TANGENT
, in Geometry, is a line that touches a curve, &c, that is, which meets it in a point without cutting it there, though it be produced both ways; as the Tangent AB of the circle BD. The point B, where the Tangent touches the curve, is called the point of contact.
The direction of a curve at the point of contact, is the same as the direction of the Tangent.
It is demonstrated in Geometry;
1. That a Tangent to a circle, as AB, is perpendicular to the radius BC drawn to the point of contact.
2. The Tangent AB is a mean proportional between AF and AE, the whole secant and the external part of it; and the same for any other secant drawn from the same point A.
3. The two Tangents AB and AD, drawn from the same point A, are always equal to one another. And therefore also, if a number of Tangents be drawn to different points of the curve quite around, and an equal length BA be set off upon each of them from the points of contact, the locus of all the points A will be a circle having the same centre C.
4. The angle of contact ABE, formed at the point of contact, between the Tangent AB and the arc BE, is less than any rectilineal angle.
5. The Tangent of an arc is the right line that limits the position of all the secants that can pass through the point of contact; though strictly speaking it is not one of the secants, but only the limit of them.
6. As a right line is the Tangent of a circle, when it touches the circle so closely, that no right line can be drawn through the point of contact between it and the arc, or within the angle of contact that is formed by them; so, in general, when any right line touches an arc of any curve, in such a manner, that no right line can be drawn through the point of contact, between the right line and the arc, or within the angle of contact that is formed by them, then is that line the Tangent of the curve at the said point; as AB.
7. In all the conic sections; if C be the centre of the figure, and BG an ordinate drawn from the point of contact and perpendicular to the axis; then is CG : CE :: CE : CA, or the semiaxis CE is a mean proportional between CG and CA.
Tangent, in Trigonometry. A Tangent of an arc, is a right line drawn touching one extremity of the arc, and limited by a secant or line drawn through the centre and the other extremity of the arc. So, AG is the Tangent of the arc AB, or of the arc ABD; and AH is the Tangent of the arc AI, or of the arc AIDK.
The same are also the Tangents of the angles that are subtended or measured by the arcs.
Hence, 1. The Tangents in the 1st and 3d quadrants are positive, in the 2d and 4th negative, or drawn the contrary way. But of 0 or 180° the semicircle, the Tangent is 0 or nothing; while those of 90° or a quadrant, and 270° or 3 quadrants, are both infinite; the former infinitely positive, and the latter infinitely negative. That is, Between 0 and 90°, or bet. 180° and 270°, the Tangents are positive. Bet. 90° and 180°, or bet. 270° and 360°, the Tangents are negative.
2. The Tangent of an arc and the Tangent of its supplement, are equal, but of contrary affections, the one being positive, and the other negative; as of a and 180° - a, where a is any arc.
| Also | 180° + a | } | have the same Tangent, and of the |
| and | a | same affection. | |
| Or | 180° + a | } | have the same Tangent, but of |
| and | 180° - a | different affections. |
3. The Tangent of an arc is a 4th proportional to the cosine the sine and the radius; that is, CN : NB :: CA : AG. Hence, a canon of sines being made or given, the canon of Tangents is easily constructed from them.
Co-Tangent, contracted from complement-tangent, is the Tangent of the complement of the arc or angle, or of what it wants of a quadrant or 90°. So LM is the Cotangent of the arc AB, being the Tangent of its complement BL.
The Tangent is reciprocally as the cotangent; or the | Tangent and cotangent are reciprocally proportional with the radius. That is Tang. is as 1/cotan., or Tang. : radius :: radius : cotan. And the rectangle of the Tangent and cotangent is equal to the square of the radius; that is, Tan. X cot. = radius2.
Artificial Tangents, or logarithmic Tangents, are the logarithms of the tangents of arcs; so called, in contradistinction from the natural Tangents, or the Tangents expressed by the natural numbers.
Line of Tangents, is a line usually placed on the sector, and Gunter's scale; the description and uses of which see under the article Sector.
Sub-Tangent, a line lying beneath the Tangent, being the part of the axis intercepted by the Tangent and the ordinate to the point of contact; as the line AG in the 2d and 3d figures above.
Method of Tangents, is a method of determining the quantity of the Tangent and subtangent of any algebraic curve; the equation of the curve being given.
This method is one of the great results of the doctrine of fluxions. It is of great use in Geometry; because that in determining the Tangents of curves, we determine at the same time the quadrature of the curvilinear spaces: on which account it deserves to be here particularly treated on.
If AE be any curve, and E any point in it, to which it is required to draw a Tangent TE. Draw the ordinate DE: then if we can determine the subtangent TD, by joining the points T and E, the line TE will be the Tangent sought.
Let dae be another ordinate indefinitely near to DE, meeting the curve, or Tangent produced, in e; and let Ea be parallel to the axis AD. Then is the elementary triangle Eae similar to the triangle TDE;
| and therefore | ea : aE :: ED : DS; |
| but | ea : aE :: flux. ED : flux. AD; |
| therefore | flux. ED : flux. AD :: DE : DT; |
| that is | , |
Hence we have this general rule: By means of the given equation of the curve, find the value either of x. or y., or of x./y., which value substitute for it in the expression , and, when reduced to its simplest terms, it will be the value of the subtangent sought. This we may illustrate in the following examples.
Ex. 1. The equation defining a circle is , where a is the radius; and the fluxion of this is ; hence ; this multi- plied by y, gives , which is a property of the circle we also know from common geometry.
Ex. 2. The equation defining the common parabola is , a being the parameter, and x and y the absciss and ordinate in all cases. The fluxion of this is ; hence ; conseq. ; that is, the subtangent TD is double the absciss AD, or TA is = AD, which is a well-known property of the parabola.
Ex. 3. The equation defining an ellipsis is , where a and c are the semiaxes. The fluxion of it is ; hence the subtangent; or by adding CD which is = a - x, it becomes , a well-known property of the ellipse.
Ex. 4. The equation defining the hyperbola is , which is similar to that for the ellipse, having only + x2 for - x2; hence the conclusion is exactly similar also, viz, .
And so on, for the Tangents to other curves.
The Inverse Method of Tangents. This is the reverse of the foregoing, and consists in finding the nature of the curve that has a given subtangent. The method of solution is to put the given subtangent equal to the general expression (yx.)/y., which serves for all sorts of curves; then the equation reduced, and the fluenta taken, will give the fluential equation of the curve sought.
Ex. 1. To find the curve line whose subtangent is = (2y2)/a. Here , and the fluents of this give , the equation to a parabola, which therefore is the curve sought.
Ex. 2. To find the curve whose subtangent is = yy/(2a - x), or a third proportional to 2a - x and y. Here , the fluents of which give , the equation to a circle, which therefore is the curve sought. |
TANTALUS's Cup, in Hydraulics, is a cup, as A, with a hole in the bottom, and the longer leg of a syphon BCED cemented into the hole; so that the end D of the shorter leg DE may always touch the bottom of the cup within. Then, if water be poured into this cup, it will rise in the shorter leg by its upward pressure, extruding the air before it through the longer leg, and when the cup is filled above the bend of the syphon at E, the pressure of the water in the cup will force it over the bend; from whence it will descend in the longer leg EB, and through the bottom at G, till the cup be quite emptied. The legs of this syphon are almost close together, and it is sometimes concealed by a small hollow statue, or figure of a man placed over it; the bend E being within the neck of the figure as high as the chin. So that poor thirsty Tantalus stands up to the chin in water, according to the fable, imagining it will rise a little higher, as more water is poured in, and he may drink; but instead of that, when the water comes up to his chin, it immediately begins to descend, and therefore, as he cannot stoop to follow it, he is lest as much tormented with thirst as ever. Ferguson's Lect. p. 72, 4to.
