# Tangents Sentence Examples

- 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. - 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. - 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. - Focus 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. - 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. - U 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 +... - Draw 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. - That 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. - He was author of the following memoirs and treatises: "Of the
**Tangents**of Curves, &c.," Phil. - 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. - In 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. - Numbers up to 1000, and log sines and
**tangents**from Gunter's Canon (1620). - 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. - Came 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. - In 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). - - 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." - A system of circles is coaxal when the locus of points from which
**tangents**to the circles are equal is a straight line. - 36 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. - 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. - With 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. - Hence, resolving along the
**tangents**to the arcs BC, CA, respectively, we have ~ (3) - It 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, - 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. - 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. - 1 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. - Two points correspond to each other when the
**tangents**at the two points again meet the cubic in the same point. - 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. - The 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. - 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. - 520) 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. - 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. - An 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. - A better process was indicated by Salmon in the " Note on the Double
**Tangents**to Plane Curves," Phil. - Two
**tangents**from any point are equally inclined to the focal distance of the point.