## 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. - A 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. - 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. - 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. - 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. - 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.