TAN
[ J 74 ]
TAN
the given Sine, A D ; and the Tangent required, EF. Since both the Sine and Tangent are perpendicular to the Radius E C, they are parallel to each other. Wherefore, as the Co- fine DC is to the Sine AD, fo is the whole Sine, to the Tangent E F. See S i n e .
Hence, a Canon of Sines being had ; a Canon of Tangents is eafily conlfru&ed therefrom. See Canon.
Artificial Tangents, are the Logarithms of the Tangents of Arches. SccLogarithm.
Line a/Tangents, is aLine ufually placed on the Seclor and Gunter's Scale ; the Defcription andUfes whereof, fee under the Articles Sector and GunterV Scale.
Tangent of a Conic Setlion, as of a Parabola, or other al-
febraic Curve, is a right Line, drawn, cutting the Axis. See Iurve, Conic, &c.
Method of Tangents, a Method of determining the Quantity of the Tangent of any algebraic Curve; the Equa- tion defining that Curve, being given.
This Method is one of the great Refults of the Calculus 2)ijferentialis. See Differential.
Its Ufe is very great in Geometry ; becaufe in determining the Tangents of Curves, we determine at the fame Time, the Quadratures of the Curvilinear Spaces : on which account it well deferves to be here particularly inlifted on. See Qua- drature.
To find the Sub-tangent in any
aic Curve.
Let the Semiordinate pm be infinitely near another PM (Tab. Analylis Fig. 13) then will P p be the Differential of rheAbfcifs; and letting fall the Perpendicular MR— Pp; Rm will be the Differential of the Semiordinate. Draw, therefore, the Tangent TM : The infinitely little Arch M m, will not differ from a Right Line ; and therefore M m R will be a right-lined, right-angled Triangle, ufually call'd the CharaBerifiic Triangle of the Curve, in regard Curve Lines are diftinguifh'd from each other hereby. See Charac- teristic.
Now, by reafon of the Parallelifm of the Right Lines PMand/;«f the Angle MmR-TMP. Wherefore the Triangle MmR is Similar to the Triangle TMP. Let, therefore A P = x, PM = y, then will Pp=MR = ^, and R m= dy. Confequently,
Rm : MR : : PM : PT
dy ■ d X ; : y : ydx
dy
If, then, from the given Equation of any Curve, you fubflitute the Value 01 dx to ydx : dy, in the general Ex- preffion of the Sub-tangent, PTj the differential Quantities will vanifli, and jthc Value of the Sub-tangent come out in common Quantities ; whence the Tangent itfelf is eafily determined. This we fhall illuftrate in a few Examples :
1° The Equation defining the common Parabola, is,
Hence, adx=z iy dy dx=z2ydy : a VT=zydx : d y=^iy 2 'dy : ady~jy z : a=.iax :a—2X- That is, the Sub-tangent is double the Abfciffe, 2° The Equation defining a Circle is, & x — xx—y y adx — ixd x—iy dy dx=.-2.ydy : (a—2x)
PT=i# :ydy=.2y l dy : (a~z x) dy=ty z : (a— *x) =;(2ax—2xx):(a—2x)~(ax—xx):( ± a — x) that is,
PC:PB::AP:PT.
Therefore AT = {ax— oex) 1 (±a— x)~ x=(a oc~xx~\ ax-\-xx) -. (\a—x)—±ax : (±a~x) that is, PC: PA:; CA:AT.
3° The Equation defining an Ellipfis, is 5
ay % = ah x — b x z Hence 2 aydy — abdx—2 bxdx iaydy : (ah — 2bx) = dx
PT=ydx :dy=2tty 7 - : (ab—2bx)=(2abx~2 bx z ) z(ab — 2b x)~{2 ax — ia x z ) : (a — 2 x) that is, as the Dittance of the Semiordinate from the Centre, is to the Half Axis, & is the Abfcifs to the Portion of the Sub-tangent in- tercepted between the Vertex of the Ellipfis and the Tangent.
Laftly, for all Algebraic Curves, the Equation is,
aym-\-bx"-\-cf
x ' + df=o
may*-
~idy-\-nbx n —
iix-\-~fcy'x'—idx-\-rcy'—i X
Ur-
=0
nbx D -
■tdx+fcy'x*-
— 1 d x= —may m -
-idy— rcy-
-1 *'
dy
£x-
=: — 7nay m ~ 1
dy ~rcy — Is
<dy
n b x n •
-1 +fcfa'~ 1
Suppofe, e. gr. y 7 — . a x = o 5 then, by comparing with the general Formula ;
b x a = — a x b~ — a n = 1
a = 1 m = 2 c y r x f = o
c = or =of=o
Thefe Values being fubftituted in the moft general Fofmula
of the Sub-tangent 5 we have the Sub-tangent of the Parabola of the firft Kind, (— 2 . I J 1 * — o.oj'i*):(i~<«i- 1 -f- o ■ o y° x°) = — 2 y z : a = iy* : tf, Suppofe y* — x s a x y = 0, then will
ay* =.yi y x n = . — x 3
dy nbx«~ i-\-fcfx',- i
tt~im=:3b = in = $ c y r x r =: — axy /^=o
c = — ar= 1 /= 1
Thefe Values being fubflituted in the general Formula of the Sub-tar,gent ; we have the Sub-tangent of the Curve, whofe Equation is given, PT— (~ 3. , ji s — 1. — ay x\ = C 3- — 1 « 3 — t+i. — ayx;— i) = (— syi'-L a-y if)
- (-,3 *-«y)=(l!y 3 —<l*y-J:(l * + ay) Confe-
quently, A I =^ y 3— axy-)-. (3i'4-«jvj_i ==( ' 3 j,3_ axy) — ( }x> _ axy) : ( ? x>- + «J'J = ( 3 axy — % a xV )
- 3 1* + ay. The Value ofy 3 — x*, that is, a xy : (3 x l +
ay) being fubflituted from the Equation to the Curve.
In the •PMlefiphical TranfaSions, we have the foilowino Method of drawing Tangents to all Geometrical Curves? without any Labour, or Calculation, by M. Shlfius.
Suppofe a Curve, as DCL (Fig. 14) whofe Points are all referable to any Right Line given, as E A B, whether the Right Line be the Diameter or not ; or whether there be more given Right Lines than one, provided their Powers do but come into the Equation. In all his Equations he puts v for the Line DA, y for B A ; and for E B, and the other given Lines, he puts b d, i$c. that is, always Confonants.
Then, fuppofing DC to be drawn touching the Curve in D,and meeting with EB produced in C; he calls the fought Line C A, by the Name of a.
To find which, he gives this general Method : 1. Rejeft out of the Equation all Members, which have not either 11 or y with them: Then put all tho r e that have y, on one Side ; and all thofe which have a, on the other ; with their Signs + or — ; and the latter, for Diftinflion andEafe fake he calls the Right, the former, the Left Side. 2. On the Right Side, let there be prefix'd to each Member, the Expo- nent of the Power, which v hath there ; or, which is all one, let that Exponent be multiplied into all the Members. 3 Let the fame be done alfo on the Left Side, multiplying each Member there by the Power of the Exponent of y. Addinu this moreover, that onejymuft, in each Member, be alwayl changed into a. This done, the Equation thus refbrm'd, will fhew the Method of drawing the required Tangent t'o the Point D : For, that being given ; as alfo y, v, and the other Quantities exprefTed by Confonants, a cannot be un- known. Suppofe an Equation by —yy=vv, in which EB is called b ■ B A =y, DA=», and let a, or A C be required fo as to find the Point C, from whence C D being drawn, ftiall be a true Tangent to rhat Curve Q_D in D. In this Example, nothing is to be rejefled out of the Equation, be- caufe y or v are in each Member : It is alfo difpofed, as required by the Rule 1 ; to each Part therefore, there muft be prefix'd the Exponent of the Powers of y or v, as in Rule 2 ; and on the Left Side, let one y be changed into a, and then the Equation will be in this Form, ba—2ya = z vv, which Equation reduced, gives eafily the Value of
a = = A C. And fo the Point C is found, from
b — iy '
whence the Tangent D C may be drawn.
To determine which Way the Tangent is to be drawn, whether towards B or E, he direfls to confider the Numera- tor and Denominator of the Fraflion. For, 1. If in bothParts of the Fraction, all the Signs are Affirmative ; or if the Affirmative ones are more in Number; then the Tangent is to run towards B. 2. If the Affirmative Quantities are greater than the Negative in the Numerator 5 but equal to them in the Denominator ; the Right Line drawn through D, and touching the Curve in that Point, will be parallel "to A B : For in this Cafe, a is of an infinite Length. 3 . If in both Parts of the Fraction, the Affirmative Quantities, are lefs than the Negative, changing all the Signs, the Tangent muft be drawn now alfo towards B : For this Cafe, after the Change, comes to be the fame as the Firft. 4. If the Affirmative Quantities are greater than the Negative in the Denominator ; but in the Numerator are lefs 5 or vice verfa $ then changing the Signs in that Part of the Fraction where they are lefs, the Tangent muft be drawn a contrary Way ;
that
