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These semicircles and the circles A'A' are joined by tangents and short arcs struck from the centre of the figure.

00Another of Roberval's discoveries was a very general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions.

00He also finished his Tabulae Directionum (Nuremberg, '475), essentially an astrological work, but containing a valuable table of tangents.

00focus by two tangents drawn from a point), and (having given the focus and a double ordinate) he uses the focus and directrix to obtain any number of points on a parabola - the first instance on record of the practical use of the directrix.

00It gives tables of sines and cosines, tangents, &c., for every to seconds, calculated to ten places.

00Draw the tangents at A and B, meeting at T; draw TV parallel to the axis of the parabola, meeting the arc in C and the chord in V; and M draw the tangent at C, meeting AT and BT in a and b.

00u m _ 4, we can form a series of trapezia by drawing the tangents at the extremities of these ordinates; the sum of the areas of these trapezia will be h(u 4 .+u 2 +...

00We have therefore B in the first place to see whether the difference can be expressed in terms of the directions of the tangents.

00Draw the tangents at A and B, meeting at T; and through T draw a line parallel to KA and LB, meeting the arc AB in C and the chord AB in V.

00that there exists a point such that the tangents from this point to the four spheres are equal, and that with this point as centre, and the length of the tangent as radius, a sphere may be described which cuts, the four spheres at right angles; this "orthotomic" sphere corresponds to the orthogonal circle of a system of circles.

00He was author of the following memoirs and treatises: "Of the Tangents of Curves, &c.," Phil.

00The table gives the logarithms of sines for every minute of seven figures; it is arranged semi-quadrantally, so that the differentiae, which are the differences of the two logarithms in the same line, are the logarithms of the tangents.

00The title of Gunter's book, which is very scarce, is Canon triangulorum, and it contains logarithmic sines and tangents for every minute of the quadrant to 7 places of decimals.

00The next publication was due to Vlacq, who appended to his logarithms of numbers in the Arithmetica logarithmica of 1628 a table giving log sines, tangents and secants for every minute of the quadrant to ro places; there were obtained by calculating the logarithms of the natural sines, &c. given in the Thesaurus mathematicus of Pitiscus (1613).

00It contains log sines (to 14 places) and tangents (to 10 places), besides natural sines, tangents and secants, at intervals of a hundredth of a degree.

00In the same year Vlacq published at Gouda his Trigonometria artificialis, giving log sines and tangents to every ro seconds of the quadrant to ro places.

00The first logarithms to the base e were published by John Speidell in his New Logarithmes (London, 1619), which contains hYPerbolic log sines, tangents and secants for every minute of the quadrant to 5 places of decimals.

00numbers up to 1000, and log sines and tangents from Gunter's Canon (1620).

00In the following year, 1626, Denis Henrion published at Paris a Traicte des Logarithmes, containing Briggs's logarithms of numbers up to 20,001 to io places, and Gunter's log sines and tangents to 7 places for every minute.

00In the same year de Decker also published at Gouda a work entitled Nieuwe Telkonst, inhoudende de Logarithmi voor de Ghetallen beginnende van r tot io,000, which contained logarithms of numbers up to io,000 to io places, taken from Briggs's Arithmetica of 1624, and Gunter's log sines and tangents to 7 places for every minute.'

00The next great advance on the Trigonometria artificialis took place more than a century and a half afterwards, when Michael Taylor published in 1792 his seven-decimal table of log sines and tangents to every second of the quadrant; it was calculated by interpolation from the Trigonometria to 10 places and then contracted to 7.

00came into very general use, Bagay's Nouvelles tables astronomiques (1829), which also contains log sines and tangents to every second, being preferred; this latter work, which for many years was difficult to procure, has been reprinted with the original title-page and date unchanged.

00In 1784 the French government decided that new tables of sines, tangents, &c., and their logarithms, should be calculated in relation to the centesimal division of the quadrant.

00I „ Logarithms of the ratios of arcs to sines from 04 00000 to 0 4.05000, and log sines throughout the quadrant 4 „ Logarithms of the ratios of arcs to tangents from 0 4 00000 to 0 4.05000, and log tangents throughout the quadrant 4 The trigonometrical results are given for every hundred-thousandth of the quadrant (to" centesimal or 3" 24 sexagesimal).

00- The " centres of similitude " of two circles may be defined as the intersections of the common tangents to the two circles, the direct common tangents giving rise to the " external centre," the transverse tangents to the " internal centre."

00A system of circles is coaxal when the locus of points from which tangents to the circles are equal is a straight line.

0036 it is seen that the line joining the points A and B is the locus of the intersection of equal tangents, for if P be any point on AB and PC and PD the tangents to the circles, then PA PB = PC 2 = PD 2, and therefore PC = PD.

00To prove this let AB, AB' be the tangents from any point on the line AX.

00Then circles having the intersections of tangents to this circle and the line of centres for centres, and the lengths of the tangents as radii, are members of the coaxal system.

00With Vieta, by reason of the advance in arithmetic, the style of treatment becomes more strictly trigonometrical; indeed, the Universales Inspectiones, in which the calculation occurs, would now be called plane and spherical trigonometry, and the accompanying Canon mathematicus a table of sines, tangents and secants.'

00The angle between a line and a curve (mixed angle) or between two curves (curvilinear angle) is measured by the angle between the line and the tangent at the point of intersection, or between the tangents to both curves at their common point.

00He was undoubtedly a clear-sighted and able mathematician, who handled admirably the severe geometrical method, and who in his Method of Tangents approximated to the course of reasoning by which Newton was afterwards led to the doctrine of ultimate ratios; but his substantial contributions to the science are of no great importance, and his lectures upon elementary principles do not throw much light on the difficulties surrounding the border-land between mathematics and philosophy.

00Hence, resolving along the tangents to the arcs BC, CA, respectively, we have ~ (3)

00It is ~asily seen graphically, or from a table of hyperbolic tangents, that the equation u tanh u = 1 has only one positive root (u = 1.200); the span is therefore 2X =2au =2A/ sinh U = 1.326 A,

00The tangents at the ends meet on the directrix, and their inclination to the horizontal is 56 30.

00It possesses thi property that the radius of gyration about any diameter is half thi distance between the two tangents which are parallel to that diameter, In the case of a uniform triangular plate it may be shown that thi momental ellipse at G is concentric, similar and similarly situatec to the ellipse which touches the sides of the triangle at their middle points.

00Line of PressuresCentres and Line of Resistance.The line of pressures is a line to which the directions of all the resistances in one polygon are tangents.

00If four fluids, a, b, c, d, meet in a point 0, and if a tetrahedron AB CD is formed so that its edge AB represents the tension of the surface of contact of the liquids a and b, BC that of b and c, and so on; then if we place this tetrahedron so that the face ABC is normal to the tangent at 0 to the line of concourse of the fluids abc, and turn it so that the edge AB is normal to the tangent plane at 0 to the surface of contact of the fluids a and b, then the other three faces of the tetrahedron will be normal to the tangents at 0 to the other three lines of concourse of the liquids, an the other five edges of the tetrahedron will be normal to the tangent planes at 0 to the other five surfaces of contact.

00This catenoid, however, is in stable equilibrium only when the portion considered is such that the tangents to the catenary at its extremities intersect before they reach the directrix.

00Hence the tangents at A and B to the upper catenary must intersect above the directrix, and the tangents at A and B to the lower catenary must intersect below the directrix.

00The condition of stability of a catenoid is therefore that the tangents at the extremities of its generating catenary must intersect before they reach the directrix.

00In the most general case two points may be chosen on the line of intersection of the diametral planes, and tangents drawn to the pitch circles of the pulleys.

00Guide pulleys are set with their diametral planes in the planes containing corresponding pairs of tangents, and a continuous belt wrapped round these pulleys in due order can then be run in either direction.

001 o, and consider as belonging to it, certain lines, which for the moment may be called " axes " tangents to the component curves n1= ol, 11 2 = o respectively.

00two points correspond to each other when the tangents at the two points again meet the cubic in the same point.

00Stating the theorem in regard to a conic, we have a real point P (called the pole) and a real line XY (called the polar), the line joining the two (real or imaginary) points of contact of the (real or imaginary) tangents drawn from the point to the conic; and the theorem is that when the point describes a line the line passes through a point, this line and point being polar and pole to each other.

00We find in it explicitly the two correlative definitions: " a plane curve is said to be of the p ith degree (order) when it has with a line m real or ideal intersections," and " a plane curve is said to be of the mth class when from any point of its plane there can be drawn to it m real or ideal tangents."

00It may be remarked that in Poncelet's memoir on reciprocal polars, above referred to, we have the theorem that the number of tangents from a point to a curve of the order m, or say the class of the curve, is in general and at most = m(m - 1), and that he mentions that this number is subject to reduction when the curve has double points or cusps.

00And we thus see how the theorem extends to curves, their points and tangents; if there is in the first figure a curve of the order m, any line meets it in m points; and hence from the corresponding point in the second figure there must be to the corresponding curve m tangents; that is, the corresponding curve must be of the class in.

00And, assuming the above theory of geometrical imaginaries, a curve such that m of its points are situate in an arbitrary line is said to be of the order m; a curve such that n of its tangents pass through an arbitrary point is said to be of the class n; as already appearing, this notion of the order and class of a curve is, however, due to Gergonne.

00It may be remarked that we cannot with a real point and line obtain the node with two imaginary tangents (conjugate or isolated point or acnode), nor again the real double tangent with two imaginary points of contact; but this is of little consequence, since in the general theory the distinction between real and imaginary is not attended to.

00The curve (1 x, y, z) m = o, or general curve of the order m, has double tangents and inflections; (2) presents itself as a singularity, for the equations dx(* x, y, z) m =o, d y (*r x, y, z)m=o, d z(* x, y, z) m =o, implying y, z) m = o, are not in general satisfied by any values (a, b, c) whatever of (x, y, z), but if such values exist, then the point (a, b, c) is a node or double point; and (I) presents itself as a further singularity or sub-case of (2), a cusp being a double point for which the two tangents becomes coincident.

00In regard to the ordinary singularities, we have m, the order, n „ class, „ number of double points, Cusps, T double tangents, inflections; and this being so, Pliicker's ” six equations ” are n = m (m - I) -2S -3K, = 3m (m - 2) - 6S- 8K, T=Zm(m -2) (m29) - (m2 - m-6) (28-i-3K)- I -25(5-1) +65K-1114 I), m =n(n - I)-2T-3c, K= 3n (n-2) - 6r -8c, = 2n(n-2)(n29) - (n2 - n-6) (2T-{-30-1-2T(T - I) -1-6Tc -}2c (c - I).

00Seeking then, for this curve, the values, n, e, of the class, number of inflections, and number of double tangents, - first, as regards the class, this is equal to the number of tangents which can be drawn to the curve from an arbitrary point, or what is the same thing, it is equal to the number of the points of contact of these tangents.

00Thirdly, for the double tangents; the points of contact of these are obtained as the intersections of the curve by a curve II = o, which has not as yet been geometrically defined, but which is found analytically to be of the order (m-2) (m 2 -9); the number of intersections is thus = m(rn - 2) (m 2 - 9); but if the given curve has a node then there is a diminution =4(m2 - m-6), and if it has a cusp then there is a diminution =6(m2 - m-6), where, however, it is to be noticed that the factor (m2 - m-6) is in the case of a curve having only a node or only a cusp the number of the tangents which can be drawn from the node or cusp to the curve, and is used as denoting the number of these tangents, and ceases to be the correct expression if the number of nodes and cusps is greater than unity.

00520) is that every singularity whatever may be considered as compounded of ordinary singularities, say we have a singularity =6' nodes, cusps, double tangents and c' inflections.

00We may further consider the inflections and double tangents, as well in general as in regard to cubic and quartic curves.

00The expression 2m(m - 2) (m - 9) for the number of double tangents of a curve of the order in was obtained by Plucker only as a consequence of his first, second, fourth and fifth equations.

00An investigation by means of the curve II = o, which by its intersections with the given curve determines the points of contact of the double tangents, is indicated by Cayley, " Recherches sur l'elimination et la theorie des courbes " (Crelle, t.

00A better process was indicated by Salmon in the " Note on the Double Tangents to Plane Curves," Phil.

00See Cayley, " On the Double Tangents of a Plane Curve " (Phil.

00The solution is still in so far incomplete that we have no properties of the curve II = o, to distinguish one such curve from the several other curves which pass through the points of contact of the double tangents.

00A quartic curve has 28 double tangents, their points of contact determined as the intersections of the curve by a curve II = o of the order 14, the equation of which in a very elegant form was first obtained by Hesse (1849).

00xlv., 1855), in respect to the triads of double tangents which have their points of contact on a conic and other like relations.

00It was assumed by Plucker that the number of real double tangents might be 28, 16, 8, 4 or o, but Zeuthen has found that the last case does not exist.

00To develop the theory, consider the curve corresponding to any particular value of the parameter; this has with the consecutive curve (or curve belonging to the consecutive value of the parameter) a certain number of intersections and of common tangents, which may be considered as the tangents at the intersections; and the so-called envelope is the curve which is at the same time generated by the points of intersection and enveloped by the common tangents; we have thus a dual generation.

00Suppose that in general the variable curve is of the order m with S nodes and cusps, and therefore of the class n with double tangents and E.

00Similarly among the common tangents of the two curves we have the double tangents each counting twice, and the stationary tangents each counting three times, and the number of the remaining common tangents is = n 2 - 27-- 3e (=m 2 -26-3K, inasmuch as each of these numbers is as was seen = m+n).

00At any one of the m 2 -26 - 3K points the variable curve and the consecutive curve have tangents distinct from yet infinitesimally near to each other, and each of these two tangents is also infinitesimally near to one of the n 2 -2T-3t common tangents of the two curves; whence, attending only to the variable curve, and considering the consecutive curve as coming into actual coincidence with it, the n 2 -2T-3c common tangents are the tangents to the variable curve at the m 2 -26-3K points respectively, and the envelope is at the same time generated by the m 2 -26-3K points, and enveloped by the n2-2T-3c tangents; we have thus a dual generation of the envelope, which only differs from Pliicker's dual generation, in that in place of a single point and tangent we have the group of m2-26-3K points and n 2 -2T-3c tangents.

00The branch, whether re-entrant or infinite, may have a cusp or cusps, or it may cut itself or another branch, thus having or giving rise to crunodes or double points with distinct real tangents; an acnode, or double point with imaginary tangents, is a branch by itself, - it may be considered as an indefinitely small re-entrant branch.

00branch may have inflections and double tangents, or there may be double tangents which touch two distinct branches; there are also double tangents with imaginary points of contact, which are thus lines having no visible connexion with the curve.

00It may be added that there are on the odd circuit three inflections, but on the even circuit no inflection; it hence also appears that from any point of the odd circuit there can be drawn to the odd circuit two tangents, and to the even circuit (if any) two tangents, but that from a point of the even circuit there cannot be drawn (either to the odd or the even circuit) any real tangent; consequently, in a simplex curve the number of tangents from any point is two; but in a complex curve the number is four, or none, - f our if the point is on the odd circuit, none if it is on the even circuit.

00A non-singular quartic has only even circuits; it has at most four circuits external to each other, or two circuits one internal to the other, and in this last case the internal circuit has no double tangents or inflections.

00A very remarkable theorem is established as to the double tangents of such a quartic: distinguishing as a double tangent of the first kind a real double tangent which either twice touches the same circuit, or else touches the curve in two imaginary points, the number of the double tangents of the first kind of a non-singular quartic is =4; it follows that the quartic has at most 8 real inflections.

00We may from each of the circular points draw tangents to a given curve; the intersection of two such tangents (belonging of course to the two circular points respectively) is a focus.

00There will be from each circular point X tangents (X, a number depending on the class of the curve and its relation to the line infinity and the circular points, 2 for the general conic, 1 for the parabola, 2 for a circular cubic, or bicircular quartic, &c.); the X tangents from the one circular point and those from the other circular point intersect in X real foci (viz.

00each of these is the only real point on each of the tangents through it), and in X 2 -X imaginary foci; each pair of real foci determines a pair of imaginary foci (the so-called antipoints of the two real foci), and the 2X(X-1) pairs of real foci thus determine the X 2 ---X imaginary foci.

00There are in some cases points termed centres, or singular or multiple foci (the nomenclature is unsettled), which are the intersections of improper tangents from the two circular points respectively; thus, in the circular cubic, the tangents to the curve at the two circular points respectively (or two imaginary asymptotes of the curve) meet in a centre.

00Imagine a curve of order m, deficiency D, and let the corresponding points P, P' be such that the line joining them passes through a given point 0; this is an (m - m-1) correspondence, and the value of k is=1, hence the number of united points is =2m-2+2D; the united points are the points of contact of the tangents from 0 and (as special solutions) the cusps, and we have thus the relation or, writing D=2(m - i)(m-2) - S - K, this is n=m(m - i)-23-3K, which is right.

00Regarding the ultimate curve as derived from a given penultimate curve, we connect with the ultimate curve, and consider as belonging to it, certain points called " summits " cn the component curves P 1 = o, P2 =o respectively; a summit / is a point such that, drawing from an arbitrary point 0 the tangents to the penultimate curve, we have OE as the limit of one of these tangents.

00Thus assuming that the penultimate curve is without nodes or cusps, the number of the tangents to it is=m2 - m, (alms+a2m2+

00Taking P 1 =o to have 3 1 nodes and K 1 cusps, and therefore its class n 1 to be=m 1 2 - m 1 25 1 -3K,, the expression for the number of tangents to the penultimate curve is = (a1 2 - a,) m1 2 + (a2 2 - a2)m2 2 +

00+ 2a,a2mlm2+ + a i(n 1+ 25 1+3 K 1)+ a 2(n 2 + 23 2 +3 K 2)+ where a term 2ala2mlm2 indicates tangents which are in the limit the lines drawn to the intersections of the curves P 1 = o, P2 = o each line 2a 1 a 2 times; a term a i (n 1 +25 1 +30 tangents which are in the No.

00limit the proper tangents to P i =o each a l times, the lines to its nodes each 2a 1 times, and the lines to its cusps each 3a 1 times; the remaining terms (a1 2 - al)m12+ (a22 - a2)m22 + ...

00indicate tangents which are in the limit the lines drawn to the several summits, that is, we have (a 1 2 - a i)m 1 2 summits on the curve P 1 = o, &c.

00On one side are placed the natural lines (as the line of chords, the line of sines, tangents, rhumbs, &c.), and on the other side the corresponding artificial or logarithmic ones.

00A line became continuous, returning into itself by way of infinity; two parallel lines intersect in a point at infinity; all circles pass through two fixed points at infinity (the circular points); two spheres intersect in a fixed circle at infinity; an asymptote became a tangent at infinity; the foci of a conic became the intersections of the tangents from the circular points at infinity; the centre of a conic the pole of the line at infinity, &c. In analytical geometry the line at infinity plays an important part in trilinear co-ordinates.

00From any point without the curve two, and only two, tangents can be drawn; if OP, OP' be two tangents from 0, and S, S' the foci, then the angles OSP, OSP' are equal and also SOP, S'OP'.

00If the tangents be at right angles, then the locus of the point is a circle having the same centre as the ellipse; this is named the director circle.

00If five points be given, Pascal's theorem affords a solution; if five tangents, Brianchon's theorem is employed.

00The principle of involution solves such constructions as: given four tangents and one point, three tangents and two points, &c. If a tangent and its point of contact be given, it is only necessary to remember that a double point on the curve is given.

00A focus or directrix is equal to two conditions; hence such problems as: given a focus and three points; a focus, two points and one tangent; and a focus, one point and two tangents are soluble (very conveniently by employing the principle of reciprocation).

00The first book deals with the generation of the three conics; the second with the asymptotes, axes and diameters; the third with various metrical relations between transversals, chords, tangents, asymptotes, &c.; the fourth with the theory of the pole and polar, including the harmonic division of a straight line, and with systems of two conics, which he shows to intersect in not more than four points; he also investigates conics having single and double contact.

00These two conditions are only compatible when the representation is made with quite narrow pencils, and where the apertures are so small that the sines and tangents are of about the same value.

00The second includes a "Method for the Quadrature of Parabolas," and a treatise "on Maxima and Minima, on Tangents, and on Centres of Gravity," containing the same solutions of a variety of problems as were afterwards incorporated into the more extensive method of fluxions by Newton and Leibnitz.

00Perhaps independently of Aryabhatta (born at Pataliputra on the Ganges 476 A.D.), he introduced the use of sines in calculation, and partially that of tangents.

00Fortunately J. John's writing style is not as prone to finding tangents as mine is.

00Note how the common tangents from the free energy curves (upper graph) " construct " the phase diagram below it.

00Corner knots, on the other hand, can have different tangents on either side of them.

00In another question connected with this, the problem of drawing tangents to any curve, Descartes was drawn into a controversy with Pierre (de) Fermat (1601-1663), Gilles Persone de Roberval (1602-1675), and Girard Desargues (1593-1661).

00In projective geometry it may be defined as the conic which intersects the line at infinity in two real points, or to which it is possible to draw two real tangents from the centre.

00If a rectangle be constructed about AA' and BB', the diagonals of this figure are the "asymptotes" of the curve; they are the tangents from the centre, and hence touch the curve at infinity.

00Two tangents from any point are equally inclined to the focal distance of the point.

00A diameter is a line through the centre and terminated by the curve: it bisects all chords parallel to the tangents at its extremities; the diameter parallel to these chords is its conjugate diameter.

00The rules then are sine of the middle part = product of tangents of adjacent parts = product of cosines of opposite parts.

00With increase of speeds this matter has become important as an element of comfort in passenger traffic. As a first approximation, the centre-line of a railway may be plotted out as a number of portions of circles, with intervening straight tangents connecting them, when the abruptness of the changes of direction will depend on the radii of the circular portions.

00The committee had not found one that did not possess grave disadvantages, but concluded that the " principle of contact of the surfaces of vertical surfaces embodied in the Janney coupler afforded the best connexion for cars on curves and tangents "; and in 1887 the Association recommended the adoption of a coupler of the Janney type, which, as developed later, is shown in fig.

00These semicircles and the circles A'A' are joined by tangents and short arcs struck from the centre of the figure.

00Another of Roberval's discoveries was a very general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions.

00He also finished his Tabulae Directionum (Nuremberg, '475), essentially an astrological work, but containing a valuable table of tangents.

00focus by two tangents drawn from a point), and (having given the focus and a double ordinate) he uses the focus and directrix to obtain any number of points on a parabola - the first instance on record of the practical use of the directrix.

00It gives tables of sines and cosines, tangents, &c., for every to seconds, calculated to ten places.

00Draw the tangents at A and B, meeting at T; draw TV parallel to the axis of the parabola, meeting the arc in C and the chord in V; and M draw the tangent at C, meeting AT and BT in a and b.

00u m _ 4, we can form a series of trapezia by drawing the tangents at the extremities of these ordinates; the sum of the areas of these trapezia will be h(u 4 .+u 2 +...

00We have therefore B in the first place to see whether the difference can be expressed in terms of the directions of the tangents.

00Draw the tangents at A and B, meeting at T; and through T draw a line parallel to KA and LB, meeting the arc AB in C and the chord AB in V.

00that there exists a point such that the tangents from this point to the four spheres are equal, and that with this point as centre, and the length of the tangent as radius, a sphere may be described which cuts, the four spheres at right angles; this "orthotomic" sphere corresponds to the orthogonal circle of a system of circles.

00Prony (1755-1839) in the formation of the great French tables of logarithms of numbers, sines, and tangents, and natural sines, called the Tables du Cadastre, in which the quadrant was divided centesimally; these tables have never been published (see Logarithms).

00He was author of the following memoirs and treatises: "Of the Tangents of Curves, &c.," Phil.

00But, as originally pointed out by Euler, the difficulty can be turned if we notice that in the ordinary trajectory of practice the quantities i, cos i, and sec i vary so slowly that they may be replaced by their mean values,, t, cos 7 7, and sec r t, especially if the trajectory, when considerable, is divided up in the calculation into arcs of small curvature, the curvature of an arc being defined as the angle between the tangents or normals at the ends of the arc.

00The table gives the logarithms of sines for every minute of seven figures; it is arranged semi-quadrantally, so that the differentiae, which are the differences of the two logarithms in the same line, are the logarithms of the tangents.

00The title of Gunter's book, which is very scarce, is Canon triangulorum, and it contains logarithmic sines and tangents for every minute of the quadrant to 7 places of decimals.

00The next publication was due to Vlacq, who appended to his logarithms of numbers in the Arithmetica logarithmica of 1628 a table giving log sines, tangents and secants for every minute of the quadrant to ro places; there were obtained by calculating the logarithms of the natural sines, &c. given in the Thesaurus mathematicus of Pitiscus (1613).

00It contains log sines (to 14 places) and tangents (to 10 places), besides natural sines, tangents and secants, at intervals of a hundredth of a degree.

00In the same year Vlacq published at Gouda his Trigonometria artificialis, giving log sines and tangents to every ro seconds of the quadrant to ro places.

00The first logarithms to the base e were published by John Speidell in his New Logarithmes (London, 1619), which contains hYPerbolic log sines, tangents and secants for every minute of the quadrant to 5 places of decimals.

00numbers up to 1000, and log sines and tangents from Gunter's Canon (1620).

00In the following year, 1626, Denis Henrion published at Paris a Traicte des Logarithmes, containing Briggs's logarithms of numbers up to 20,001 to io places, and Gunter's log sines and tangents to 7 places for every minute.

00In the same year de Decker also published at Gouda a work entitled Nieuwe Telkonst, inhoudende de Logarithmi voor de Ghetallen beginnende van r tot io,000, which contained logarithms of numbers up to io,000 to io places, taken from Briggs's Arithmetica of 1624, and Gunter's log sines and tangents to 7 places for every minute.'

00The next great advance on the Trigonometria artificialis took place more than a century and a half afterwards, when Michael Taylor published in 1792 his seven-decimal table of log sines and tangents to every second of the quadrant; it was calculated by interpolation from the Trigonometria to 10 places and then contracted to 7.

00came into very general use, Bagay's Nouvelles tables astronomiques (1829), which also contains log sines and tangents to every second, being preferred; this latter work, which for many years was difficult to procure, has been reprinted with the original title-page and date unchanged.

00In 1784 the French government decided that new tables of sines, tangents, &c., and their logarithms, should be calculated in relation to the centesimal division of the quadrant.

00I „ Logarithms of the ratios of arcs to sines from 04 00000 to 0 4.05000, and log sines throughout the quadrant 4 „ Logarithms of the ratios of arcs to tangents from 0 4 00000 to 0 4.05000, and log tangents throughout the quadrant 4 The trigonometrical results are given for every hundred-thousandth of the quadrant (to" centesimal or 3" 24 sexagesimal).

00- The " centres of similitude " of two circles may be defined as the intersections of the common tangents to the two circles, the direct common tangents giving rise to the " external centre," the transverse tangents to the " internal centre."

00A system of circles is coaxal when the locus of points from which tangents to the circles are equal is a straight line.

0036 it is seen that the line joining the points A and B is the locus of the intersection of equal tangents, for if P be any point on AB and PC and PD the tangents to the circles, then PA PB = PC 2 = PD 2, and therefore PC = PD.

00To prove this let AB, AB' be the tangents from any point on the line AX.

00Then circles having the intersections of tangents to this circle and the line of centres for centres, and the lengths of the tangents as radii, are members of the coaxal system.

00With Vieta, by reason of the advance in arithmetic, the style of treatment becomes more strictly trigonometrical; indeed, the Universales Inspectiones, in which the calculation occurs, would now be called plane and spherical trigonometry, and the accompanying Canon mathematicus a table of sines, tangents and secants.'

00The angle between a line and a curve (mixed angle) or between two curves (curvilinear angle) is measured by the angle between the line and the tangent at the point of intersection, or between the tangents to both curves at their common point.

00He was undoubtedly a clear-sighted and able mathematician, who handled admirably the severe geometrical method, and who in his Method of Tangents approximated to the course of reasoning by which Newton was afterwards led to the doctrine of ultimate ratios; but his substantial contributions to the science are of no great importance, and his lectures upon elementary principles do not throw much light on the difficulties surrounding the border-land between mathematics and philosophy.

00duced by euclidian methods from the definition include the following: the tangent at any point bisects the angle between the focal distance and the perpendicular on the directrix and is equally inclined to the focal distance and the axis; tangents at the extremities of a focal chord intersect at right angles on the directrix, and as a corollary we have that the locus of the intersection of tangents at right angles is the directrix; the circumcircle of a triangle circumscribing a parabola passes through the focus; the subtangent is equal to twice the abscissa of the point of contact; the subnormal is constant and equals the semilatus rectum; and the radius of curvature at a point P is 2 (FP) 4 /a 2 where a is the semilatus rectum and FP the focal distance of P.

00Hence, resolving along the tangents to the arcs BC, CA, respectively, we have ~ (3)

00It is ~asily seen graphically, or from a table of hyperbolic tangents, that the equation u tanh u = 1 has only one positive root (u = 1.200); the span is therefore 2X =2au =2A/ sinh U = 1.326 A,

00The tangents at the ends meet on the directrix, and their inclination to the horizontal is 56 30.

00It possesses thi property that the radius of gyration about any diameter is half thi distance between the two tangents which are parallel to that diameter, In the case of a uniform triangular plate it may be shown that thi momental ellipse at G is concentric, similar and similarly situatec to the ellipse which touches the sides of the triangle at their middle points.

00Line of PressuresCentres and Line of Resistance.The line of pressures is a line to which the directions of all the resistances in one polygon are tangents.

00If four fluids, a, b, c, d, meet in a point 0, and if a tetrahedron AB CD is formed so that its edge AB represents the tension of the surface of contact of the liquids a and b, BC that of b and c, and so on; then if we place this tetrahedron so that the face ABC is normal to the tangent at 0 to the line of concourse of the fluids abc, and turn it so that the edge AB is normal to the tangent plane at 0 to the surface of contact of the fluids a and b, then the other three faces of the tetrahedron will be normal to the tangents at 0 to the other three lines of concourse of the liquids, an the other five edges of the tetrahedron will be normal to the tangent planes at 0 to the other five surfaces of contact.

00This catenoid, however, is in stable equilibrium only when the portion considered is such that the tangents to the catenary at its extremities intersect before they reach the directrix.

00Hence the tangents at A and B to the upper catenary must intersect above the directrix, and the tangents at A and B to the lower catenary must intersect below the directrix.

00The condition of stability of a catenoid is therefore that the tangents at the extremities of its generating catenary must intersect before they reach the directrix.

00In the most general case two points may be chosen on the line of intersection of the diametral planes, and tangents drawn to the pitch circles of the pulleys.

00Guide pulleys are set with their diametral planes in the planes containing corresponding pairs of tangents, and a continuous belt wrapped round these pulleys in due order can then be run in either direction.

001 o, and consider as belonging to it, certain lines, which for the moment may be called " axes " tangents to the component curves n1= ol, 11 2 = o respectively.

00two points correspond to each other when the tangents at the two points again meet the cubic in the same point.

00Stating the theorem in regard to a conic, we have a real point P (called the pole) and a real line XY (called the polar), the line joining the two (real or imaginary) points of contact of the (real or imaginary) tangents drawn from the point to the conic; and the theorem is that when the point describes a line the line passes through a point, this line and point being polar and pole to each other.

00We find in it explicitly the two correlative definitions: " a plane curve is said to be of the p ith degree (order) when it has with a line m real or ideal intersections," and " a plane curve is said to be of the mth class when from any point of its plane there can be drawn to it m real or ideal tangents."

00It may be remarked that in Poncelet's memoir on reciprocal polars, above referred to, we have the theorem that the number of tangents from a point to a curve of the order m, or say the class of the curve, is in general and at most = m(m - 1), and that he mentions that this number is subject to reduction when the curve has double points or cusps.

00And we thus see how the theorem extends to curves, their points and tangents; if there is in the first figure a curve of the order m, any line meets it in m points; and hence from the corresponding point in the second figure there must be to the corresponding curve m tangents; that is, the corresponding curve must be of the class in.

00And, assuming the above theory of geometrical imaginaries, a curve such that m of its points are situate in an arbitrary line is said to be of the order m; a curve such that n of its tangents pass through an arbitrary point is said to be of the class n; as already appearing, this notion of the order and class of a curve is, however, due to Gergonne.

00It may be remarked that we cannot with a real point and line obtain the node with two imaginary tangents (conjugate or isolated point or acnode), nor again the real double tangent with two imaginary points of contact; but this is of little consequence, since in the general theory the distinction between real and imaginary is not attended to.

00The curve (1 x, y, z) m = o, or general curve of the order m, has double tangents and inflections; (2) presents itself as a singularity, for the equations dx(* x, y, z) m =o, d y (*r x, y, z)m=o, d z(* x, y, z) m =o, implying y, z) m = o, are not in general satisfied by any values (a, b, c) whatever of (x, y, z), but if such values exist, then the point (a, b, c) is a node or double point; and (I) presents itself as a further singularity or sub-case of (2), a cusp being a double point for which the two tangents becomes coincident.

00In regard to the ordinary singularities, we have m, the order, n „ class, „ number of double points, Cusps, T double tangents, inflections; and this being so, Pliicker's ” six equations ” are n = m (m - I) -2S -3K, = 3m (m - 2) - 6S- 8K, T=Zm(m -2) (m29) - (m2 - m-6) (28-i-3K)- I -25(5-1) +65K-1114 I), m =n(n - I)-2T-3c, K= 3n (n-2) - 6r -8c, = 2n(n-2)(n29) - (n2 - n-6) (2T-{-30-1-2T(T - I) -1-6Tc -}2c (c - I).

00Seeking then, for this curve, the values, n, e, of the class, number of inflections, and number of double tangents, - first, as regards the class, this is equal to the number of tangents which can be drawn to the curve from an arbitrary point, or what is the same thing, it is equal to the number of the points of contact of these tangents.

00Thirdly, for the double tangents; the points of contact of these are obtained as the intersections of the curve by a curve II = o, which has not as yet been geometrically defined, but which is found analytically to be of the order (m-2) (m 2 -9); the number of intersections is thus = m(rn - 2) (m 2 - 9); but if the given curve has a node then there is a diminution =4(m2 - m-6), and if it has a cusp then there is a diminution =6(m2 - m-6), where, however, it is to be noticed that the factor (m2 - m-6) is in the case of a curve having only a node or only a cusp the number of the tangents which can be drawn from the node or cusp to the curve, and is used as denoting the number of these tangents, and ceases to be the correct expression if the number of nodes and cusps is greater than unity.

00520) is that every singularity whatever may be considered as compounded of ordinary singularities, say we have a singularity =6' nodes, cusps, double tangents and c' inflections.

00We may further consider the inflections and double tangents, as well in general as in regard to cubic and quartic curves.

00The expression 2m(m - 2) (m - 9) for the number of double tangents of a curve of the order in was obtained by Plucker only as a consequence of his first, second, fourth and fifth equations.

00An investigation by means of the curve II = o, which by its intersections with the given curve determines the points of contact of the double tangents, is indicated by Cayley, " Recherches sur l'elimination et la theorie des courbes " (Crelle, t.

00A better process was indicated by Salmon in the " Note on the Double Tangents to Plane Curves," Phil.

00See Cayley, " On the Double Tangents of a Plane Curve " (Phil.

00The solution is still in so far incomplete that we have no properties of the curve II = o, to distinguish one such curve from the several other curves which pass through the points of contact of the double tangents.

00A quartic curve has 28 double tangents, their points of contact determined as the intersections of the curve by a curve II = o of the order 14, the equation of which in a very elegant form was first obtained by Hesse (1849).

00xlv., 1855), in respect to the triads of double tangents which have their points of contact on a conic and other like relations.

00It was assumed by Plucker that the number of real double tangents might be 28, 16, 8, 4 or o, but Zeuthen has found that the last case does not exist.

00To develop the theory, consider the curve corresponding to any particular value of the parameter; this has with the consecutive curve (or curve belonging to the consecutive value of the parameter) a certain number of intersections and of common tangents, which may be considered as the tangents at the intersections; and the so-called envelope is the curve which is at the same time generated by the points of intersection and enveloped by the common tangents; we have thus a dual generation.

00Suppose that in general the variable curve is of the order m with S nodes and cusps, and therefore of the class n with double tangents and E.

00Similarly among the common tangents of the two curves we have the double tangents each counting twice, and the stationary tangents each counting three times, and the number of the remaining common tangents is = n 2 - 27-- 3e (=m 2 -26-3K, inasmuch as each of these numbers is as was seen = m+n).

00At any one of the m 2 -26 - 3K points the variable curve and the consecutive curve have tangents distinct from yet infinitesimally near to each other, and each of these two tangents is also infinitesimally near to one of the n 2 -2T-3t common tangents of the two curves; whence, attending only to the variable curve, and considering the consecutive curve as coming into actual coincidence with it, the n 2 -2T-3c common tangents are the tangents to the variable curve at the m 2 -26-3K points respectively, and the envelope is at the same time generated by the m 2 -26-3K points, and enveloped by the n2-2T-3c tangents; we have thus a dual generation of the envelope, which only differs from Pliicker's dual generation, in that in place of a single point and tangent we have the group of m2-26-3K points and n 2 -2T-3c tangents.

00The branch, whether re-entrant or infinite, may have a cusp or cusps, or it may cut itself or another branch, thus having or giving rise to crunodes or double points with distinct real tangents; an acnode, or double point with imaginary tangents, is a branch by itself, - it may be considered as an indefinitely small re-entrant branch.

00branch may have inflections and double tangents, or there may be double tangents which touch two distinct branches; there are also double tangents with imaginary points of contact, which are thus lines having no visible connexion with the curve.

00It may be added that there are on the odd circuit three inflections, but on the even circuit no inflection; it hence also appears that from any point of the odd circuit there can be drawn to the odd circuit two tangents, and to the even circuit (if any) two tangents, but that from a point of the even circuit there cannot be drawn (either to the odd or the even circuit) any real tangent; consequently, in a simplex curve the number of tangents from any point is two; but in a complex curve the number is four, or none, - f our if the point is on the odd circuit, none if it is on the even circuit.

00A non-singular quartic has only even circuits; it has at most four circuits external to each other, or two circuits one internal to the other, and in this last case the internal circuit has no double tangents or inflections.

00A very remarkable theorem is established as to the double tangents of such a quartic: distinguishing as a double tangent of the first kind a real double tangent which either twice touches the same circuit, or else touches the curve in two imaginary points, the number of the double tangents of the first kind of a non-singular quartic is =4; it follows that the quartic has at most 8 real inflections.

00We may from each of the circular points draw tangents to a given curve; the intersection of two such tangents (belonging of course to the two circular points respectively) is a focus.

00There will be from each circular point X tangents (X, a number depending on the class of the curve and its relation to the line infinity and the circular points, 2 for the general conic, 1 for the parabola, 2 for a circular cubic, or bicircular quartic, &c.); the X tangents from the one circular point and those from the other circular point intersect in X real foci (viz.

00each of these is the only real point on each of the tangents through it), and in X 2 -X imaginary foci; each pair of real foci determines a pair of imaginary foci (the so-called antipoints of the two real foci), and the 2X(X-1) pairs of real foci thus determine the X 2 ---X imaginary foci.

00There are in some cases points termed centres, or singular or multiple foci (the nomenclature is unsettled), which are the intersections of improper tangents from the two circular points respectively; thus, in the circular cubic, the tangents to the curve at the two circular points respectively (or two imaginary asymptotes of the curve) meet in a centre.

00Imagine a curve of order m, deficiency D, and let the corresponding points P, P' be such that the line joining them passes through a given point 0; this is an (m - m-1) correspondence, and the value of k is=1, hence the number of united points is =2m-2+2D; the united points are the points of contact of the tangents from 0 and (as special solutions) the cusps, and we have thus the relation or, writing D=2(m - i)(m-2) - S - K, this is n=m(m - i)-23-3K, which is right.

00Regarding the ultimate curve as derived from a given penultimate curve, we connect with the ultimate curve, and consider as belonging to it, certain points called " summits " cn the component curves P 1 = o, P2 =o respectively; a summit / is a point such that, drawing from an arbitrary point 0 the tangents to the penultimate curve, we have OE as the limit of one of these tangents.

00Thus assuming that the penultimate curve is without nodes or cusps, the number of the tangents to it is=m2 - m, (alms+a2m2+

00Taking P 1 =o to have 3 1 nodes and K 1 cusps, and therefore its class n 1 to be=m 1 2 - m 1 25 1 -3K,, the expression for the number of tangents to the penultimate curve is = (a1 2 - a,) m1 2 + (a2 2 - a2)m2 2 +

00+ 2a,a2mlm2+ + a i(n 1+ 25 1+3 K 1)+ a 2(n 2 + 23 2 +3 K 2)+ where a term 2ala2mlm2 indicates tangents which are in the limit the lines drawn to the intersections of the curves P 1 = o, P2 = o each line 2a 1 a 2 times; a term a i (n 1 +25 1 +30 tangents which are in the No.

00limit the proper tangents to P i =o each a l times, the lines to its nodes each 2a 1 times, and the lines to its cusps each 3a 1 times; the remaining terms (a1 2 - al)m12+ (a22 - a2)m22 + ...

00indicate tangents which are in the limit the lines drawn to the several summits, that is, we have (a 1 2 - a i)m 1 2 summits on the curve P 1 = o, &c.

00On one side are placed the natural lines (as the line of chords, the line of sines, tangents, rhumbs, &c.), and on the other side the corresponding artificial or logarithmic ones.

00A line became continuous, returning into itself by way of infinity; two parallel lines intersect in a point at infinity; all circles pass through two fixed points at infinity (the circular points); two spheres intersect in a fixed circle at infinity; an asymptote became a tangent at infinity; the foci of a conic became the intersections of the tangents from the circular points at infinity; the centre of a conic the pole of the line at infinity, &c. In analytical geometry the line at infinity plays an important part in trilinear co-ordinates.

00From any point without the curve two, and only two, tangents can be drawn; if OP, OP' be two tangents from 0, and S, S' the foci, then the angles OSP, OSP' are equal and also SOP, S'OP'.

00If the tangents be at right angles, then the locus of the point is a circle having the same centre as the ellipse; this is named the director circle.

00If five points be given, Pascal's theorem affords a solution; if five tangents, Brianchon's theorem is employed.

00The principle of involution solves such constructions as: given four tangents and one point, three tangents and two points, &c. If a tangent and its point of contact be given, it is only necessary to remember that a double point on the curve is given.

00A focus or directrix is equal to two conditions; hence such problems as: given a focus and three points; a focus, two points and one tangent; and a focus, one point and two tangents are soluble (very conveniently by employing the principle of reciprocation).

00The first book deals with the generation of the three conics; the second with the asymptotes, axes and diameters; the third with various metrical relations between transversals, chords, tangents, asymptotes, &c.; the fourth with the theory of the pole and polar, including the harmonic division of a straight line, and with systems of two conics, which he shows to intersect in not more than four points; he also investigates conics having single and double contact.

00These two conditions are only compatible when the representation is made with quite narrow pencils, and where the apertures are so small that the sines and tangents are of about the same value.

00The second includes a "Method for the Quadrature of Parabolas," and a treatise "on Maxima and Minima, on Tangents, and on Centres of Gravity," containing the same solutions of a variety of problems as were afterwards incorporated into the more extensive method of fluxions by Newton and Leibnitz.

00Perhaps independently of Aryabhatta (born at Pataliputra on the Ganges 476 A.D.), he introduced the use of sines in calculation, and partially that of tangents.

00Fortunately J. John 's writing style is not as prone to finding tangents as mine is.

00Note how the common tangents from the free energy curves (upper graph) " construct " the phase diagram below it.

00Corner knots, on the other hand, can have different tangents on either side of them.

00An essay that goes off on unrelated tangents can indicate a lack of focus and can be a huge turnoff for the reader.

00If it is too brief or it goes off on too many unrelated tangents, this can be a huge turnoff for the reader.

00It's easy to get sidetracked because as you gain experience, you're tempted to go off on tangents as "opportunities" present themselves.

00But tangents can kill a business by diluted focus and stretching operating funds to a point where the link snaps and the business folds in upon itself and fails.

00In another question connected with this, the problem of drawing tangents to any curve, Descartes was drawn into a controversy with Pierre (de) Fermat (1601-1663), Gilles Persone de Roberval (1602-1675), and Girard Desargues (1593-1661).

02In projective geometry it may be defined as the conic which intersects the line at infinity in two real points, or to which it is possible to draw two real tangents from the centre.

02Two tangents from any point are equally inclined to the focal distance of the point.

02The rules then are sine of the middle part = product of tangents of adjacent parts = product of cosines of opposite parts.

02With increase of speeds this matter has become important as an element of comfort in passenger traffic. As a first approximation, the centre-line of a railway may be plotted out as a number of portions of circles, with intervening straight tangents connecting them, when the abruptness of the changes of direction will depend on the radii of the circular portions.

02

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