the article CONTACT, we shall first take a 

 rough method of illustration grounded on the notions we derive from 

 experience, and show how the accuracy of mathematics is introduced 

 into the definition. We must suppose the reader to have gained the 

 ideas which are introduced in the articles DIRECTION and VELOCITY. 

 Two points, A and a, are describing two curves, the directions o 



inch, and BK by I of an inch, the limit towards which cxc-rfc 

 approaches, as B is made to approach towards A, is the diameter of the 

 circle of the curvature, or double of its radius. Or if through A and B 

 a circle be always conceived to pass, which touches AT at A, the 

 limiting position of that circle is the circle of curvature. 



If the curve be referred to rectangular co-ordinates, and if x and y 

 be those of the point A, }/ = <j>x the equation of the curve, and <t>' and 

 <t>" the first and second differential co-efficients of f.r, then 



motion ,->t A and being A T and at. That the firot curve is nmre 

 curved than the second, we may easily see ; and if we wished to give 

 Rome notion of the comparative degree of curvature, we might proceed 

 as follows : Measure off equal arcs A B and a b (remember this equality 

 throughout), say of one inch each ; ascertain the directions B v and 

 b r. in which A and a are proceeding when they arrive at B and b and 

 measure the angles B v T and * r t. If we find the first to be twice as 

 great as the second, then the phenomenon by which we recognise 

 curvature (change of direction) is twice as rapid in the first as in the 

 second. Hence we say that the curvature of the first is twice as great 

 as that of the second. 



But here arises a difficulty of the same kind as that explained in 

 V KI.OCITY. The preceding ratio of two to one is that of the whole 

 effects of curvature from A to B, and from a to 6. If the change of 

 direction were uniform, that is, if every tenth (or other) part of AB 

 gave a tenth (or similar other) part of the change of direction, it would 

 then be indifferent whether we chose an inch for A B and a b, or any 

 other length. But if the curvature do not continue uniform, the 

 comparison of the united effects of all the curvatures from A to B, and 

 from a, to b, does not give a proper ratio of the comparative curvatures 

 at A and a. The only curve in which this is the case is the circle, 

 which is the curve of uniform curvature, just as the straight line is 

 the line of uniform direction. But we immediately perceive that if 

 A B and o 6 had each been the hundredth part of an inch instead of an 

 inch, this objection would have held in a less degree ; if the thousandth 

 I>art <if an inch, still less, and so on. Hence, as in VELOCITY, it is not 

 liy the proportion of the angles B v T and b r t that we get an exact and 

 unalterable notion of what is taking place at A and a, but by the limit 

 to which that proportion approaches, as B and b move back towards 

 A and a, and the angles in question diminish without limit. This is 

 the accurate terminal notion on which the mathematical theory of 

 curvature is founded. 



If the two curves had been circles (or curves of uniform curvature) 

 found that the limit of the proportion of these angles (or the 

 proportion of the angles themselves, which for circles is the same as 

 limit) is inversely as the radii of the circles ; that is, doubling the 

 radius of a circle halves its curvature, and so on. This suggests an 

 solute measure of the curvature at A. Let the second curve be a 

 cle, so taken that B v T and b r t shall have a limiting proportion of 

 to l,or shall continually approximate to equality. The circle ab 

 then at a the same curvature as the curve at A, and its radius 

 "lius of curvature) being determined, the degree of cur- 

 it A u known, as compared with that of any point of any other 

 binse radius of curvature is known. This radius of curvature 

 .us determined : Draw the perpendicular BK and the chord A B. 

 lie chord AB be always represented by the fraction c of an 



ARTS AND SCI. DIV. VOL. Iir. 



Radius of curvature at A is '_ 





If (p be the angle made by the tangent with the axis of x, and s the 

 arc from any given point to A, 



Rad. of curv. at A is 

 dp 



When the curve is not a plane curve, imagine it orthographically 

 projected upon a plane. Its change of direction is then partly parallel 

 to the plane, partly perpendicular to it, and the curve is said to have 

 double curvature. But as this subject, together with that of the 

 curvature of surfaces, is not of an elementary character, we shall not 

 proceed further here, but refer to works which treat largely of the 

 Differential Calculus. 



The circle of curvature is also the circle of 'contact, or the nearest 

 circle which can be drawn to the curve, just as the straight line of 

 direction is also the line of contact, or the tangent. An infinite 

 number of circles can be made to present the appearance of touching 

 the curve at A [CONTACT], of all of which the circle of curvature comes 

 the closest. Moreover, it always cuts the curve which it also touches 

 (in the mathematical sense), except only at particular points ; that is 

 to say, the circle of curvature has in general a contact of the second 

 order with its curve, in which case the circle always passes through the 

 curve. But if at any particular point the contact should be of a 

 higher and odd order, the circle of curvature does not pass through 

 the curve. 



CURVE, and CURVES, THEORY OF. A curve is a line which 

 has curvature. Though the second of these terms be derived from the 

 first, yet it is the notion explained in the preceding article which is 

 preliminary to the explanation of the general term cum. Let a point 

 move with a perfectly gradual change of direction, and it describes a 

 curve. 



Curves are said to be of the same species, in which the motion of 

 the describing point is regulated by the same mathematical law. 

 Thus the general law of the circle is, that all its points are equidistant 

 from a given point. This law is the characteristic of the species; one 

 circle is distinguished from another by the length of the constant 

 distance supposed in the law of formation. 



And in like manner as 0, or nothing, is classed under the general 

 name of number or quantity, so the straight line itself (or the line 

 without curvature) is, in algebra, spoken of under the general term 

 curve. Or, in the last-mentioned science, the word means any line 

 which is described by a point moving under one and the same law 

 through every part of space which is consistent with the law. 



The connexion of algebra with the doctrine of curves depends upon 

 the method of co-ordinates (ABSCISSA, ORDINATE, CO-ORDINATES), by 

 means of which every algebraical function whatsoever is connected 

 with the properties of a curve. This is the point of greatest utility in 

 the theory, namely, the power which it gives of representing to the 

 eye all the varieties of magnitude which an algebraical function under- 

 goes, while one of its letters passes through every state of numerical 

 magnitude. 



The number of curves which have received distinct names, out of 

 the infinite number which may be drawn, is very small ; we subjoin 

 the names of those which are of most usual occurrence, referring to 

 the several articles for further information, for the cases which are not 

 presently mentioned. 



1. Circle. 



2. Ellipse. 



3. Hyperbola. 



4. Parabola. 



5. Semi-Cubical Parabola. 



6. Cusoid of Diodes. 



7. Conchoid of Nicomcdes. 



8. Trisectrix. 



9. Lemniscata. 



10. Cycloid. 



1 1 . Companion of the Cycloid. 



1 2. Harmonic Curve. 



13. Troehoid. 



14. Epicycloid and 



Cardioide. 



15. Hypocycloid. 

 10. Epitrochoid. 

 17. Hypotrochoid. 



18. Curve of Sines, cosines, tan. 



gents, &c. [SINKS, TAN- 

 GENTS, &c., CURVES or.] 



19. Exponential or Logarithmic 



Curve. 



20. Spiral of Archimedes. 



21. Logarithmic Spiral or Equi- 



angular Spiral. 



22. Reciprocal Spiral. 



23. Liluus. 



24. duadratrix of Dinostrntus. 



25. of Tcliii nhauscn. 



26. Catenary. 



27. Tractory. 



28. Syntractory. 



29. Tractrix. 



30. Ovals of Cas-ini. 



31. Watt's Parallel motion curre. 



The general characteristics of curves are extremely varied, and very 

 ew of them have received names. We subjoin a diagram, which will 

 show all the varieties of figure most commonly considered in works on 

 the Differential Calculus, premising, however, that we do not actually 



A A 



