# Tangents sentence example

These semicircles and the circles A'A' are joined by

**tangents**and short arcs struck from the centre of the figure.Another 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.He also finished his Tabulae Directionum (Nuremberg, '475), essentially an astrological work, but containing a valuable table of

**tangents**.It gives tables of sines and cosines,

**tangents**, &c., for every to seconds, calculated to ten places.Draw 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.AdvertisementWe have therefore B in the first place to see whether the difference can be expressed in terms of the directions of the

**tangents**.The 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**.The 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.The 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).It 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.AdvertisementIn 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.The 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.In 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.In 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.'The 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.AdvertisementIn 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.I „ 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).A system of circles is coaxal when the locus of points from which

**tangents**to the circles are equal is a straight line.To prove this let AB, AB' be the

**tangents**from any point on the line AX.Then 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.AdvertisementWith 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.'The 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.He 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.The

**tangents**at the ends meet on the directrix, and their inclination to the horizontal is 56 30.It 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.AdvertisementLine 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**.If 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.This 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.Hence 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.The 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.In 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.Guide 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.Stating 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.It 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.And 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.And, 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.It 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.In 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).Seeking 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**.Thirdly, 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.We may further consider the inflections and double

**tangents**, as well in general as in regard to cubic and quartic curves.The 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.A better process was indicated by Salmon in the " Note on the Double

**Tangents**to Plane Curves," Phil.The 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**.A 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).It 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.To 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.Similarly 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).At 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**.The 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.It 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.A 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.We 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.There 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.Imagine 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.Regarding 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**.Taking 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 +On 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.A 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.From 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'.If 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.If five points be given, Pascal's theorem affords a solution; if five

**tangents**, Brianchon's theorem is employed.The 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.These 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.The 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.Perhaps 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**.Fortunately J. John's writing style is not as prone to finding

**tangents**as mine is.Note how the common

**tangents**from the free energy curves (upper graph) " construct " the phase diagram below it.Corner knots, on the other hand, can have different

**tangents**on either side of them.If 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.The 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.Prony (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).But, 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.Fortunately J. John 's writing style is not as prone to finding

**tangents**as mine is.If it is too brief or it goes off on too many unrelated

**tangents**, this can be a huge turnoff for the reader.It's easy to get sidetracked because as you gain experience, you're tempted to go off on

**tangents**as "opportunities" present themselves.But

**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.In 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).In 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.Two

**tangents**from any point are equally inclined to the focal distance of the point.The rules then are sine of the middle part = product of

**tangents**of adjacent parts = product of cosines of opposite parts.With 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.