Tangent (geometry)

“ Let us return to the subject of Analytical Geometry.
The tangent to an ellipse is an imaginary straight line, representing the path which would be followed by the body tracing the ellipse, if its connection with the attracting focus were suddenly to cease. In its essence, the tangent is a sublime effort of the scientific imagination; it pictures the result of a sudden cessation of the action of gravity. In practice the tangent is a convenient line for indicating the curvature at any given point. ”

“ A tangent to a circumference is perpendicular to a radius at the point of contact.
Let the straight line A B be tangent at the point D to the circumference of the circle whose centre is C.
145Join the centre C with the point of contact D, the tangent will be perpendicular to the radius C D.
For draw any other line from the centre to the tangent, as C F.
As the point D is the only one in which the tangent touches the circumference, any other point, as F, must be without the circumference. ”

“ In plane geometry it is easily proved that if two circles intersect, the tangents from any point on their common chord produced are equal. For if the common chord is AB and the point P is taken on AB produced, then the square on any tangent from P is equal to PB × PA. The line PBA is sometimes called the radical axis.
Similarly in this proposition concerning spheres, if from any point in the plane of the circle formed by the intersection of the two spherical surfaces lines are drawn tangent to either sphere, these tangents are equal. ”

“ Modern textbook makers find it convenient to define tangent circles as those that are tangent to the same straight line at the same point, and to define external and internal tangency by reference to their position with respect to the line, although this may be characterized as open to about the same objection as Euclid's. ”

“ It is easily shown that any line of the one system intersects every line of the other system.
Considering any curve (of double curvature) whatever, the tangent lines of the curve form a singly infinite system of lines, each line intersecting the consecutive line of the system,—that is, they form a developable, or torse; the curve and torse are thus inseparably connected together, forming a single geometrical figure. An osculating plane of the curve (see § 38 below) is a tangent plane of the torse all along a generating line. ”

“ Consequently a quadric surface is covered by two sets of straight lines, a pair through every point on it; these are imaginary for the ellipsoid, hyperboloid of two sheets, and elliptic paraboloid.
A surface of any order is covered by two singly infinite systems of curves, a pair through every point, the tangents to which are all chief tangents at their respective points of contact. These are called chief-tangent curves; on a quadric surface they are the above straight lines. ”

“ What part of the diameter o d is the radius o c?
Name a chord that is produced without the circle.
It is called a “secant.”
A secant is a chord produced.
Name two secants.
If the chord d i were made a secant, would it become longer or shorter?
In how many points does the straight line l m touch the circumference?
It is called a “tangent.”
A tangent is a straight line that touches a circumference in only one point. ”

“ This is seen directly in the fundamental problems—of rectifications, quadratures, and cubatures—which constitute, as has been shown, the final object of geometry. But this remark is no less incontestable, even for investigations which relate to the different properties of lines and of surfaces, and of which the most essential, such as the question of tangents or of tangent planes, the theory of curvatures, [Pg 206] &c., are evidently common to all figures whatever. The very few investigations which are truly peculiar to particular figures have only an extremely secondary importance. ”

“ A fundamental property of the curve is that the line at infinity is a tangent (see Geometry, Projective) , and it follows that the centre and the second real focus and directrix are at infinity. It also follows that a line half-way between a point and its polar and parallel to the latter touches the parabola, and therefore the lines joining the middle points of the sides of a self-conjugate triangle form a circumscribing triangle, and also that the nine-point circle of a self-conjugate triangle passes through the focus. ”

“ The demonstration of the converse is like unto the former. For if the tangent, or touch-line ae, be not perpendicular to the diameter iou, let oe, from the center o, be drawne perpendicular; Then shall the angle oei, be right angle: And oie an acutangle: And therefore by the 22 e vj, oi, that is oy, shall be greater then oye, that is the part, then the whole.
Therefore
20 If a right line doe passe by the center and touch-point, it is perpendicular to the tangent or touch-line. ”

“ For when points and tangents and the centre are given, as many other points and as many other tangents are given at an equal distance on the other side of the centre. And an asymptote is to be considered as a tangent, and its infinitely remote extremity (if we may say so) is a point of contact. ”

“ The tangent PT at P is regarded as identical with the secant PQ, and the position of the tangent is determined by the similarity of the triangles PTM, PQR. The increments QR, PR of the ordinate and abscissa are denoted by a and e; and the ratio of a to e is determined by substituting x + e for x and y + a for y in the equation of the curve, rejecting all terms which are of order higher than the first in a and e, and omitting the terms which do not contain a or e. This process is equivalent to differentiation. ”

“ But we have all along supposed the angle of contact to be neither infinitely greater nor infinitely less than the angles of contact made by circles and their tangents; that is, that the curvature at the point A is neither infinitely small nor infinitely great, or that the interval AJ is of a finite magnitude. For DB may be taken as AD3: in which case no circle can be drawn through the point A, between the tangent AD and the curve AB, and therefore the angle of contact will be infinitely less than those of circles. ”

“ For if A and P are supposed to be the points of contact situated any where else in the tangents, and through any of the points H, I, K, L, as I, situated in either tangent HI a right line IT is drawn, parallel to the other tangent KL, and meeting the curve in X and I and in that right line there be taken IZ equal to a mean proportional between IX and IT ”

“ The line of Artificial Tangents, or the Logarithmical Tangent-line, beginning at 45 deg. and taking every half degree for a whole one, is found to agree pretty near with the Meridian-line of the Sea-Charte; they both growing, as it were, after the same Proportion. But the Table of Meridional degrees being calculated only to every Sexagesimal minute of a degree, shews some small difference from the said Logarithmical Tangent-line. ”

“ Conceive the point of contact of any tangent removed in infinitum, and the tangent will degenerate into an asymptote, and the constructions of the preceding problem will be changed into the constructions of those problems wherein the asymptote is given. ”

“ In reality, however, we have only to do with the case in which the arc and the tangent do coincide, and in which the law that the deflection is equal to the versed sine of the angle is absolutely true. Here, in observing this most familiar thing, we are, at a single step, taken to that which is utterly beyond our comprehension. The angles we have to consider disappear, not only from our sight, but even from our conception. ”

“ We will suppose in the first instance that the curve is plane. It is often convenient to resolve the forces on an element PQ (= δs) in the directions of the tangent and normal respectively. If T, T + δT be the tensions at P, Q, and δψ be the angle between the directions of the curve at these points, the components of the tensions along the tangent at P give (T + δT) cos ψ − T, or δT, ultimately; whilst for the component along the normal at P we have (T + δT) sin δψ, or Tδψ, or Tδs/ρ, where ρ is the radius of curvature. ”

“ In Figure 2108b, A, B, and C are the angles of a plane right triangle, with the right angle at C. The sides are labeled a, b, c, with opposite angles labeled A, B, and C respectively.
Figure 2108b. A right triangle.
The six principal trigonometric functions of angle B are:
Since A and B are complementary, these relations show that the sine of an angle is the cosine of its complement, the tangent of an angle is the cotangent of its complement, and the secant of an angle is the cosecant of its complement. Thus, the co–function of an angle is the function of its complement. ”