## 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. - 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. - Two
**tangents**from any point are equally inclined to the focal distance of the point. - 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. - 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). - He was author of the following memoirs and treatises: "Of the
**Tangents**of Curves, &c.," Phil. - 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. - 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. - Duced 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. - 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.